by Selim Elias Tawﬁkblog.math.toronto.edu/GraduateBlog/files/2020/12/main.pdfSelim Elias Tawﬁk Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2020

FUSION PRODUCT OF D=G-VALUED MOMENT MAPS

by

Selim Elias Tawfik

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

© Copyright 2020 by Selim Elias Tawfik

Abstract

Fusion Product of D=G-valued Moment Maps

Selim Elias TawfikDoctor of Philosophy

Graduate Department of MathematicsUniversity of Toronto

2020

A fusion product is defined for Hamiltonian spaces with moment maps valued in a Lie group

D generalizing those of Alekseev-Malkin-Meinrenken. An analogous theory for these general

Hamiltonian spaces is developed and, among other results, versions of symplectic reduction,

duality and the shifting trick are derived. The Hamiltonian spaces with moment maps valued

in a homogeneous spaceD=G of Alekseev-Kosmann-Schwarzbach are shown to be equivalent

to certain Hamiltonian spaces with group-valued moment maps. The aforementioned theory is

consequently brought to bear on that of D=G-valued moment maps, thereby defining a fusion

product for these. This fusion product affords many new examples of D=G-valued moment

maps, of which there was hitherto a paucity. Among said examples are moduli spaces of flat

principal bundles over certain surfaces with boundary.

ii

Contents

1 Introduction 11.1 Context and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Moment map theories . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Initial problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Dirac-geometric approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Courant algebroids and Dirac structures . . . . . . . . . . . . . . . . . 5

1.2.2 Moment maps in Dirac geometry . . . . . . . . . . . . . . . . . . . . 6

1.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Dirac geometry 102.1 Linear aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Courant algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Primitive notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Involutive subbundles and Dirac structures . . . . . . . . . . . . . . . 15

2.2.3 Exact Courant algebroids . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Morphisms of Courant algebroids . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 General morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 Dirac morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Exact morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Coisotropic reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Reduction of Courant algebroids . . . . . . . . . . . . . . . . . . . . . 25

2.4.2 Reduction of Courant morphisms . . . . . . . . . . . . . . . . . . . . 30

3 L-Hamiltonian spaces 363.1 Hamiltonian spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 L-Hamiltonian spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

iii

3.2.1 Manin pairs determined by quadratic Lie algebras . . . . . . . . . . . . 383.2.2 Multiplicative structures . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.3 Fusion product of L-Hamiltonian spaces . . . . . . . . . . . . . . . . 44

3.3 Dual, symplectic reduction and shifting trick . . . . . . . . . . . . . . . . . . . 523.3.1 Dualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.2 Symplectic reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 D=G-valued moment maps 634.1 Hamiltonian spaces for action Courant algebroids . . . . . . . . . . . . . . . . 63

4.1.1 Fusion product of L-Hamiltonian spaces . . . . . . . . . . . . . . . . 684.2 D=G-valued moment maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2.2 Fusion product and dual . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Comparison with Poisson g-spaces . . . . . . . . . . . . . . . . . . . . . . . . 744.3.1 Poisson g-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3.2 Fusion product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.A Appendix: Classicism near the coset of the group identity . . . . . . . . . . . . 84

4.A.1 Gauge-theoretic preliminaries . . . . . . . . . . . . . . . . . . . . . . 844.A.2 Classicism of D=G-valued moment maps . . . . . . . . . . . . . . . . 88

5 Moduli space examples 925.1 Moduli spaces of flat D-bundles on coloured surfaces . . . . . . . . . . . . . . 92

5.1.1 Coloured surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.1.2 Fusion of coloured surfaces . . . . . . . . . . . . . . . . . . . . . . . 1005.1.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.1.4 Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2 Severa’s coloured surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.3 D=G-valued moment maps examples . . . . . . . . . . . . . . . . . . . . . . 105

Bibliography 108

iv

Chapter 1

Introduction

1.1 Context and motivation

1.1.1 Moment map theories

In the classical theory of moment maps, one is given a function J W X ! g�, called the moment

map, from a manifold X to the dual g� of the Lie algebra g of a Lie group G. The manifoldX carries a G-action and a 2-form ! 2 �2.X/ which, denoting the corresponding g-action by% W g! X.X/, are required to satisfy the following three axioms:

(a) ! is closed,

(b) ! is non-degenerate,

(c) �%. /! D �d h ; J i for all 2 g.

In that case the manifold X is called a G-Hamiltonian space. It was first observed by SophusLie himself in his seminal Theorie der Transformationsgruppen that the dual of a Lie algebracarries a canonical Poisson structure. The notion of a G-Hamiltonian space can be recast interms of the canonical Poisson structure on g� and a Poisson structure on the manifold X thatthe moment map J is required to intertwine. On the other hand, the canonical Poisson structureon g� can be seen as dual to the trivial Poisson structure on the group G; putting

D D T �G D g� ÌG;

where G acts on g� via the coadjoint representation, the Lie group D is a Poisson Lie group,i.e. it carries a Poisson bivector field � 2 X2.D/ that is multiplicative in the sense that

.Ld1/��jd2 C .Rd2/��jd1 D �jd1d2 for every d1; d2 2 G;

1

CHAPTER 1. INTRODUCTION 2

In general, a triple .d; g; h/ where d is a Lie algebra equipped with an ad-invariant metricand g; h � d are Lagrangian subalgebras is called a Manin triple. According to Drinfeld [25],the connected and simply connected Poisson Lie groups are in one-to-one correspondence withManin triples. A group D integrating the Lie algebra d inherits a Poisson Lie group structureand the connected subgroups G;G� � D integrating g and h respectively are Poisson Liesubgroups of D. The subgroup G� may be regarded as the dual of the subgroup G (hence thenotation) and this suggests a more general moment map theory, where moment maps may takevalue in the dual G� arising from any Manin triple. This was undertaken by A. Weinstein’sstudent J.-H. Lu [44]. In Lu’s theory, the action on G� is the so-called dressing action of G,which, assuming mult W G �G� ! D is a diffeomorphism, is characterized by

gh D .g:h/g0 for g 2 G and h 2 G�;

where g:h 2 G� and g0 2 G are the unique elements such that their product (in that order) isequal to gh. (In the case of the Manin triple .g� Ì g; g; g�/, the dressing action coincides withthe coadjoint representation of course.)

Still more generally, one can consider instead of Manin triples so-called Manin pair .d; g/,which consist of a Lie algebra d equipped with an ad-invariant metric and a Lagrangian sub-algebra g � d. Although in practice a Lagrangian subalgebra h � d complementary to g

may often be found thus completing the Manin pair .d; g/ to a Manin triple (see the work ofP. Delorme [22] for far-reaching results on the matter), this is not always the case; specificcounterexamples are given in this thesis in Chapter 4. This said, a Lagrangian (but not neces-sarily closed under the Lie bracket) complement h � d of g may always be found and the triple.d; g; h/ in this case is called a Manin quasi-triple. Y. Kosmann-Schwarzbach first investigatedManin quasi-triples and showed, much like Drinfeld for the special case of Manin triples, thatthey classify the so-called quasi-Poisson Lie groups [33, 34]. A quasi-Poisson Lie group Kcarries a multiplicative bivector field whose Schouten-Nijenhuis bracket is no longer requiredto vanish and is instead required to satisfy conditions depending on a choice of tensor � 2

V3 k

in the exterior algebra of the Lie algebra of K.

Suppose .D;G/ is pair of Lie groups where D integrates the Lie algebra d and G � D isa closed and connected subgroup integrating the subalgebra g � d; the pair .D;G/ is calleda group pair for the Manin pair .d; g/. Motivated by the theory of quasi-Poisson Lie groups,A. Alekseev and Y. Kosmann-Schwarzbach [2] have developed a theory with moment mapsvalued in the homogeneous space D=G. The action on D=G, also called the dressing action,


is the action by G that descends from multiplication on the left, i.e.

g:Œd � D Œgd �:

In this theory, after a choice of Lagrangian complement h � d of g, a Hamiltonian spaceX is required to carry a bivector field �h 2 X2.X/ satisfying a moment map condition aswell as an integrability and an equivariance condition (this is made precise in Chapter 4).If h0 � d is another Lagrangian complement of g then the bivector fields �h and �h0 arerelated to each other in function of the “twist” t 2

V2 g relating h and h0. The choice ofLagrangian complement is therefore essentially immaterial, but it must be made nonethelessin the framework adopted in [2]. The theory of D=G-valued moment maps as introduced byAlekseev and Kosmann-Schwarzbach thus leaves something to be desired: since the theorydoes not ultimately depend on the choice of a Lagrangian complement of g, one should like tofind an invariant formulation of the theory. Exactly this was achieved in works of H. Bursztyn,M. Crainic, P. Severa and D. Iglesias Ponte [16, 15, 14, 13], taking a Dirac-geometric approach.The discussion will turn to these ideas in the next section.

As a separate line of development of the idea of a moment map, is the theory of momentmaps valued in a group G introduced by A. Alekseev, A. Malkin and E. Meinrenken. In thiscontext, the group action on G is the adjoint action of G on itself and Hamiltonian spaces areG-spaces required to satisfy a list of three axioms, much like in the classical theory. The Dirac-geometric approach has also proven effective in recasting this theory in less computational andmore abstract terms [1].

1.1.2 Initial problem

The theories of G�- and G-valued moment maps have many of the familiar trappings of theclassical theory of g�-valued moment maps, with notions of fusion products (see [27] in thecase of G�-valued moment maps), duality, symplectic reduction etc. These notions have beenunderstandably challenging to extend to the theory of D=G-valued moment maps due to theabsence of a group structure on the target space. A fusion product was defined by P. Severa[55] and results on symplectic reduction were derived by Bursztyn, Crainic, Iglesias Ponte andSevera [13, 16]. The definitions and results in question are however ostensibly quite disparateand Severa’s fusion product in particular relies on structures far afield. It is therefore notclear how to bring them into a framework where they may be combined into e.g. a “shiftingtrick”-like result. Developing such a framework for the theory of D=G-valued moment mapswherein some of the most salient features of the classical theory of g�-valued moment mapsmay be brought over has been the chief preoccupation of this author.


The initial impetus of this project came from a simple observation. Given a Hamiltonianspace X with a moment map J W X ! D=G valued in D=G, let Xlift be the pullback along Jof the groupD seen as a principal bundle overD=G. The map J lifts to a map Jlift W Xlift ! D

so that there is a commutative diagram

(1.1)Xlift D

X D=G

Jlift

�=G �=G

J

;

where the vertical arrows are the quotient maps. In this way the Hamiltonian space X is liftedto a kind of “Hamiltonian space” with moment map valued in the group D; note however thatthe dressing action on D=G lifts to the action of G �G on D via

.g1; g2/:d D g1dg�12

and not the usual action ofD on itself by conjugation found in the prior theory of group-valuedmoment maps [4]. On the other hand, if X 0 is another Hamiltonian space with moment mapJ 0 W X 0 ! D=G then there is an analogous lift X 0lift and diagram (1.1). The direct productXlift �X

0lift of the lifts is a “Hamiltonian space” with moment map Jlift � J

0lift valued in D �D.

Composing the latter with the group multiplication mult W D �D gives a map

Xlift �X0lift ! D:

Quotienting the left-hand-side by the action of e � G on X 0lift and the right-hand-side by theaction of G on D by inverse multiplication on the right gives a G-equivariant map

Xlift �X0! D=G:

The space Xlift � X0 may thus be naively taken, as it were, to be the fusion product of X and

X 0. Actually, the space Xlift �X0 is not a Hamiltonian space at all because its auxiliary 2-form

does not satisfy the appropriate minimal degeneracy condition. Rather, it must be quotientedby the residual e �G�-action (G� the diagonal of G �G). The space

X ~X 0 WDXlift �X

0

e �G�

will reveal to be the correct definition of the fusion product of X and X 0.

This picture also suggests a notion of duality. Namely, the two G-actions on the lift Xlift


are traded; equivalently the dual of X is the quotient of Xlift by the G � e-action. Unlikethe classical and Alekseev-Malkin-Meinrenken theories, dual Hamiltonian spaces need not beisomorphic as G-spaces.

1.2 Dirac-geometric approach

1.2.1 Courant algebroids and Dirac structures

The language of Dirac geometry has been remarkably effective in organizing, simplifying andelucidating the subjects of symplectic and Poisson geometry. The fundamental objects of Diracgeometry are Courant algebroids, the basic example of which is the generalized tangent bundleTM D TM ˚ T �M of a manifold M . The bundle TM possesses two natural structures ofinterest; aside from the metric h�; �i provided by the pairing of TM and T �M , there is a bracketon the sections of TM given by

Jv1 C �1; v2 C �2K D Œv1; v2�C Lv1�2 � �v2d�1:

for vi 2 X.M/ and �i 2 �1.M/. This bracket was essentially introduced by T. Courant [20]and was later revised by I. Dorfman [24], giving its current version. More general Courantalgebroids, first defined by Z.-J. Liu, A. Weinstein, and P. Xu [41], generalize the structuresfound in the generalized tangent bundle TM ; a Courant algebroid is a vector bundle A! M

together with a metric h�; �i, a Courant bracket J�; �K on its sections and a map a W A ! TM

called the anchor that satisfy the properties C1-C3 listed below for all sections �i 2 �.A/

C1. J�1; J�2; �3KK D JJ�1; �2K; �3KC J�2; J�1; �3KK,

C2. a.�3/h�1; �2i D hJ�3; �1K; �2i C h�1; J�3; �2Ki,C3. a�d h�1; �2i D J�1; �2KC J�2; �1K, where A� has been identified with A via h�; �i.

Most of the interesting geometric data in Dirac geometry is encoded by Dirac structures –involutive and fiberwise Lagrangian subbundles of Courant algebroids. Indeed, it is from theobservation that a 2-form ! 2 �2.M/ or bivector field � 2 X2.M/ is closed (resp. Poisson)if and only if its graph is a Dirac structure of TM [20, 18] that the subject owes its inception.

The most ubiquitous class of Courant algebroids after generalized tangent bundles (andtheir “twists” [51, 56]) is that of the action Courant algebroids first defined by D. Li-Bland andE. Meinrenken [37]. Given an action of a Lie algebra d equipped with an ad-invariant metricon a manifold M , the product M � d inherits the structure of a Courant algebroid providedthe stabilizing subalgebras are all coisotropic. In that case the Courant bracket is the uniqueone which extends the Lie bracket of d identified with the constant sections of M � d and the


anchor is the d-action. If g � d is a Lagrangian algebra, then the trivial vector subbundleM �gis a Dirac structure of the action Courant algebroid M � d.

1.2.2 Moment maps in Dirac geometry

Let .D;G/ be a group pair for a Manin pair .d; g/. Of particular interest among action Courantalgebroids are

D � .d˚ d/; D=G � d

arising from the D � D-action on D via .d1; d2/:d D d1dd�12 and the action of D on the

homogeneous space D=G via d:Œd 0� D Œdd 0� respectively. The group-valued moment maptheory of Alekseev-Malkin-Meinrenken turns out to be encoded by a relation between theDirac structures TM � TM , where M is a Hamiltonian space, and D � d� � D � .d˚ d/.Likewise, the D=G-moment map theory of Alekseev and Kosmann-Schwarzbach is encodedby the same type relation between the tangent bundle of a Hamiltonian space and the Diracstructure D=G � g � D=G � d [16, 15, 14, 13].

The kind of relation between Dirac structures in question is called a Dirac morphism andwill in general be denoted by

.A1; E1/Ü .A2; E2/

for Dirac structures Ei of Courant algebroids Ai so that a Hamiltonian space with a momentmap valued inD in the sense of Alekseev-Malkin-Meinrenken is the data of a Dirac morphism

(1.2) .TX; TX/Ü .D � .d˚ d/;D � d�/

and a Hamiltonian space with a moment map valued in D=G is the data of Dirac morphism

.TX; TX/Ü .D=G � d;D=G � g/:

There is a diagram at the level of Dirac morphisms analogous to the diagram (1.1)

(1.3).TXlift; TXlift/ .D � .d˚ d/;D � .g˚ g//

.TX; TX/ .D=G � d;D=G � g/

:

The foregoing considerations behoove one to try to develop a theory of group-valued momentmaps where the Dirac structure D � .g ˚ g/ is, so to speak, substituted for D � d� in (1.2).This theory could then be marshalled in the service of the programme of expanding the theory


of D=G-valued moment maps.

1.3 Results and discussion

1.3.1 Overview

The following represents a quick summary of the constructions and results found in this thesis.

General Hamiltonian spaces. It should be stated that this thesis goes beyond its title, Fusion

product of D=G-valued moment maps. Indeed, in wanting to develop a new theory of group-valued moment maps, it was realized that the subalgebra d� � d ˚ d appearing in the priortheory of Alekseev-Malkin-Meinrenken [4] could be substituted not only by the subalgebrag ˚ g � d ˚ d but also by an arbitrary Lagrangian subalgebra l � d ˚ d so that generalHamiltonian spaces

(1.4) .TX; TX/Ü .D � .d˚ d/; E � l/

could be considered.

Fusion product. Given two Lagrangian subalgebras l1; l2 � d˚ d, their product l1 ı l2 in thepair groupoid

(1.5) d˚ d� d

is also a Lagrangian subalgebra of d˚ d. Consequently one can consider Hamiltonian spaces

(1.6) .TXi ; TXi/Ü .D � .d˚ d/;D � li/

and define a sensible notion of fusion whereby a new Hamiltonian space

(1.7) .T .X1 ~X2/; T .X1 ~X2//Ü .D � .d˚ d/;D � .l1 ı l2//

is constructed. Since.g˚ g/ ı .g˚ g/ D g˚ g;

this fusion product restricts to the category of D=G-valued moment maps as initially desired.

Duality. There is a natural notion of duality for a general Hamiltonian spaces (1.4) consistingin substituting l by its image l�1 under the inversion relation of the pair groupoid (1.5).


Synthesis. A number of results are proven regarding these general Hamiltonian spaces, notablya version of symplectic reduction and a shifting trick. The multiplicative Lagrangian subalgeb-ras l � d ˚ d, i.e. with the property l ı l D l, are classified: they are equivalent to pairs ofcoisotropic subalgebras of d.

D=G-valued moment maps. As regards D=G-valued moment maps, the fusion productherein defined is shown to coincide with that found in Lu’s moment map theory in the pres-ence of a Lagrangian subalgebra h � d complementary to g. It is also argued that if g admitsan adg-invariant Lagrangian complement h � d then D=G-valued moment maps are classicalg�-valued moment maps near the coset Œe� of the group identity e 2 D. In view of this, twoexamples of Manin pairs .d; g/ are given for which there is no Lagrangian complement h � d

of g that is either a subalgebra or adg-invariant – thus showing that the D=G-valued momentmap theory is not redundant.

Moduli space examples. Finally, inspired by work of Severa [54], moduli spaces of flat D-bundles are given as examples of general Hamiltonian spaces (1.4). These include Hamiltonianspaces with moment maps valued inD=G; they correspond to 2-manifolds with corners wheresome boundary segments have been decorated with the Lagrangian subalgebra g � d (e.g.Figure 1.1).

Figure 1.1: A boundary segment is coloured orange to indicate it is decorated with the Lagrangiansubalgebra g � d; otherwise it is drawn with a dotted line.

1.3.2 Future directions

It is obviously impossible to pursue all lines of investigation that may present themselves in thecourse of a research project such as this one. The following are ideas regarding possible futureendeavors.

Volume forms. The Hamiltonian spaces (1.4) carry canonical (up to a constant) volumeforms, which follows from the machinery of Clifford modules and pure spinors developedby Alekseev and Meinrenken [1] provided some mild topological condition is satisfied. Aquestion that remains to be answered entirely is that of the relationship between the volume


forms of Hamiltonian spaces (1.6) and that of their fusion product (1.7). This is already knownfor the Lagrangian subalgebra l1 D l2 D d� [6, Thm. 3.5] [1, Prop. 5.15], and partially forl1 D l2 D g˚ g to this author, but the general case remains unsettled.

Weil algebras and Duistermaat-Heckman measures. It is expected that the general Hamilto-nian spaces (1.4) carry Liouville-like measures and that these define Duistermaat-Heckmanmeasures on the target Lie group D. One motivation behind the search for a fusion product ofD=G-valued moment maps is the expectation that a Weil algebra could be defined so that a ver-sion of equivariant cohomology could be used to frame results on such Duistermaat-Heckmanmeasures in the spirit of [5, 6].

General CA-groupoids. By virtue of the pair groupoid structure (1.5), the Courant algebroidD � .d ˚ d/ is a CA-groupoid [38]: it possesses a multiplicative structure compatible withthe multiplication of their base Lie groups. More general CA-groupoids exist [40] and sincethe fusion product (1.7) depends on the pair groupoid structure (1.5), the question would bewhether an even more general moment map theory and auxiliary notions of fusion and dualitycould be developed. This would presumably not be as simple as the case considered here sincemorphisms of Courant algebroids do not generically take Dirac structures to Dirac structures;one has in mind the product E1 �E2 of Dirac structures E1; E2 � A of a CA-groupoid and itsmultiplication morphism Mult W A �AÜ A.

Chapter 2

Dirac geometry

A short but somewhat comprehensive introduction to the subject of Dirac geometry is given. Ifnothing else, this chapter serves to fix the notation for the remainder of the thesis.

2.1 Linear aspects

All vector spaces considered are real and finite-dimensional. A metric on a vector space V isunderstood to be a non-degenerate symmetric bilinear form h�; �i W V � V ! R. The notationV will be used for V equipped with its opposite metric.

Given a subspaceW of V , its orthogonal relative to the metric of V will be denoted byW ?.A subspace W � V is called isotropic if W � W ?, coisotropic if W � W ? and Lagrangian

if W D W ?. Clearly, the orthogonal of a coisotropic subspace is isotropic and vice-versa.A Lagrangian subspace E is both isotropic and coisotropic, and is respectively maximal andminimal in those regards; if C � E � I where C is coisotropic and I is isotropic thenequality on either side holds. From the equality dim.W / C dim.W ?/ D dim.V /, it is seenthat the dimension of a Lagrangian subspace is 1

2dim.V /. A pair .V;E/, where E � V is

Lagrangian subspace, is called a linear Manin pair.

Proposition 2.1.1. Let C � V be a coisotropic subspace. Then there is an isotropic subspace

I � V such that V D C ˚ I .

Proof. LetD � V be an arbitrary complement of C , i.e V D C ˚D. The orthogonalD? is acomplement of C? in V . Let prC? W V ! C? be the projection of V onto C? along D? andput I D fv � 1

2prC? v W v 2 Dg. Then V D C C I as C? � C . By a dimension count, this

is a direct sum. Moreover, for v;w 2 D one has

hv � 12

prC? v;w �12

prC? wi D hv;wi �12hv; prC? wi �

12hprC? v;wi

10

CHAPTER 2. DIRAC GEOMETRY 11

D hv;wi � 12hv;wi � 1

2hv;wi D 0;

in other words I is isotropic. This completes the proof.

If C � V is coisotropic then the quotient Vred D C=C? inherits the metric

hx C C?; y C C?i D hx; yi. A Lagrangian subspace E � V descends to the Lagrangiansubspace

Ered D .E \ C/=.E \ C?/

of Vred. Conversely the preimage of a Lagrangian subspace of Vred by the quotient map C !C=C? is a Lagrangian subspace of V . Note however that the preimage of Ered is not equal toE unless unless C? � E.

Suppose now V1 and V2 are metrized vector spaces. A Lagrangian relation between V1and V2 is a relation1 R � V2 � V1that is Lagrangian as a subspace of V2 � V1. One writesv1 �R v2 to indicate that .v2; v1/ 2 R. Given subsets S � V1 regarded as unitary relation, thecomposition of relations R ı S � V2 will be called the forward image of S by R. Likewise fora subset S � V2, the composition of relations S ı R � V1 will be called the backward image

of S by R. The following subspaces are also introduced

ker.R/ D 0 ıR;

ran.R/ D R ı V1;

ker�.R/ D R ı 0;

ran�.R/ D V2 ıR;

where the notation was chosen for obvious reasons. Note that ker.R/ is isotropic with coiso-tropic orthogonal ran�.R/, likewise for ker�.R/ and ran.R/.

Definition 2.1.1. If E1 � R and E2 � V2 are Lagrangian subspaces, one will say that Ris a Dirac relation from the linear Manin pair .V1; E1/ to the linear Manin pair .V2; E2/ ifE1 \ ker.R/ D 0 and R ıE1 D E2.

Note that a Dirac relation R induces a linear map % W E2 ! E1 whose graph is contained in R.

Proposition 2.1.2 ([1]). Let E1 � V1 and E2 � V2 be Lagrangian subspaces. Then

(a) The forward image R ıE1 � V2 and backward image E2 ıR � V1 are Lagrangian.

1By convention a relation from A to B is a subset of B � A. This order is chosen so that the composition ofrelations from left to right reads more naturally.


(b) If R is a Dirac relation from .V1; E1/ to .V2; E2/ and F2 is a Lagrangian complement of

E2, then F1 D F2 ıR is a Lagrangian complement of E1. Furthermore, each element of

F1 is R-related to a unique element of F2 and the induced map F1 ! F2 is dual to the

induced map % W E2 ! E1 after identifying Fi D E�i via the metrics.

Proof. (a) Regarding E1 as a Lagrangian relation between 0 and V1, then R ı E1 is thecomposition of Lagrangian relations and is therefore Lagrangian. Similarly for E1 ıE.

(b) The backward image F1 D F2 ı R is a Lagrangian subspace of V1 by part (a). Theintersection E1 \ F1 must be trivial since an element of this intersection is at once onlyR-related to elements of E2 and R-related to at least one element of F2. Thus F1 is aLagrangian complement of E1. Since R ı E1 D E2, it follows that ker�.R/ � E2 andthen that each element of F1 is R-related to exactly one element of F2. Finally, the lastpart of the statement follows from the fact that R is Lagrangian in V2 � V1.

Say now, in the context of Proposition 2.1.2, that a Lagrangian complement F � V1 (notnecessarily the backward image of a Lagrangian subspace of V2) of E1 has been fixed, therebyidentifying V with E1 ˚ E�1 equipped with the metric induced from the natural pairing of E1and its dual. Then the backward image F1 D F2 ıR of a Lagrangian complement of E2 can beregarded as a bivector �F2 2

V2E1: its contraction with an element � 2 E�1 is the necessarily

unique element v 2 E1 such that2 v � � 2 F1. Retain % to denote its extension to a map ofexterior algebras

V�E2 !

V�E1.

Proposition 2.1.3 ([1]). If F 02 is another Lagrangian complement of E2, then

(2.1) �F02 D �F2 � %.t/

where t 2V2

E2 is the twist defined by

(2.2) t ] W F2 ' E�2 ! E2; v 7! prF 02 v � v:

Proof. Let %�F2 and %�F 02

denote the maps E�1 ! E�2 and E�1 ! E�2 dual to %, respectively. Theduality of %�F2 , alternatively %�

F 02, and % means that

h%�F2.�/; vi D h%�

F 02.�/; vi D h�; %.v/i H) h.%�F 02

� %�F2/.�/; vi D 0

for � 2 E�1 and v 2 E2. As E2 is Lagrangian, this implies that .%�F 02� %�F2/.�/ 2 E2, in

other words %�F 02D prF 02 ı %

�F2

so that one can write %�F 02.�/ D %�F2.�/C t

].%�F2.�//. Note that

2The negative sign is due to the convention adopted in other literature, cf. [2].


%.t ].%�F2.�/// D ��%.t/. Adding .��F2; �/ �R %�F2.�/ and ��%.t/ �R t ].%�F2.�//, oneobtains

.��.�F2 � %.t//; �/ �R %

�

F 02.�/:

Finally, subtracting .��F02; �/ �R %

�

F 02.�/ from the above gives

.��.�F2 � %.t/ � �F

02/; 0/ �R 0:

However, the intersection ker.R/\E1, according to the definition of a Dirac structure, is trivialand therefore ��.�F2 � %.t/ � �F

02/ D 0, i.e. (2.1) holds.

2.2 Courant algebroids

2.2.1 Primitive notions

Consider the generalized tangent bundle TM D TM ˚ T �M of a finite-dimensional smoothmanifold M . It comes with a natural fiber metric given by the pairing of TM and its dual, aswell as the following bracket [24] on its sections:

(2.3) Jv1 C �1; v2 C �2K D Œv1; v2�C Lv1�2 � �v2d�1;

where vi 2 X.M/ and �i 2 �1.M/. Let a W TM ! TM be the projection onto thefirst factor. Studying the properties of the structures on TM just introduced, the followingdefinition is abstracted.

Definition 2.2.1 ([41]). A Courant algebroid is a vector bundle A! M together with a non-degenerate fiber metric h�; �i, a R-bilinear bracket J�; �K W �.A/ � �.A/ ! �.A/, called theCourant bracket, and a bundle map a W A! TM , called the anchor, such that for all sections�1; �2; �3 2 �.A/

C1. J�1; J�2; �3KK D JJ�1; �2K; �3KC J�2; J�1; �3KK,

C2. a.�3/h�1; �2i D hJ�3; �1K; �2i C h�1; J�3; �2Ki,C3. a�d h�1; �2i D J�1; �2KC J�2; �1K, where A� has been identified with A via h�; �i.

Properties 1-3 imply the further properties

C4. a.J�1; �2K/ D Œa.�1/; a.�2/�,C5. J�1; f �2K D f J�1; �2KC .a.�1/f /�2 where f 2 C1.M/.


The definition of a Courant algebroid first appeared in [41]. Properties C4 and C5 appearalongside C1-C3 in the definition, the redundancy seemingly not having been realized beforesome time [52].

Example 2.2.1. The first and quintessential example of a Courant algebroid is of course thegeneralized tangent bundle TM equipped with the above bracket and anchor. It will aptly becalled the standard Courant algebroid of M from here on.

Example 2.2.2. More generally the bracket (2.3) can be twisted by a closed 3-form � 2 �3.M/

to obtain a new bracket J�; �K� on TM :

(2.4) Jv1 C �1; v2 C �2K� D Œv1; v2�C Lv1�2 � �v2d�1 C �v2�v1�

where xi 2 X1.M/ and �i 2 �1.M/. With the fiber metric and anchor otherwise being thesame, the bracket J�; �K� determines a new Courant algebroid structure on TM [56]. It will becalled the �-twisted standard Courant algebroid of M and denoted by TM�. Of course when� D 0 this corresponds to the previous example and � is omitted.

Conversely, if (2.4) is a Courant bracket for some 3-form � 2 �3.M/ then � is necessarilyclosed by virtue of the equation

(2.5) J�1; J�2; �3K�K� C �a.�3/�a.�2/�a.�1/d� D JJ�1; �2K�; �3K� C J�2; J�1; �3K�K�:

Example 2.2.3. Let g be a quadratic Lie algebra, i.e. g is equipped with an Ad-invariant metric.Suppose % W M � g ! TM is a g-action on the manifold M . If the stabilizer algebrasgm D f 2 g W %.x; / D 0g are all coisotropic, then M � g possesses the structure of aCourant algebroid with anchor a D % and Courant bracket the unique one extending the Liebracket of g identified with the constant sections [37]. The product M � g is called an action

Courant algebroid.

Example 2.2.4 (Standard constructions). For a Courant algebroid A, another Courant algebroidA is obtained by reversing the sign of the metric. The direct product A D A1 � A2 of twoCourant algebroids A1 ! M1 and A2 ! M2 is also a Courant algebroid with its anchor andmetric being the respective direct products of those of A1 and A2. Its Courant bracket J�; �Kis defined as the unique one which restricts to the Courant bracket of Ai on �.Ai/ and suchthat J�; �K D 0 whenever � 2 �.A1/ and � 2 �.A2/. Its existence can be established locally;given a local frame �1; : : : ; �k of A1 and a local frame �1; : : : ; �l of A2, let �1; : : : ; �kl be theresulting local frame for A1 � A2. The brackets J�i ; �j K are all determined and Property C5


forces

JXi

fi�i ;Xj

gj �j K DXi;j

.fi.a.�i/gj /�j � gj .a.�j /fi/�i C figj J�i ; �j K/

for functions fi ; gj 2 C1.M1 �M2/ and this completely determines J�; �K. Properties C1-C3are then easily verified.

2.2.2 Involutive subbundles and Dirac structures

A subbundle V ! S of a Courant algebroid A ! M (or more generally a collection V � A

consisting of a choice of subspace in each fiber of A over a base manifold S � M ) is calledisotropic, coisotropic or Lagrangian provided its fibers (the subspaces constituting V ) possessthose respective properties. Let �.A; V / denote the space of sections � 2 �.A/ that takevalues in V on its base manifold S . Then V is called involutive if �.A; V / is closed underthe Courant bracket J�; �K of A and a.V / � TS . Note that, by C4, in this case a.V / is aninvolutive singular distribution of S in the sense of Stefan-Suessmann and thus V defines asingular foliation on S .

Example 2.2.5. Given a Lie algebra g acting on a manifold M , one can form the action Liealgebroid M � g (not to be confused with an action Courant algebroid). The bracket Œ�; ��X onC1.M; g/ is the unique one extending the Lie bracket of g identified with the constant sectionsof M � g and its anchor is the g-action M � g! TM .

Suppose now g is quadratic and acts onM with coisotropic stabilizers. Then g also definesa Courant bracket J�; �K on C1.M; g/. The brackets Œ�; �� and J�; �K are related by [37, Lemma4.1]

J�1; �2K D Œ�1; �2�X C a�hd�1; �2i:

For a subspace s � g, let E.s/ D M � s. By virtue of the above equation, the subbundleE.s/ is involutive if s is an isotropic subalgebra of g. Note that E.s/ is isotropic, coisotropic orLagrangian if and only if s possesses those respective properties.

Example 2.2.6. The kernel ker.a/ � A of the anchor a W A ! TM of a Courant algebroidA is involutive in view of C4. By C3 and C4, the composition a ı a� is trivial, in otherwords ran.a�/ � ker.a/. As ha��; xi D h�; a.x/i, one sees at once that ker.a/ is coisotropicwith ran.a�/ as its isotropic complement. Thus ker.a/ is an involutive coisotropic singulardistribution of A.

Remark 2.2.1. The definition of involutivity given here is more restrictive than the one alreadyfound in the literature [39, 57]; there it is not required that a.V / � TS .


Proposition 2.2.1. Let V � A be an involutive subbundle along S �M . Then:

(a) [57] If V is not isotropic then it contains ran.a�jS/.(b) If V is coisotropic and S 0 ,! S is a submanifold of S that is tangent to the singular

distribution a.V / � TS , i.e. a.V jS 0/ � TS 0, then V 0 D V jS 0 is also involutive.

(c) Suppose V 0 � A is an involutive subbundle with base manifold S 0 such that the intersec-

tion V \ V 0 is clean, i.e. V \ V 0 is a submanifold of A and T .V \ V 0/ D T V \ T V 0.

Then the intersection S \ S 0 is clean and V \ V 0 is an involutive subbundle as well.

(d) [37] If V is isotropic and S is open then V inherits the structure of a Lie algebroid3.

(e) [12] If V is coisotropic then V ? is also involutive and is an “ideal” of V in the sense that

the bracket of a section in �.A; V ?/ and one in �.A; V / (in either order) is in �.A; V ?/.

Moreover, the quotient V=V ? inherits the structure of a Courant algebroid and ifE � A

is a Lagrangian involutive subbundle of A transversal to V , then .E \ V /=.E \ V ?/ is

a Lagrangian involutive singular distribution of V=V ?.

Proof. (a) Suppose V is not isotropic. Then the restriction of the metric to V has non-trivialsignature and thus, at least near any point in S , there is a section � 2 �.A; V / withh�; �i ¤ 0 along S . By C3 one has

f a�d h�; �i C h�; �ia�df D Jf�; �KC J�; f �K

for an arbitrary function f 2 C1.M/. From the special case where f vanishes alongS , one sees that h�; �ia�df 2 �.A; V /, or equivalently a�df 2 �.A; V /. As f wasarbitrary, this means that ran.a�/jS � V .

(b) For i D 1; 2, let � 0i 2 �.A; V0/ and let �i 2 �.A; V / be sections coinciding with � 0i on

S 0. One knows that J�1; �2K � �.A; V / and would like to show J� 01; �02K 2 �.A; V

0/.Since J�1; �2K � J� 01; �

02K D J�1; �2 � � 02K C J�1 � � 01; �

02K, it is sufficient to show

that J�1; f �K 2 �.A; V 0/ and Jf �; � 02K 2 �.A; V 0/ for any section � 2 �.A/ andfunction f 2 C1.M/ vanishing along S 0. From C5 and the assumption a.V 0/ �TS 0 it follows that J�1; f �K vanishes along S 0. On the other hand, from C3 one hasJf �; � 02K D �J� 02; f �K C a�d hf �; � 02i, which is equal to h�; � 02ia�df along S 0. Thenfor any � 0 2 �.A; V 0/ one has ha�df; � 0i D hdf; a.� 0/i D 0 along S 0. As V 0 is alsocoisotropic, it follows that Jf �; � 02K 2 �.A; V

0/. One thus concludes that V 0 is indeedinvolutive.

3Recall that a Lie algebroid is a vector bundle A ! Q together with a Lie bracket on its space of sections�.A/ and a bundle map a W A! TQ satisfying the properties analogous to C4 and C5. See for instance [46] forthe general theory of Lie algebroids along with that of their integral counterparts, Lie groupoids.


(c) A result of Grabowski and Rotkievicz [28] states that a submanifold of a vector bundle isa vector subbundle if and only if it is closed under scalar multiplication. In particular, theintersection V \ V 0 is a vector subbundle of A and thus its base S \ S 0 is a submanifoldof M . Seeing S , S 0 and M as the zero-sections of V , V 0 and A respectively, one has

T .S \ S 0/ D T .V \ V 0/ \ TM D T V \ T V 0 \ TM D TS \ TS 0;

and thus S and S 0 intersect cleanly. This shows the first part of the claim.Now as V and V 0 intersect cleanly, the map �.A; V /\�.A; V 0/! �.A; V \V 0/ is

surjective4. The involutivity of V \ V 0 then immediately follows from that of V and V 0.

(d) Let �1; �2 2 �.A; V /. Since V is isotropic and S is open, C3 implies that J�1; �2K D�J�2; �1K along S . Now, similarly to the proof of (a), if f 2 C1.M/ vanishes along Sthen J�1; f �2K also vanishes along S . It follows that J�; �K restricts to an anti-symmetricbracket on �.V /. This last bracket along with the restriction of the anchor ajV W V ! TS

turn V into a Lie algebroid by virtue of C1, C4 and C5.

(e) If V ? D V there is nothing to prove and it is therefore assumed that they are not equal.One knows that V=V ? inherits a metric. Furthermore, by part (a), one has ran.a�/jS �V or equivalently V ? � ker.a/jS . In particular, the anchor a descends to a map ared W

V=V ? ! TS .As part of the proof of the statement, it will be shown that J�; �K descends to a bracket

J�; �Kred on �.V=V ?/. This entails showing that for � 2 �.A; V ?/ and �; �1; �2 2�.A; V / that (1) J�; �K; J�; �K 2 �.A; V ?/ and in particular that V ? is involutive andan ideal of V ; and (2) J�1; f �2K; Jf �1; �2K 2 �.A; V ?/ whenever f 2 C1.M/ is afunction vanishing on S .

(1) That J�; �K 2 �.A; V ?/ follows from C2. Now

ha�d h�; �i; �1i D hd h�; �i; a.�1/i D 0

along S since h�; �ijS D 0. Thus a�d h�; �i 2 �.A; V /, so C3 gives J�; �K 2 �.A; V ?/.Thus V ? is involutive and an ideal of V .

(2) Similarly as in part (a), the claim J�1; f �2K 2 �.A; V ?/ is trivial and it sufficesto argue that a�df 2 �.A; V ?/. This is obviously the case since a.�3/f D 0 for any�3 2 �.A; V /.

Taking V=V ? with its metric, the map ared as its anchor and J�; �Kred as its bracket, itis readily seen that C1-C3 are inherited from their counterparts for A. Finally, suppose

4This follows from the fact that around any point in V \ V 0 local coordinates exist in which V and V 0 arelinear subspaces, see e.g. [30].


E is a Lagrangian involutive subbundle of A. Then the quotient .E \ V /=.E \ V ?/is fiberwise Lagrangian as observed in Section 2.1. It represents a subbundle of V=V ?

provided E is transversal to V and thus a fortiori to V ?. In this case, its involutivity isguaranteed by that of E \ V , which is a subbundle of A by part (c).

Example 2.2.7. Let g andM be as in Example 2.2.3. Suppose s � g is an isotropic subalgebra.Then E.s/ � M � g inherits the structure of a Lie algebroid according to part (c) above.Its bracket is the restriction of the bracket Courant bracket J�; �K to E.s/. By definition J�; �Kcoincides with the Lie bracket of s embedded in E.s/ as the constant sections. That is to sayJ�; �K extends the Lie bracket of s to a bracket on �.E.s//, turning E.s/, along with the s-actionE.s/ ! M as its anchor, into a Lie algebroid. In other words, the Lie algebroid structure ofE.s/ is that of the action Lie algebroid M � s.

On the other hand, suppose s � g is coisotropic. Then s? is isotropic and Œs?; s� � s?.

According to the previous proposition, involutive Lagrangian subbundles of A are maximalamong the Lie algebroids naturally associated to A. It stands to reason that they deserve specialinterest. As will be appreciated in the sequel, they constitute the foundation of Dirac geometry.

Definition 2.2.2 ([53, 59]). A Dirac structure supported on S is an involutive Lagrangiansubbundle E � AjS . When S D M , it is simply called a Dirac structure and the pair .A; E/is called a Manin pair.

If E ! S is a Lagrangian subbundle of a Courant algebroid A satisfying a.E/ � TS , thedegree to which it fails to be involutive may be measured by its so-called Courant tensor

‡E 2 �.V3

E�/

(2.6) ‡E .�1; �2; �3/ D hJe�1;e�2K;e�3ijS ;where �i 2 �.E/ and e� i 2 �.A; E/ extends �i . One verifies with C1-C3 that ‡E is well-defined and ipso facto tensorial in its arguments. Note that ‡E vanishes if and only if E is aDirac structure supported on S .

Example 2.2.8. Suppose A D TM�. Given a 2-form ! 2 �2.M/, its graph Gr.!/ is Lag-rangian in A and a direct computation shows that ‡Gr.!/ 2 �3.M/ (with the identificationGr.!/ ' TM ) is d!��. On the other hand, if � 2 X2.M/ is an anti-symmetric bivector fieldthen ‡Gr.�/ 2 X3.M/ (with the identification Gr.�/ ' T �M ) is 1

2Œ�; ��

V3�].�/ [56],

where Œ�; �� is the Schouten-Nijenhuis bracket.

Example 2.2.9. If E � A is a Dirac structure then its restriction to a submanifold S ,!

M tangent to the generalized distribution a.E/ � TM is a Dirac structure supported on S .


However, not all Dirac structures supported on a submanifold arise in this way. For instance,consider A included in A � A diagonally. Then A is a Dirac structure of A � A supportedon the diagonal of M �M and the Courant bracket of A � A has a well-defined restrictionto �.A/, which is just the Courant bracket of A. As such, the Courant algebroid A cannot bethe restriction of a Dirac structure of A � A as that would mean its bracket is anti-symmetric,which in view of C3 is never the case.

2.2.3 Exact Courant algebroids

A Courant algebroid is called exact [51, 56] if the sequence

(2.7) 0 T �M A TM 0a� a

is exact. In this case ran.a�/ D ker.a/ is a Dirac structure of A (Example 2.2.6). Accordingto Proposition 2.1.1, there exists a splitting j W TM ! A such that ran.j / is a Lagrangiancomplement of ker.a/ D ran.a�/. Such a splitting will be called isotropic. The map a � j �

identifies A ' TM as metrized vector bundles but not as Courant algebroids. The curvaturetensor � 2 �3.M/

�v2�v1� D Jj.v1/; j.v2/K � Œv1; v2�;

where v1; v2 2 X.M/ and J�; �K is the Courant bracket of A, is then introduced to measure thediscrepancy between the Courant algebroid structures of A and TM . One has

(2.8) �v3�v2�v1� D hJj.v1/; j.v2/K; j.v3/i

where v1; v2; v3 2 X.M/, in other words � is the Courant tensor‡ ran.j / whence its tensoriality.As Jv1; v2K D Œv1; v2� C �v2�v1�, it may be concluded that J�; �K has the form (2.4). Thus � isclosed and A ' TM� as Courant algebroids.

Suppose now j 0 W TM ! A is another isotropic splitting with corresponding 3-form �0.With A identified with TM�0 via j 0, the image ran.j / is the graph of a 2-form $ 2 �2.M/,i.e.

(2.9) j.v/ D j 0.v/C a�.�v$/:

Conversely, the image ran.j 0/ in A ' TM� is the graph of �$ . In practice, the 2-form $ iscomputed by noticing that

(2.10) �v2�v1$ D ha�.�v1$/; j.v2/i D �hj

0.v1/; j.v2/i D hj.v1/; j0.v2/i:


The corresponding 3-forms on the other hand are related by

(2.11) � D �0 C d$:

Conversely, a 2-form $ 2 �2.M/ defines an isotropic splitting j 0 W TM ! A via (2.9). Thespace of isotropic splittings of A is therefore an affine space modelled on�2.M/ and the spaceof corresponding 3-forms � is a cohomology class in H 3.M/.

2.3 Morphisms of Courant algebroids

2.3.1 General morphisms

A morphism R of Courant algebroids is a fiberwise Lagrangian relation compatible with theCourant brackets.

Definition 2.3.1 ([1, 15]). A Courant morphism from A1 to A2, denoted R W A1Ü A2, is aDirac structure of A2 � A1 supported on the graph Gr.f / � M2 �M1 of a smooth base mapf WM1 !M2.

Example 2.3.1. Let f WM1 !M2 be a smooth map. Define a relation R � TM2 � TM1 by

v1 C �1 �R v2 C �2 ” v2 D f�v1; �1 D f��2;

for vi 2 TMi and �i 2 T �Mi . Then R is a Courant morphism R W TM1Ü TM2. This anexample of an exact Courant morphism, see Section 2.3.3.

Example 2.3.2. Suppose gi (i D 1; 2) are quadratic Lie algebras acting on a manifoldsMi thusgiving action Courant algebroids A1 D M1 � g1 and A2 D M2 � g2. Suppose further thatR � g2 � g1 is a Lagrangian subalgebra and that f W M1 ! M2 is a smooth map such thatx1 �R x2 H) f�a.x1/ D a.x2/ where xi 2 Ai . Then Gr.f / � R � A2 � A1 is a Courantmorphism A1Ü A2 [37, Prop. 4.7], which will also be denoted by R.

The choice of a dotted arrow as opposed to a solid one reflects the fact that a Courantmorphism R is merely a relation and not generally a function. The notation introduced inSection 2.1 is reinterpreted in the obvious way so that one may speak of forward images R ıS1 � f

�A2 (in the pullback bundle of A2), backward images S2 ıR � A1 as well as singulardistributions ker.R/; ran�.R/ � A1 and ker�.R/; ran.R/ � f �A2.

If R0 W A2Ü A3 is another morphism of Courant algebroids, by part (e) of Proposition2.2.1 composingR andR0 as relations gives a Courant morphismR0 ıR W A1Ü A3 provided


R0 �R � A3 �A2 �A2 �A1 is transversal to 0�A2;� � 0 where A2;� is the diagonal of A2

since

R0 ıR D.R0 �R/ \ .A3 �A2;� �A1/

.R0 �R/ \ .0 �A2;� � 0/:

Example 2.3.3. A Dirac structure E � A is equivalent to a Courant morphism E W A Ü 0.In particular if R W A0Ü A is a Courant morphism then the backward image E ıR is a Diracstructure of A0 provided the composition is transverse.

The Courant morphism R W A1 Ü A2 will be called a Courant isomorphism if its basemap f is a diffeomorphism. In that case, one can also consider R as a Dirac structure ofA1 � A1 supported on Gr.f �1/, in which case it is denoted by R�1 and called the inverse ofR. Note that R ıR�1 D IdA1 and R�1 ıR D IdA2 where IdAi are the identity morphisms.

2.3.2 Dirac morphisms

For Manin pairs .Ai ; Ei/, i D 1; 2, one is led to consider morphisms A1Ü A2 that relate E1and E2 in some way.

Definition 2.3.2. A Dirac morphism from .A1; E1/ to .A2; E2/ is a Courant morphism R W

A1 Ü A2 such that, for every m 2 M1, each element in E2jf .m/ is R-related to a uniqueelement of E1jm. The notation R W .A1; E1/Ü .A2; E2/ is used.

In other words, the morphism R W .A1; E1/Ü .A2; E2/ is a Dirac morphism provided thepullback bundle f �E2 is equal to the forward image R ı E1 and the intersection E1 \ ker.R/is trivial. Analogously to the case of linear Dirac morphisms, if F2 � A2 is a Lagrangiancomplement of E1 then the backward image F1 D F2 ıR is a Lagrangian complement of E1.Furthermore, the Dirac morphism R induces a bundle map

(2.12) % W f �E2 ! E1

and, in the presence of the Lagrangian complement F2, a bundle map

(2.13) %�F2 W F1 ! f �F2

dual to % according to Proposition 2.1.2.The composition of Dirac morphisms is categorical; for supposeR W .A1; E1/Ü .A2; E2/

andR0 W .A2; E2/Ü .A3; E3/ are Dirac morphisms. The equality .f 0ıf /�E3 D R0ıRıE1is immediate. If x 2 E1 \ ker.R0 ı R/, then x �R y �R0 0 for some y 2 E2 and sinceE2 \ ker.R0/ D 0 and E1 \ ker.R/ D 0 it follows that y D 0 and in turn x D 0. ThusR0 ıR W .A1; E1/Ü .A2; E3/ is a Dirac morphism.


Lemma 2.3.1 ([1]). Suppose R W .A1; E1/Ü .A2; E2/ is a Dirac morphism and that F2 �

A2 is a Lagrangian complement ofE2 so that the backward image F1 D F2ıR is a Lagrangian

complement according to Proposition 2.1.2. Then the Courant tensors ‡F1 2 �.V3

E1/ and

‡F2 2 �.V3

E2/ are related by

(2.14) ‡F1 D %.f �‡F2/;

where % has been retained to denote the extension of the bundle map (2.12) to a map of exterior

algebrasV�

f �E2 !V�

E1.

Proof. Let �i (i D 1; 2; 3) be sections in �.F2�F1/\�.A2�A1; R/. Form 2M1, the valueof �i at .f .m/;m/ is .%�F2.xi/; xi/ for some xi 2 F1jm. Now

%.‡F2/.x1; x2; x3/jm � ‡F1.x1; x2; x3/jm D ‡

F2.%�F2.x1/; %�F2.x2/; %

�F2.x3//jf .m/ �

‡F1.x1; x2; x3/jm

D ‡F2�F1.�1; �2; �3/j.f .m/;m/

D hJ�1; �2K; �3ij.f .m/;m/;

where the duality of % and %�F2 was used in the first equality and the fact that ‡F2�F1 is thedirect sum ‡F2 ˚‡F1 in the second. On the other hand, the Courant tensor of R vanishes andthus

0 D ‡R.�1; �2; �3/j.f .m/;m/

D hJ�1; �2K; �3ij.f .m/;m/:

The upshot is that%.‡F2/.x1; x2; x3/ � ‡

F1.x1; x2; x3/ D 0;

which is what was needed.

2.3.3 Exact morphisms

Suppose now A1 and A2 are exact and suppose R W A1 Ü A2 is a morphism with basemap f W M1 ! M2. Let Gr.f�/ � TM2 � TM1 and Gr.f �/ � T �M2 � T

�M1 be therelations defined by the tangent map f� and the cotangent map f � of f respectively. Theimage a�.Gr.f �//, where a D a2� a1, is orthogonal to the preimage a�1.Gr.f�// in A2�A1.


The former is thus contained in R. The morphism R will be called exact [51] if the sequence

(2.15) 0 Gr.f �/ R Gr.f�/ 0a� a

is exact. In fact, only exactness at Gr.f�/ must hold for the above sequence to be exact sincea� is injective and, by a dimension count, one has ker.a/\R D a�.Gr.f �// if a.R/ D Gr.f�/.In particular, the morphism R is exact if and only if a1.ran�.R// D TM1.

Proposition 2.3.1. The morphism R is exact if and only if it is a Dirac morphism

R W .A1; ran.a�1//Ü .A2; ran.a�2//:

Proof. In one direction, suppose R is exact. Let C D a�1.Gr.f�//, which is coisotropicin A2 � A1 with isotropic orthogonal C? D a�.Gr.f �//. Since R � C , it follows thatC? � R and in particular R relates each element of ran.a�2/ to an element of ran.a�1/, thusestablishing existence. Now if � �R 0 for some non-zero � 2 ran.a�1/ then by orthogonalitythe image under a1 of the projection of R onto A1 must be contained in ker.�/ � TM1. Sincea.R/ D Gr.f�/, this must mean that � D 0, establishing uniqueness.

In the other direction, suppose R W .A1; ran.a�1//Ü .A2; ran.a�2// is a Dirac morphism. Itmust only be shown that a1.ran�.R// D TM1. The intersection ran.a�1/ \ ker.R/ is trivial bydefinition, and taking its orthogonal one has

ker.a1/C ran�.R/ D A1 H) a1.ran�.R// D TM1:

This completes the proof.

Proposition 2.3.2 ([17]). Suppose R is exact. Let j2 W TM2 ! A2 be an isotropic splitting of

A2. Then there exists a unique isotropic splitting j1 W TM1 ! A1 of A1 such that R ı j1 D

j2 ı f�. Furthermore, the corresponding 3-forms are related by �1 D f ��2.

Proof. According to the previous proposition and Proposition 2.1.2, the backward imageran.j2/ ı R is a Lagrangian complement of ran.a�1/. Defining j1 W TM1 ! A1 by ran.j1/ Dran.j2/ ıR, it is clear the condition R ı j1 D j2 ı f� holds. Conversely, if j1 W TM1 ! A1 isan isotropic splitting satisfying this condition then ran.j1/ � ran.j2/ ıR, which is actually anequality since both sides are Lagrangian. This shows the first part of the statement. The secondpart follows immediately from (2.8).

Proposition 2.3.1 shows that composition of exact morphisms is also exact. On the otherhand, Proposition 2.3.2 is key in giving a more explicit description of exact morphisms. To-wards this, start by choosing arbitrary isotropic splittings ji W TMi ! Ai identifying Ai '


TMi;�i for i D 1; 2. If R W A1 Ü A2 is exact then it determines an identification A1 '

TM1;f ��2 . By (2.11) then, the image ran.j1/ � TM1;f ��2 is the graph Gr.!/ of a 2-form! 2 �2.M1/ satisfying

(2.16) d! D �1 � f��2:

This leads to the explicit description

(2.17) v1 C �1 �R v2 C �2 ” v2 D f�v1; �1 C �v1! D f��2:

Conversely, suppose f W M1 ! M2 is a smooth map and ! 2 �2.M1/ is a 2-form satisfying(2.16). Define a Lagrangian subbundle R � A2 � A1 along Gr.f / via (2.17). It is easilychecked that R is a Dirac morphism5, obviously an exact one. The upshot is that an exactmorphism TM1;�1 Ü TM2;�2 is equivalent to a pair .f; !/ as above. The notation Tf! , orTf if ! D 0, will be used to denote the exact morphism corresponding to .f; !/ from hereon.

If the composition of functions f 0 ıf is defined, the composition of exact Courant morph-isms Tf! and Tf 0!0 with base maps f and f 0 is also defined with

(2.18) Tf 0!0 ı Tf! D T .f 0 ı f /!Cf �!0 :

Example 2.3.4. Let G be a Lie group with Lie algebra g and denote the group multiplicationby multG W G � G ! G. Then the tangent bundle TG of G is also a Lie group wheremultiplication is given by the differential .multG/� W TG � TG ! TG. On the other hand,the cotangent bundle carries a symplectic Lie groupoid structure T �G � g� where the sourceand target maps are respectively the left and right trivializations and multiplication is given by

.g1; �1/ ı .g2; �2/ D .g; �/ ” g D g1g2; mult�G� D .�1; �2/:

Multiplication in the direct sum of Lie groupoids TG D TG ˚ T �G is therefore given by

.g1; v1; �1/ ı .g2; v2; �2/ D .g; v; �/ ”

g D g1g2; v D .multG/�.v1; v2/; mult�G� D .�1; �2/:

This is precisely the exact morphism TmultG W TG � TGÜ TG.

5One may ipso facto assume that �1 D f ��2, making this verification quite simple.


2.4 Coisotropic reduction

2.4.1 Reduction of Courant algebroids

Suppose C � A is a coisotropic subbundle along M . The quotient bundle C=C? inherits afiber metric from A, though it will rarely inherit a bracket. In an attempt to rectify this, thesymmetries of a Courant algebroid are briefly considered.

The infinitesimal counterpart of an automorphism of vector bundles of A is a derivation,i.e. a R-linear operator D W �.A/! �.A/ together with a vector field v 2 X1.M/ satisfyingthe Leibniz rule: D.f �/ D D� C .vf /� for all f 2 C1.M/ and � 2 �.A/. The spaceDer.A/ of derivations is informally the Lie algebra of Aut.A/, the space of vector bundleautomorphisms. Its Lie bracket is the commutator of operators. A Courant automorphism is aCourant isomorphism AÜ A. The infinitesimal counterpart of a Courant automorphism is aCourant derivation, i.e. a derivation .D; v/ 2 Der.A/ possessing the properties

vh�1; �2i D hD�1; �2i C h�1;D�2i;

DJ�1; �2K D JD�1; �2KC J�1;D�2K;

a.D�/ D Œv; a.�/� ;

for all �; �1; �2 2 �.A/. The space DerCA.A/ � Der.A/ of Courant derivations is informallythe Lie algebra of the group AutCA.A/ � Aut.A/ of Courant automorphisms. A section� 2 �.A/ determines a Courant derivation � 7! .J�; �K; a.�// and this assignment is bracketpreserving.

Definition 2.4.1 ([17]). Suppose G is a Lie group, with Lie algebra g, acting on A by Courantautomorphisms. Suppose moreover that % W g! �.A/ is a bracket-preserving linear map suchthat

(2.19)@

@t

ˇtD0

exp.�t /:� D J%. /; �K

for all 2 g and � 2 �.A/. The elements %. / 2 �.A/ will be called generators for theG-action on A.

In other words, the map % W g ! �.A/ defines generators for a G-action on A if the infin-itesimal g-action on A factors through the assignment �.A/ ! DerCA.A/ composed with %.Note that G acts on M (embedded in A as the zero-section) as well, with a ı % W g ! X.M/

as the infinitesimal action. The symbol % will be retained to denote the corresponding bundlemap M � g! A.


Example 2.4.1. Suppose G is a Lie group acting on a manifold M with infinitesimal action% W g ! X.M/ and that � 2 �3.M/ is a G-invariant closed 3-form. Then % defines isotropicgenerators for the action of G on TM� by g:.v C �/ D g�v C .g

�1/�� for v 2 TM and� 2 T �M .

Example 2.4.2. In the context of Example 2.2.3, suppose G is a Lie group integrating the Liealgebra g and that the g-action a W g ! M � g integrates to a G-action. Consider the actionof G on M � g given by g:.m; / D .gm;Adg /, which preserves the metric as well as thebracket of constant sections. Since the constant sections generate �.M � g/, one sees from C3and C5 that the G-action on M � g is bracket-preserving for all pairs of sections in �.M � g/,i.e G acts on M � g by Courant automorphisms. The anchor a W g ! �.M/ then definesgenerators for this G-action.

Suppose G acts on A with % W g! �.A/ defining generators for the G-action. If E ! S

is a Lagrangian subbundle of A satisfying a.E/ � TS , one may may in addition to the Couranttensor ‡E consider the Dirac tensor ƒE 2 g� ˝ �.

V2E�/ defined by

ƒE . ; �1; �2/ D hJ%. /;e�1K;e�2ijSwhere 2 g and �i 2 �.E/ with e� i 2 �.A; E/ extending �i . The Dirac tensor measuresthe degree to which E fails to be g-invariant, or G-invariant if G is connected. Here again oneverifies from C1-C3 that ƒE is well-defined and tensorial in �1 and �2.

Example 2.4.3. Suppose A D TM and � 2 X2.M/ is a bivector field. Then

ƒGr.�/. ; �1; �2/ D hJ%. /; ��1� C �1K; ��2� C �2i

D hŒ%. /; ��1��C L%. /�1; ��2� C �2i

D ��2L%. /��1� � ��2�L%. /�1�

D L%. /��2��1� � �L%. /�2��1� � ��2�L%. /�1�

D ��2��1L%. /�:

In other words ƒGr.�/. / D L%. /� . Similarly one finds that ƒGr.!/. / D L%. /! for a 2-form! 2 �2.M/.

Lemma 2.4.1. Suppose G is a connected Lie group acting on A by Courant automorphisms

and % W g! �.A/ defines generators for the G-action. Then the space �.V /G of G-invariant

sections of V is closed under the Courant bracket.


Proof. As G is connected, the G-invariant sections of V are those whose bracket with thegenerators %. / are trivial. Then the closure of �.V /G under the Courant bracket followsdirectly from C1.

For the remainder of this subsection, suppose % W g ! �.A/ defines generators for a G-action by Courant automorphisms on A where G is a connected Lie group, that C? D ran.%/is an isotropic subbundle and that the induced G-action on M is free and proper. Recall thatC? is a Lie algebroid by part (d) of Proposition 2.2.1. In fact, since the G-action is free themap % is injective and thus gives an isomorphism of Lie algebroids M � g! C?.

Lemma 2.4.2. The subbundle C? is involutive, its orthogonal C isG-invariant. Moreover, the

Courant bracket descends to a bracket on �.C=C?/G .

Proof. For 1; 2 2 g, one has %.Œ 1; 2�/ D J%. 1/; %. 2/K by (2.19). Thus J%. 1/; %. 2/K 2�.C?/. For f 2 C1.M/, Property C5 gives J%. 1/; f 2K 2 �.C?/ and C3 gives

Jf%. 1/; %. 2/K D �J%. 2/; f%. 1/K

since C? is isotropic. Thus Jf%. 1/; %. 2/K 2 �.C?/ as well. It follows that C? is involutive.

The G-invariance of C follows from the G-invariance of C? and the preservation of themetric by the G-action. Since �.C?/G and �.C /G are closed under the Courant bracket byLemma 2.4.1, the same argument regarding the descent of the Courant bracket given in theproof of part (e) of Proposition 2.2.1, mutatis mutandis, can be employed to show that theCourant bracket descends to �.C=C?/G .

Consider the reduced space Ared DCC?=G, which is a vector bundle over Mred DM=G.

Theorem 2.4.1 ([17]). With everything as above, supposeMred and Ared are quotient manifolds

so that Ared is a smooth vector bundle over Mred. Then Ared inherits the structure of a Courant

algebroid.

Proof. Since Ared is a smooth vector bundle, the base manifold Mred is also smooth and thequotient map M ! Mred is a submersion. As C=C? inherits a metric so does Ared. Thefundamental vector fields of the G-action on M span a.C?/ and, writing TMred D

TMa.C?/=G,

let ared W Ared ! TMred be the map to which a W C ! TM descends.

The previous lemma implies that the Courant bracket J�; �K descends to a bracket J�; �Kred on�.Ared/. Equipping Ared with the metric h�; �ired, the anchor ared and the bracket J�; �Kred, it isreadily seen that C1-C3 are inherited from A.


Example 2.4.4. In the context of Example 2.2.3, suppose h � g is an isotropic subalgebraintegrating to a Lie group H and that the h-action on M integrates to a free and proper H -action. Now H acts on M � g by Courant automorphisms via s:.m; / D .s:m;Ads / as inExample 2.4.2. Assuming the coisotropic orthogonal h? is also a subalgebra of g, the quotienth?=h is a Lie algebra (Proposition 2.2.1) and .M � g/red and M=H � h?=h are equal as vectorbundles. Surely, they should also be equal as Courant algebroid.

To verify this, first note the anchors coincide since both are defined as the map ared W

M=H ! TMa.h/=G to which the anchor a ofM �g descends. Next let �1; �2 2 �.M=H �h?=h/

be constant sections. Then �1 and �2 lift to H -invariant sections of M � g of the form Adx i ,i D 1; 2, respectively, for some fixed elements i 2 h?. Then

JAdx 1;Adx 2K D AdxJ 1; 2K D Adx Œ 1; 2� ;

from which one sees that

J�1; �2Kred D Œ 1; 2�C h D Œ�1; �2�red ;

where Œ�; �� is the Lie bracket of h?=h, i.e. the Courant bracket J�; �K agrees with the bracket ofh?=h on constant sections. Since this property uniquely determines the bracket of the actionCourant groupoid M=H � h?=h, it follows that .M � g/red and M=H � h?=h are equal asCourant algebroids.

Remark 2.4.1. With the same assumptions as Theorem 2.4.1, if S is aG-invariant submanifoldofM and a.C /jS � TS then .Ared/jS=G D

C jSC?jS

=G is also a Courant algebroid, i.e. a “Courantsubalgebroid” of Ared.

The process of passing from A to Ared will be called the coisotropic reduction of A by C .The reduction morphism q W AÜ Ared is defined by x �q Œx� for all x 2 C . Its involutivityis manifest from the definition of J�; �Kred in the proof of the preceding theorem.

Proposition 2.4.1. Suppose E � A is a G-invariant Dirac structure and that E \ C?, or

equivalently E \ C , has constant rank. Then its forward image q ı E, which can be taken as

a subset of Ared as E is G-invariant, is a Dirac structure of Ared. Conversely, if F is a Dirac

structure of Ared then its backward image F ı q � A1 is a G-invariant Dirac structure of A1

and E \ C? has constant rank.

Proof. In the first direction, note first that q ı E is subbundle of Ared since its fiber over Œm� 2Mred does not depend on the representative m 2 M (smoothness follows from the constantrank of E \ C?). By Proposition 2.1.2, it is Lagrangian. As �.q ı E/ D �.E/G \ �.C /G ,the involutivity of q ıE follows from Lemma 2.4.1.


The second direction is trivial as the backward image F ı q is obviously G-invariant andcontains C?, as well as being a Dirac structure by Example 2.3.3.

The previous proposition can be extended to encompass G-invariant Dirac structures sup-ported on aG-invariant submanifold S ,! P in one direction and its quotient S=G in the otherprovided a.C /jS � TS ; indeed, in view of Remark 2.4.1, this is in essence the same result.

Proposition 2.4.2. Suppose, under the assumptions of Theorem 2.4.1, that A is exact and

ran.a�/ \ C? D 0. Then Ared is also exact and the reduction morphism q W A Ü Ared is

exact.

Proof. Clearly, one has rank.Ared/ D 2 dim.Mred/ since rank.A/ D 2 dim.M/. It thus sufficesto show that a�red W TMred ! Ared is injective to establish the exactness of Ared. The reducedcotangent bundle T �Mred canonically identifies with Ann.a.C?//=G, where Ann.a.C?// �T �M is the annihilator of a.C?/. Then since ran.a�/ \ C? D 0 and a� W TM ! A isinjective, the map T �Mred ! Ared to which a� descends, which is none other than a�red, mustbe injective as well.

From the foregoing and Proposition 2.3.1 the exactness of the reduction morphism q W

AÜ Ared is immediate.

With A exact and ran.a�/\C? D 0, an isotropic splitting jred W TMred ! Ared determinesa G-invariant isotropic splitting j W TM ! A with C? � ran.j / according to Proposition2.3.2. The corresponding 3-forms are then related by � D .�=G/��red, where �=G WM !Mred

is the quotient map. Conversely, one recovers jred from j by observing that

(2.20) ran.jred/ D q ı ran.j / Dran.j /C?

=G:

On the other hand, if j W TM ! A is an arbitrary G-equivariant isotropic splitting withC? � ran.j /, in which case it is called a G-basic splitting, then (2.20) defines an isotropicsplitting jred of Ared.

Example 2.4.5. Suppose j W TM ! A is a G-basic splitting with corresponding 3-form �.One has C? � TM under the identification A ' TM�. Then (Example (2.4.1)) the G-actionon TM� in terms of the G-action on M is g:.v C �/ D g�v C .g

�1/��, for v 2 TM and� 2 T �M . The identification Ared ' TM�red has a simple interpretation: it descends from theidentification A ' TM� by virtue of

Ared DCC?=G ' .TM

C?˚ Ann.C?//=G D TMred ˚ T

�Mred:


2.4.2 Reduction of Courant morphisms

Having described a process for reducing Courant algebroids, the question of the reduction ofCourant morphisms naturally arises. Let A1 and A2 be Courant algebroids. For i D 1; 2,assume that %i W gi ! �.Ai/ define generators for Gi -actions on Ai where Gi are connectedLie groups. Assume furthermore that Ci D ran.%i/? are coisotropic so that C?i D ran.%i/ areisotropic, and that the Gi -actions on the base manifolds Mi are free and proper. Let

R W A1Ü A2

be a Courant morphism with base map f W M1 ! M2. Fix a homomorphism of Lie algebras� W g1 ! g2. The symbol � will also be used for the corresponding bundle map

C?1 'M1 � g1 !M2 � g2 ' C?2

where the identifications have been made through the infinitesimal actions %i .

Definition 2.4.2 ([17]). The morphism R is said to intertwine the generators (with respect to�) if

(2.21) %1. / �R %2.�. //

for all Lie algebra elements 2 g1.

A few remarks are in order ifR intertwines the generators. First, one has the useful containment

C?1 � C?2 ıR

or equivalentlyC2 ıR � C1:

Moreover, if %1. 1/ �R %2. 2/ for elements i 2 gi then 0 �R %2.�. 1/ � 2/ and thusa2.%2.�. 1/� 2// D 0. As the G2-action on M2 is free and a2 ı %2 is its infinitesimal action,the elements 2 and %2.�. 1// must be equal. Thus the intersection R\ .C?2 �C

?1 / is, taking

%i WMi � gi ! Ai and � WM1 � g1 !M2 � g2 as bundle maps, the image .%2 � %1/.Gr.�//.

Assume now that � integrates to homomorphism of Lie groups ˆ W G1 ! G2.

Definition 2.4.3. The Courant morphism R will be called equivariant (with respect to ˆ, orsimply G if G1 D G2 D G and ˆ is the identity map) if for all group elements g 2 G1 and


elements xi 2 Ai one has

(2.22) x1 �R x2 H) g:x1 �R ˆ.g/:x2:

Equivalently, R is equivariant if it is invariant under the action of Gr.ˆ/ on A2 �A1.

Note that if R is equivariant then the base map f WM1 !M2 is necessarily ˆ-equivariant.

Lemma 2.4.3. Suppose R intertwines the generators. Then:

(a) It is equivariant.6

(b) If R is also a Dirac morphism .A1; E1/ Ü .A2; E2/ where Ei � Ai are Dirac

structures then, given a Lagrangian complement F2 � A2 of E2, the Dirac tensor

ƒF1 2 g� �V2

E1 of the backward image F1 D F2 ı R is related to the Dirac tensor

ƒF2 2 g� �V2

E2 of F2 by

ƒF1 D .�� %/.f �ƒF2/:

Proof. (a) As seen above, the Courant morphismR contains .%2�%1/.Gr.�//. In particular,the contraction of the Courant tensor

‡R..%2 � %1/.�. /; /; �1; �2/

vanishes for all 2 g1 and �1; �2 2 �.R/. But this expression is equal to ƒR. ; �1; �2/,so the Dirac tensor of R vanishes. As G1 is connected, this means that R is equivariant.

(b) One proceeds much like in the proof of Lemma 2.3.1. Let �1; �2 2 �.F2�F1/\�.A2�

A1; R/ and 2 g. For m 2 M1, the value of �i at .f .m/;m/ is .%�F2.xi/; xi/ for somexi 2 F1jm. Then

.�� %/.f �ƒF2/. ; x1; x2/jm �

ƒF1. ; x1; x2/jm D ƒF2.�. /; %�F2.x1/; %

�F2.x2// �

ƒF1. ; x1; x2/

D ƒF2�F1.�. /; ; �1; �2/j.f .m/;m/

D hJ.%2 � %1/.�. /; /; �1K; �2ij.f .m/;m/:

On the other hand

�.�. /; /ƒR.�1; �2/ D hJ.%2 � %1/.�. /; /; �1K; �2ijGr.f /

6It must be emphasized that this is not true in the more general case where the group G1 is not connected.


vanishes as seen in part (a) and thus

.�� %/.f �ƒF2/. ; x1; x2/ �ƒF1. ; x1; x2/ D 0

as needed.

This chapter culminates with the following theorem, appearing in [17] except for parts (b)and (d) which are strengthened here.

Theorem 2.4.2. Suppose R W A1Ü A2 is a that intertwines the generators (and is therefore

equivariant). Suppose furthermore that Ai (i D 1; 2) satisfy the conditions of Theorem 2.4.1.

Then:

(a) The Courant morphism R descends to a Courant morphism Rred W A1;red Ü A2;red such

that the diagram

(2.23)A1 A2

A1;red A2;red

R

q1 q2Rred

commutes, where qi W Ai Ü Ai;red are the reduction morphisms, in the sense that

R ı q1 D q2 ı R. Its base map is fred W M1;red ! M2;red, the map which descends from

f WM1 !M2.

(b) Let E1 � A1 and E2 � A2 be Dirac structures such that E1;red D q1 ı E1 and E2;red D

q2 ı E2 are Dirac structures of A1;red and A2;red respectively. Suppose � W C?1 ! C?2

restricts to a bijective map

(2.24) � W E1 \ C?1 ! E2 \ C

?2 :

If R is a Dirac morphism R W .A1; E1/Ü .A2; E2/ then Rred is a Dirac morphism

Rred W .A1;red; E1;red/Ü .A2;red; E2;red/.

(c) Suppose Ai are exact Courant algebroids and that ran.a�i / \ C?i D 0 so that Ai;red

are exact by Proposition 2.4.2. Let fred W M1;red ! M2;red be the map which descends

from the base map f W M1 ! M2. Then the reduction procedure gives a one-to-one

correspondence between Courant morphisms A1 Ü A2 intertwining the generators

with base map f and Courant morphisms A1;red Ü A2;red with base map fred.

(d) The correspondence in (c) preserves the exactness of Courant morphisms. If ji W TMi !

Ai are Gi -basic splittings with corresponding 3-forms �i 2 �3.Mi/ then, after identify-


ing Ai ' TAi;�i , the 2-forms ! 2 �2.M1/ and !red 2 �2.M1;red/ such that R D Tf!

and Rred D T .fred/!red are related by .�=G1/�!red D ! .

Proof. (a) Let eR be the flow-out of R by G D G2 � G1. Then eR D [g2Gg:R and eachtranslation g:R is a Dirac structure of A2 � A1 supported on g:Gr.f /. From this itfollows that eR is a G-invariant Dirac structure of A D A2 � A1 supported on eGr.f /,the flow-out of Gr.f / by G. As the intersection R \ .C?2 \ C

?1 has constant rank as

previously observed, so does the intersection eR \ .C?2 � C?1 /. Thus eR descends to aDirac structure Rred of A2;red�A1;red supported on Gr.ef /=G according to the discussionfollowing Proposition 2.4.1. The submanifold Gr.ef /=G ,!M2;red �M1;red is the graphof fred, in other words Rred is a Courant morphism A1;red Ü A2;red with base map fred.

Now say x1 �R x2 �q2 y2. Then x2 2 C2 and, since R intertwines the generators,one has x1 2 C1. So x1 �q1 y1 �Rred y2 where y1 D Œx1�. Conversely, if x1 �q1y1 �Rred y2, then x1 �R x2 for some x2 2 C2 with x2 �q2 y2. It follows that x1 �q2ıRy2. This shows the diagram (2.23) commutes.

(b) It is claimed is that every element of .E2;red/jfred.Œm�/ is Rred-related to a unique elementof .E1;red/jŒm� for any m 2 M1,. Now suppose y2 2 .E2;red/jfred.Œm�/ and let x2 2 .E2 \C2/jf .m/ be a lift x2 �q2 y2. Let x1 be the unique element in E1jm that is R-related tox2. Since R intertwines generators, one has x1 2 C1. So y1 D Œx1� is Rred-related to y2.This shows existence.

Say now y1 �Rred 0 for some y1 2 E1;red and let x1 2 E1 \ C1 be a lift x1 �q1y1. Then x1 �R x2 for some x2 2 E2 \ C?2 since R ı E1 D f �E2. As x1 is theunique element of E1jm that is R-related to x2 and the restriction (2.24) is a bijection, onconcludes that x1 2 E1 \ C?1 . Thus y1 D 0, showing uniqueness.

(c) It is shown how, starting with Rred W A1;red Ü A2;red, to obtain a Courant morphismA1 Ü A2 intertwining generators, whereby R is recovered. Let G D G2 � G1 andM D M2 �M1. The preimage eR of Rred under the quotient map C ! C

C?=G, where

C D C2 �C1, is a G-invariant Lagrangian subbundle of A D A2 �A1 supported on theflow-out of Gr.f / according to the discussion following Proposition 2.4.1.

Recall that, as the Gi -actions on Mi are free, the spaces Ci and Mi � gi are equalas Lie algebroids. Let D D a�1.Gr.f�//, which is an involutive coisotropic subbundlealong Gr.f / with isotropic orthogonal D? D a�.Gr.f �//. As Gr.�/ � D, the sub-bundle D is invariant under the action of the diagonal G� D f.g;ˆ.g// W g 2 G1g.Consider the Lagrangian subbundle

(2.25) eR \D CD?:


As JD;D?K � D? (Proposition 2.2.1), the space (2.25) is a Dirac structure supportedon Gr.f /, i.e. a Courant morphism A1 Ü A2. Furthermore, clearly (2.25) containsGr.�/ and thus intertwines the generators.

Conversely, if the Courant morphism R W A1Ü A2 is the starting point, then eR isthe flow-out of R by G and, as R � D, the morphism R contains both summands of(2.25) and equality follows.

(d) Suppose R is exact. One knows that R is a Dirac morphism

R W .A1; ran.a�1//Ü .A2; ran.a�2//

by Proposition 2.3.1. The condition (2.24) then trivially holds and thus Rred is a Diracmorphism

Rred W .A1;red; ran.a�1;red//Ü .A2;red; ran.a�2;red//;

meaning that Rred is exact.Conversely, suppose Rred is exact. The morphism R is exact provided every element

in f � ran.a�2/ is R-related to a unique element in ran.a�1/. Existence is guaranteed sincea�.Gr.f �// � R in all case. For uniqueness, suppose x �R 0 where x 2 ran.a�1/. Thenx 2 C1 as R intertwines generators and consequently Œx� 2 ran.a�1;red/ is Rred-related to0. Since Rred is exact, Œx� D 0, i.e. x 2 C?1 . But then x 2 ran.a�1/ \ C?1 , meaning thatx D 0. Thus R is also exact.

Next recall that j2 induces an isotropic splitting j 01 W TM1 ! A1 of A1 whose rangeis ran.j2/ ıR. Since R is equivariant and intertwines the generators, it follows that j 01 isG1-basic. Now

q1 ı ran.j 01/ D q1 ı .ran.j2/ ıR/ D .q2 ı ran.j2// ıRred;

where the second equality follows from the commutativity of (2.23). In other words, theisotropic splitting of A1;red to which j 01 descends coincides with the isotropic splittingof A1;red induced by j2;red. This means that, after identifying A1 ' TM1;f ��2 andA1;red ' T .M1;red/f �red�2;red , the containment

ran.!/ D ran.j1/ � TM1;f ��2

descends to the containment

ran.!red/ D ran.j1;red/ � T .M1;red/f �red�2;red;


showing that ! D .�=G1/�!red.

Chapter 3

L-Hamiltonian spaces

A theory of general group-valued moment maps is developed. Unlike the prior theory ofAlekseev-Malkin-Meinrenken [4], the Hamiltonian action comes from an arbitrary Lagrangiansubalgebra of the double d˚ d, where d is a quadratic Lie algebra. Notions of fusion, dualityand symplectic reduction are defined and various results are derived regarding these.

3.1 Hamiltonian spaces

Suppose A is a Courant algebroid over M and E � A is a Dirac structure. A Hamiltonian

space of .A; E/ is a manifold X together with a Dirac morphism R W .TX; TX/Ü .A; E/.The base map J W X ! M is then called the moment map of the Hamiltonian space X . SinceJ �E D R ı TX and TX \ ker.R/ D 0 by definition, the morphism R defines a vector bundlemap J �E ! TX . Consider the induced map

(3.1) �.E/! X.X/; � 7! �X :

Note that �X �R � and, with R being a Courant morphism, this means that � 7! �X is aLie algebra homomorphism. Moreover, the section �X is J -related to a.�/, where a is theanchor of A. Thus R defines a comorphism of Lie algebroids TXÜ E, in other words a Liealgebroid action of E on X along J .

Suppose now A is exact. A Hamiltonian space of A will be called exact if the Courantmorphism R W TXÜ A is exact. Choosing an isotropic splitting j W TM ! A identifyingA ' TM�, exact Hamiltonian spaces are characterized as follows.

Proposition 3.1.1 ([17]). An exact Hamiltonian space X of .A; E/ is equivalently a manifold

X together with an L-equivariant map J W X ! M , a 2-form ! 2 �2.X/ and a E-action

TXÜ E on X along J such that

36

CHAPTER 3. L-HAMILTONIAN SPACES 37

(a) d! D �J ��,

(b) ker.!/ \ ker.J�/ D 0,

(c) ��X! D J�.j ��/, where �.E/! X.X/, � 7! �X is the E-action.

Proof. Condition (a) holds if and only if .J; !/ defines an exact Courant morphism TJ! W

TXÜ A. It must simply be argued, with (a) holding, that (b) and (c) are equivalent to TJ!

being a Dirac morphism .TX; TX/Ü .A; E/, i.e. each element of EjJ.m/ is TJ!-related toa unique element of TmX for every m 2 X .

Suppose now that (b) and (c) hold. From (2.17), one has the expression

ker.R/ D fv � �v! W v 2 ker.J�/g:

Uniqueness then immediately follows. On the other hand, the section �X being J -relatedto a.�/ together with (c) imply that .a � j �/.�/ is TJ!-related to �X , establishing existence.Conversely if TJ! W .TX; TX/Ü .A; E/ is a Dirac morphism, then a symmetrical argumentshows that (b) and (c) hold.

Example 3.1.1. Let � W O ,! M be a leaf of the singular distribution a.E/ � TM . Then foreach tangent vector v 2 TO there exists some � 2 T �M such that v C � 2 E. Define a2-form on O by putting

(3.2) �v!O D ��:

To see that !O is well-defined, suppose �1; �2 2 T �M are such that .v; �i/ 2 E for i D 1; 2.Then .0; �2 � �1/ 2 E and it follows that �2 � �1 2 Ann.TO/, i.e. ��.�2 � �1/ D 0.According to Proposition 2.2.1, the restriction of E to O � M is involutive and from this itfollows that d!O D �d�

��. Thus !O defines an exact morphism T �!O W TOÜ TM�. It isobvious from the definition of !O and (2.17) that each element .v; �/ 2 EjO is T �!O-relatedto a unique element of TO, namely v. Thus O is an exact Hamiltonian space of .A; E/.

Remark 3.1.1. In the context Proposition 3.1.1, if the E-action on X is transitive, that is to saythe assignment (3.1) is surjective onto X.X/, then the 2-form ! 2 �2.X/ is necessarily uniquefor this particular choice of E-action and moment map since it is then completely determinedby (c). In particular, the 2-form !O defined in Example 3.1.1 is the unique one for whichthe leaf O together with the natural E-action and inclusion as the moment map is an exactHamiltonian space for TM�.


3.2 L-Hamiltonian spaces

3.2.1 Manin pairs determined by quadratic Lie algebras

Of particular interest in the category of Hamiltonian spaces are those where the target Maninpair .A; E/ arises from a quadratic Lie algebra d and Lagrangian subalgebra l of d˚ d. Let Dbe a connected Lie group integrating d. Consider the quadratic Lie algebra of split signatured˚ d, which integrates to the Lie group D �D. The group D �D acts transitively on D as.d1; d2/ � d D d1dd

�12 . The corresponding d˚ d-action is given by aD W d˚ d! X.D/ with

aD.�1; �2/ D �L2 � �R1 where �Li and �Ri are respectively the left- and right-invariant vector

fields corresponding to �i . The stabilizer subalgebra at the identity e 2 D is the diagonald� � d˚ d, which is Lagrangian. Since D �D acts transitively, it follows that the stabilizersubalgebras at every group element d 2 D is Lagrangian and in fact given explicitly by

(3.3) ker.aDjd / D f.Add �; �/ W � 2 dg:

The above thus determines an action Courant algebroidD�.d˚d/ with anchor aD. It is exact,admitting as a D �D-invariant isotropic splitting

(3.4) jD.v/ D1

2.��v�

R; �v�L/:

where v 2 X.D/ and �L; �R W X.D/! d are respectively the left- and right-invariant Maurer-Cartan forms . The corresponding 3-form is the Cartan 3-form

(3.5) �D D1

12hŒ�L; �L�; �Li;

where h�; �i is the metric of d.Consider now a Lagrangian subalgebra l � d ˚ d integrating to the connected subgroup

L � D � D. The subbundle E.l/ D D � l is a Dirac structure of D � .d ˚ d/ as wellas an action Lie algebroid (Examples 2.2.5 and 2.2.7).If X is an exact Hamiltonian space of.D � .d ˚ d/; E.l// with moment map J W X ! D then restricting the assignment (3.1) tol (identified with the constant sections) gives a Lie algebra action % W l ! X1.X/. As E.l/

is an action Lie algebroid, the pullback bundle J �E.l/ D X � l is the action Lie algebroidcorresponding to % and J is l-equivariant by construction.

Remark 3.2.1. The Dirac structure E.d�/ has a special name in the literature: it is the so-calledCartan-Dirac structure [1, �3.4].

Definition 3.2.1. Let l � d˚ d be a Lagrangian subalgebra integrating to the closed and con-


nected subgroup L � D2. An L-Hamiltonian space (forD � .d˚ d/) is an exact Hamiltonianspace

R W .TX; TX/Ü .D � .d˚ d/; E.l//

such that the induced l-action on X integrates to an L-action.

Note that, since the Lie group L is connected and R intertwines the generators of the l-actionson X and D by definition, the morphism R must be L-equivariant by Lemma 2.4.3.

Proposition 3.2.1 ([17]). A L-Hamiltonian space is equivalently a triple .X; J; !/ where X is

a L-manifold, the moment map J W X ! D is an L-equivariant map and ! 2 �2.X/ is an

L-invariant 2-form such that

(a) d! D �J ��D,

(b) ker.!/ \ ker.J�/ D 0,

(c) �%.�1;�2/! D �12J �.h�2; �

Li C h�1; �Ri/ for all .�1; �2/ 2 l, where % W l ! X.X/ is the

infinitesimal action.

Proof. This is simply a rewording of Proposition 3.1.1, save for the L-invariance of !. But thelatter simply follows from the fact that the isotropic splitting (3.4) is D � D-invariant. Thusit must only be shown that (c) here is equivalent to (c) in Proposition 3.1.1. Consider dualj �D W D � .d ˚ d/ ! T �D of the isotropic splitting (3.4). Then for .�1; �2/ 2 d ˚ d andv 2 X.D/ one has

hj �D.�1; �2/; vi D h.�1; �2/; jD.v/i D �1

2.h�2; �v�

Li C h�1; �v�

Ri/;

which is what needed to be shown.

Example 3.2.1. Adapting Example 3.1.1, an l-orbit O ,! D is a L-Hamiltonian space.

Example 3.2.2. With l D d�, the previous proposition gives precisely the axioms defining aquasi-HamiltonianD-space [4]. In other words, q-HamiltonianD-spaces andD�-Hamiltonianspaces are the same objects.

3.2.2 Multiplicative structures

Multiplicative structure of D � .d ˚ d/. In [1], a natural multiplication morphism1 Mult WD2 � .d ˚ d ˚ d ˚ d/ ! D � .d ˚ d/ is defined over the group multiplication map mult W

1By conventionD2� .d˚d˚d˚d/ D D� .d˚d/�D� .d˚d/ where the i th d-factor on the left-hand-sidecorresponds to the i th d-factor on the right-hand-side.


D�D ! D. In terms of the procedure for manufacturing Courant morphisms between actionCourant algebroids given in Example 2.3.2, the morphism Mult is determined by the graph ind� d� d of the multiplication of pair groupoid d˚ d� d. It may also be seen as the quotientmorphism corresponding to a coisotropic reduction ofD2 � .d˚ d˚ d˚ d/ and this approachwill prove quite profitable.

Let d�.2;3/ D 0˚d�˚0, which is an isotropic subalgebra of d˚d˚d˚d with coisotropicorthogonal d?

�.2;3/D d˚ d�˚ d. Then the elements E.d�.2;3// are generators for the action of

D�.2;3/ D e�D��e onD2�.d˚d˚d˚d/ The coisotropic reduction ofD2�.d˚d˚d˚d/

by E.d�.2;3// gives

(3.6) D2=D�.2;3/ � d�.2;3/?=d�.2;3/ ' D � .d˚ d/;

where D has been identified with the second factor of D2. Define

Mult W D2� .d˚ d˚ d˚ d/Ü D � .d˚ d/

as the quotient morphism corresponding to the coisotropic reduction (3.6). Since .d1; d2/ �D�.2;3/.d1d2; e/ for d1; d2 2 D, the base map of Mult is simply the group multiplication, denoted bymult W D �D ! D.

Note that, omitting base points,

.�1; �2; �3; �4/ �Mult .�01; �02/ ” �1 D �

01; �2 D �3; �4 D �

02

for �i ; � 0i 2 d considered as constant sections ofD�d. Thus Mult can alternatively be describedas the product mult � Gr.ı/, where ı is the multiplication in the pair groupoid

(3.7) d˚ d� d:

In the sequel, the groupoid multiplication ıwill also be denoted by Mult W d˚d�d˚dÜd˚ d when one wishes to see it as a Courant morphism.

Proposition 3.2.2 ([1]). The Courant morphism Mult W D2 � .d˚ d˚ d˚ d/Ü D � .d˚ d/

is exact. Furthermore, the 2-form & 2 �2.D2/

(3.8) & D 12h�L;1; �R;2i

is the 2-form (2.9). In particular, it relates �1D C �2D and mult��D via

(3.9) d& D �1D C �2D �mult��D:


Proof. In view of Proposition 2.4.2, to show that Mult is exact it suffices to show that theintersection of ran.a�/˚ran.a�/ andE.d�.2;3// is trivial. Recall that the space ran.a�/ D ker.a/at d 2 D is given by

ran.a�jd / D f.Add �; �/ W � 2 dg:

For d1; d2 2 D and �; � 0; � 00 2 d, the equation

.Add1 �; �;Add2 �0; � 0/ D .0; � 00; � 00; 0/

has the unique solution � D � 0 D � 00 D 0, whence Mult is exact.

Next, let jmult W TD2 ! D2 � .d ˚ d ˚ d ˚ d/ be the induced isotropic splitting as in

Proposition 2.3.2. IdentifyD�.d˚d/ ' TD�D via jD andD2�.d˚d˚d˚d/ ' TD2mult��D

via jMult. Consider the image ran.jD � jD/ of the isotropic splitting jD � jD W TD2 !

D2 � .d ˚ d ˚ d ˚ d/ in TD2mult��D

, which is the graph of a 2-form & 2 �2.D2/ satisfying(3.9). It will be shown that & is given by (3.8). To compute & , notice that it is invariant underthe action of D �D on D2 via

.d1; d2/:.c1; c2/ D .d1c1; c2d�12 /

since both mult��D and �1D C �2D are D �D-invariant. From Proposition 2.3.2 and its proof,

Gr.�& j.d1;d2// D Multj.d1;d2/ ı ran.jDjd1d2/ D f.�Add1 �; �0; � 0;Add�12 �/ W �; � 0 2 dg;

which, together with (3.4) and D �D-invariance, allows one to compute (recall (2.10))

�v0�v& D �hjmult.v/; .jD � jD/.v0/i D 1

2h�v�

L;1; �v0�R;2i �

12h�v0�

L;1; �v�R;2i

for v; v0 2 X.D2/, which is to say & is given by (3.8).

Multiplicative structure of the Lagrangian subalgebras of d˚d. For subspaces a; b � d˚d,let a ı b denote set of all elements of d˚ d resulting from the product of elements in a and b,in that order. Note that

a ı b D.a � b/ \ d?

�.2;3/

d�.2;3/:

In particular, if a and b are Lagrangian subalgebras of d˚d then so is aıb according to Section2.1.

The variety L.d ˚ d/ of Lagrangian subalgebras of d ˚ d may thus be said to be “closed


under multiplication”. Expanding on this idea, one has

(3.10) d� ı l D l ı d� D l; l1 ı .l2 ı l3/ D .l1 ı l2/ ı l3

for any Lagrangian subalgebras l; l1; l2; l3 � d˚d, which means that L.d˚d/ is a monoid. Themost obvious examples of multiplicative Lagrangian subalgebras are the diagonal d� � d˚ d

and the direct sum g ˚ h of Lagrangian subalgebras g; h � d. In general, multiplicativeLagrangian subalgebras are classified by pairs of coisotropic subalgebras of d.

Theorem 3.2.1 (Classification of multiplicative Lagrangian subalgebras). A subset l � d˚ d

is a multiplicative Lagrangian subalgebra if and only if

(3.11) l D ..c1 ˚ c2/ \ d�/C .c?1 ˚ c?2 /;

where c1; c2 � d are coisotropic subalgebras of d.

Proof. In one direction, assume c1; c2 � d are coisotropic subalgebras. Then the orthogonal of(3.11) is

..c?1 ˚ c?2 /C d�/ \ .c1 ˚ c2/:

Since c?i � ci (i D 1; 2), the above is actually equal to (3.11), which is ipso facto Lagrangian.By part (d) of Proposition 2.2.1, the isotropic orthogonals c?1 ; c

?2 � d are subalgebras of d and

are ideals in their respective coisotropic orthogonals, i.e.�c?i ; ci

�� c?i . From this it follows

that (3.11) is a (Lagrangian) subalgebra of d˚ d. Finally, as d� is multiplicative and

.c?1 ˚ c?2 / ı .c?1 ˚ c?2 / D c?1 ˚ c?2 ;

one concludes that (3.11) is multiplicative as well.

In the other direction suppose l � d˚ d is a multiplicative Lagrangian subalgebra. Let c1and c2 be the projections of l onto the first and second factors of d˚d respectively. Then c?1 ˚0

is orthogonal to l and thus contained in l; the same goes for 0˚ c?2 . From this it follows thatc1 and c2 are coisotropic in d. Now suppose .�; �/ 2 .c1˚ c2/\ d�. Then there exist elements� 0; � 00 2 d such that .�; � 0/; .� 00; �/ 2 l. As l is multiplicative, it contains the pair

.� 00; �/ ı .�; � 0/ D .� 00; � 0/:

But it also contains the pair

.�; � 0/C .� 00; �/ � .� 00; � 0/ D .�; �/;


and thus .c1 ˚ c2/ \ d� � l. So l contains the right-hand-side of (3.11) and, as both areLagrangian, equality must hold.

Example 3.2.3. The example l D d� corresponds to the choices c1 D c2 D d in (3.11), i.e.c1 and c2 have maximal dimensions. Incidentally, this shows that d� is the only multiplicativeLagrangian subalgebra of d ˚ d if the metric of d is positive- or negative-definite, which isthe case when d is simple and compact for instance. The other extreme, where c1 and c2 areof minimal dimension, corresponds to l D g˚ h where g; h � d are Lagrangian subalgebrasof d. (Of course, the metric of d must be of split signature in that case.) There are a numberof known coisotropic subalgebras and therefore ways of generating multiplicative Lagrangiansubalgebras, see [60].

Corollary 3.2.1. Let l � d˚ d be a multiplicative Lagrangian subalgebra and let � W O ,! D

be the leaf through the group identity e 2 D of the singular distribution a.E.l// � TD. Then

the 2-form (3.2) defined in Example 3.1.1 vanishes. In particular O, provided the connected

subgroup L � D � D integrating l is closed, is a L-Hamiltonian space with corresponding

Courant morphism T � W TOÜ TD�.

Proof. It is sufficient to show that the 2-form (3.2) vanishes at the group identity e 2 O. Bythe previous theorem, the subalgebra l is equal to .c1˚ c2/\d�C c?1 ˚ c?2 where c1; c2 � d arecoisotropic subalgebras. Now suppose .�i ; �i/C .� 0i ; �

00i / 2 l (i D 1; 2) are elements of l where

�i 2 c1 \ c2 and � 0i 2 c?1 and � 00i 2 c?2 . Then the anchor aD W d˚ d ! d sends these elementsto the vectors � 00i � �

0i 2 d and the dual j �D W d˚ d! d� of the splitting (3.4) sends them to the

1-forms �12h2�i C �

0i C �

00i ; �i, all respectively. But it is clear that

�1

2h2�1 C �

01 C �

001 ; �002 � �

02i D 0;

which implies that the 2-form (3.2), by way of its definition, vanishes.

At the level of Lie groups, if l1; l2 � d ˚ d are Lagrangian subalgebras integrating to theconnected subgroups L1; L2 � D2 respectively then the connected subgroup Lie.l1 ı l2/ is theidentity component of the quotient group

(3.12) L1 ı L2 D.L1 � L2/ \ .D �D� �D/

.L1 � L2/ \ .e �D� � e/�D �D� �D

e �D� � eD D2;

where D� � D2 is the diagonal of D. In particular, if L1 and L2 are closed then so isLie.l1 ı l2/.


3.2.3 Fusion product of L-Hamiltonian spaces

The existence of a monoidal structure on the space of Lagrangian subalgebras of d˚d (and con-sequently on the space of connected subgroups of D2 integrating these) motivates the searchfor an analogous monoidal structure on the category of L-Hamiltonian spaces. Suppose, fori D 1; 2, that li � d ˚ d are Lagrangian subalgebras integrating to closed and connectedsubgroups Li � D and that Li -Hamiltonian spaces

Ri W .TXi ; TXi/Ü .D � .d˚ d/; E.li //

with moment maps Ji W Xi ! D are given. A reasonable multiplication procedure for L-Hamiltonian spaces should produce a Lie.l1 ı l2/-Hamiltonian space from X1 and X2. As afirst attempt, consider the direct product X1 �X2 together with the composition

Mult ı .R1 �R2/ W TX1 � TX2Ü D � .d˚ d/:

Unfortunately, this last Courant morphism may fail to be a Dirac morphism (see Definition2.3.2)

.TX1 � TX2; TX1 � TX2/Ü .D � .d˚ d/; E.l1ıl2//:

Indeed, although R ı .TX1 � TX2/ D E.l1˚l2/, the intersection of E.l1˚l2/ andker.Mult/ D E.d�.2;3// may not be trivial. This intersection is of course E..l1˚l2/\d�.2;3// andthis pathology therefore rests on the size of the subalgebra

.l1 ˚ l2/ \ d�.2;3/:

For l1 D l2 D d�, for instance, the subalgebra (3.2.3) is trivial and X1 � X2 is indeed a D�-Hamiltonian space, equivalently a quasi-Hamiltonian D-space [4]. For l1 D l2 D g˚ g on theother hand, where g � d is a Lagrangian subalgebra, it is equal to 0˚ g� ˚ 0. The followingconstruction, which is central to this thesis, deals with this pathology.

Definition 3.2.2 (Fusion product). Let Lie..l1˚l2/\d�.2;3// be the connected (and necessarilyclosed) subgroup of D4 integrating .l1˚ l2/\ d�.2;3/. The fusion product X1~X2 of X1 andX2 is the quotient

(3.13) X1 ~X2 DX1 �X2

Lie..l1 ˚ l2/ \ d�.2;3//:

Lemma 3.2.1. The space X1 ~X2 is a smooth manifold.

Proof. The Lie..l1 ˚ l2/ \ d�.2;3//-action on D �D is clearly free and proper. As J1 � J2 W


X1 � X2 ! D �D is L1 � L2-equivariant, it follows that the Lie..l1 ˚ l2/ \ d�.2;3//-actionon X1 �X2 is also be free and proper, whence X1 ~X2 is a smooth manifold.

Theorem 3.2.2. Let X1 and X2 as the previous definition. Then the fusion product X1~X2 is

a Lie.l1 ı l2/-Hamiltonian space.

Proof. Let % W X1 � X2 � .l1 ˚ l2/! TX1 ˚ TX2 be the l1 ˚ l2-action on X1 � X2, and letC? be the image of X1 �X2 � ..l1 ˚ l2/ \ d�.2;3// under %. Then by definition

R ıE.l1˚l2/ D J � ran.%/ H) R ıE..l1˚l2/\d�.2;3// D J �C? H)

J �C? � R ıE.d�.2;3//

where R D R1 �R2 and J D J1 � J2. Thus R intertwines the generators of the .L1 � L2/ \D�.2;3/-action on TX1 � TX2 and the D�.2;3/-action on D2 � .d˚ d˚ d˚ d/. Since R isL1�L2-equivariant, it is also equivariant with respect to the inclusion of Lie..l1˚l2/\d�.2;3//

into D�.2;3/.Now part (a) of Theorem 2.4.2 gives a commutative diagram

(3.14)TX1 � TX2 D2 � .d˚ d˚ d˚ d/

T .X1 ~X2/ D � .d˚ d/

q

R

MultRred

where the vertical arrows are the reduction morphisms. Since E..l1˚l2/\d�.2;3// � E.l1˚l2/ andC? � TX , parts (b)-(d) of Theorem 2.4.2 imply that Rred is exact in addition to being a Diracmorphism

Rred W .T .X1 ~X2/; T .X1 ~X2//Ü .D � .d˚ d/; E.l1ıl2//;

where the fact that E.l1ıl2/ D E.l1˚l2/red was used. Thus X1~X2 is an exact Hamiltonian spacefor .D � .d˚ d/; E.l1ıl2//.

The l1 ı l2-action on X1~X2 coincides with the map %red W l1 ı l2 ! X.X1~X2/ to whichthe restriction of % to .l1˚l2/\d

?�.2;3/

descends (now seeing % as a map l1˚l2 ! X.X1�X2/).Thus the Lie.l1 ı l2/-action on X1 ~ X2 to which the Lie..l1 � l2/ \ .d � d� � d//-action onX1 �X2 descends integrates the l1 ı l2-action on X1 ~X2. The claim is thus proven.

Associativity. Suppose X1, X2 and X2 are L1-, L2- and L3-Hamiltonian spaces, respectively.


Theorem 3.2.3. The fusion product is associative, that is .X1~X2/~X3 andX1~ .X2~X3/are canonically isomorphic as Lie.l1 ı l2 ı l2/-Hamiltonian spaces.

Proof. Let D�.2;3/ D e �D� � e � e � e and D�.4;5/ D e � e � e �D� � e. Rewrite

.X1 ~X2/~X3 DX1�X2

Lie..l1�l2/\d�.2;3//�X3

Lie...l1 ı l2/ � l3/ \ d�.2;3//

DX1 �X2 �X3

Lie..l1 � l2 � l3/ \ d�.2;3//

�Lie..l1 � l2 � l3/ \ .0 � d� � d� � 0//

Lie..l1 � l2 � l3/ \ d�.2;3//

DX1 �X2 �X3

Lie..l1 � l2 � l3/ \ .e � d� � d� � e//:

Similarly, one finds that

X1 ~ .X2 ~X3/ DX1 �X2 �X3

Lie..l1 � l2 � l3/ \ d�.4;5//

�Lie..l1 � l2 � l3/ \ .0 � d� � d� � 0//

Lie..l1 � l2 � l3/ \ d�.4;5//

DX1 �X2 �X3

Lie..l1 � l2 � l3/ \ .0 � d� � d� � 0//:

Sincel1 ı l2 ı l3 D

l1 � l2 � l3

.l1 � l2 � l3/ \ .0 � d� � d� � 0/;

it follows that .X1 ~X2/~X3 D X1 ~ .X2 ~X3/ as Lie.l1 ı l2 ı l3/-manifolds.Next, let

q W TX1 � TX2Ü T .X1 ~X2/;

q0 W TX2 � TX3Ü T .X2 ~X3/;

qred W T .X1 ~X2/ � TX3Ü T .X1 ~X2 ~X3/;

q0red W TX1 � T .X2 ~X3/Ü T .X1 ~X2 ~X3/;

be the reduction morphisms. In view of the above, one has qredı.q�IdTX3/ D q0redı.IdTX1�q/,

where IdTXi W TXi Ü TXi are the identity isomorphisms. The diagram (3.14) then gives adiagram in the shape of a triangular prism


(3.15)

TX1 � TX2 � TX3

TX1 � TX2 � TX3 D3 � .d˚ d˚ d˚ d˚ d˚ d/

TX1 � T .X2 ~X3/

T .X1 ~X2/ � TX3 D2 � .d˚ d˚ d˚ d/

T .X1 ~X2 ~X3/

T .X1 ~X2 ~X3/ D � .d˚ d/

R1 �R2 �R3

IdTX1 � q0

q � IdTX3

R1 �R2 �R3

Mult � IdD

q0red

R1 � .R2 ~R3/

qred

.R1 ~R2/ �R3

MultR1 ~ .R2 ~R3/

.R1 ~R2/~R3

;

known to commute along all faces (and their subdivisions) except the bottom face. A simplediagram chase then shows that it commutes along the latter as well. This completes the proof.

Commutativity. The fusion product of L-Hamiltonian spaces cannot be commutative in gen-eral. For instance, one has .g˚ h/ ı .h˚ g/ D g˚ g, whereas .h˚ g/ ı .g˚ h/ D h˚ h forLagrangian subalgebras g; h � d, equality thus not holding if g and h are distinct. This said,the fusion product is indeed commutative in the subcategory of q-Hamiltonian spaces [4, Thm.6.2], cf. Example 3.2.2. One might wonder if commutativity also holds in the subcategoryof G � G-Hamiltonian spaces. As will be seen in the next chapter, the answer is negative; inthat subcategory, the fusion products X1 ~ X2 and X2 ~ X1 are generally not isomorphic asG �G-spaces.Monoidal structure. The singleton fegwhere e 2 D is the identity element is a q-Hamiltonianspace, alternatively a D�-Hamiltonian space. As should be clear from (3.10), one has

(3.16) X ~ feg D feg~X D X;

for anyL-Hamiltonian spaceX . Thus the category ofL-Hamiltonian spaces carries a monoidalstructure.Fusion product without Dirac geometry. In view of part (d) of Theorem 2.4.2, the fusionproduct X1 ~ X2 is described at the level of 2-forms. For i D 1; 2, let .Xi ; Ji ; !i/ be a Li -


Hamiltonian space. Denote by J1~J2 the moment map ofX1~X2 and by !1~!2 the 2-formin �2.X1 ~X2/ such that, in the notation of the proof of Theorem 3.2.2, one has

Rred D T .J1 ~ J2/!1~!2 :

By changing isotropic splittings TD2

�1DC�2D

' TD2mult��D

, the morphismR D T .J1�J2/!11C!22changes to

T .J1 � J2/!11C!22C.J1�J2/�& ;

where & 2 �2.D2/ is the 2-form (3.8). According to part (d) of Theorem 2.4.2 then

(3.17) .�=Lie..l1 ˚ l2/ \ d�.2;3///�.!1 ~ !2/ D !11 C !

22 C .J1 � J2/

�&:

Internal fusion product. In addition to the fusion product of L-Hamiltonian spaces, one mayalso consider the “internal” fusion product of a L-Hamiltonian space .X; J; !/ for D2 � .d˚

d ˚ d ˚ d/, where L � D4 is a closed and connected subgroup integrating a Lagrangiansubalgebra l � d˚ d˚ d˚ d. The object in question is the quotient

X DX

Lie.l \ d�.2;3//;

which is seen to be a L-Hamiltonian space by adjusting the proof of Theorem 3.2.2, where L

is the closed and connected subgroup of D integrating the Lagrangian subalgebra of d˚ d

l Dl \ .d \ d�.2;3//

?

d \ d�.2;3/:

(The fusion product considered above corresponds to the special case where l is the direct sumof two Lagrangian subalgebras of d˚d.) Its moment map J W X! D descends from multıJand the corresponding 2-form ! descends from ! C J �& .

It will turn out to be of great usefulness to recast the relationship between ! and ! in termsof a certain group structure introduced by P. Severa [54]. For any smooth manifold M , thespace C1.M;D/ ��2.M/ possesses a group structure given by

.d1; �1/ � .d2; �2/ D .d1d2; �1 C �2 C .d1 � d2/�&/

and when equipped with this group structure will be denoted by C1.M;D/ ‰ �2.M/. If Nis another manifold and f WM ! N is a smooth map, define the pullback by f of an element


.d; �/ 2 C1.M;D/‰ �2.M/ by f , denoted by f �.d; �/, to be the element

.d ı f; f ��/ 2 C1.N;D/‰ �2.N /:

This next lemma is a simple observation, it is recorded here for later use.

Lemma 3.2.2.

(a) The pullback map f � W C1.M;D/ ‰ �2.M/ ! C1.N;D/ ‰ �2.N / is a group

homomorphism, i.e.

f �..d1; �1/ � .d2; �2// D f�.d1; �1/ � f

�.d2; �2/:

(b) One has

.d; �1/.d�1; �2/ D .d

�1; �2/.d; �1/ D .e; �1 C �2/:

Returning now to the L-Hamiltonian space X , let J1 and J2 denote the composition ofthe moment map J W X ! D � D with the projections on the first and second D-factorsrespectively. Let � W X ! X be the quotient map. Then the relationship between ! and !takes the form

(3.18) .J1J2; ��!/ D .J1; !/ � .J2; 0/;

where the product is taken in the group C1.X;D/ ‰ �2.X/. Note that (3.18) amounts to(3.17) in the special where X is the direct product of two L-Hamiltonian spaces.

In full generality, one can consider an L-Hamiltonian space .X; J; !/ for a power2 of D �.d˚ d/

D � .d˚ d/ � � � � �D � .d˚ d/„ ƒ‚ …n times

D Dn� d˚ d˚ � � � ˚ d˚ d„ ƒ‚ …

n times

and perform the fusion product along any two D � .d ˚ d/-factors. This may be repeatedindefinitely until an L-Hamiltonian space .X ;J ;!/ for .D � .d ˚ d//m, where m < n, isobtained. The particular order in which this “fusion product in stages” is carried out among afixed list of D � .d˚ d/-factors is inconsequential following the same line of argument givenfor the associativity of the fusion product. Denote by Ji the composition of the moment mapJ W X ! Dn with the projection onto the i th factor of Dn. Let � W X ! X be again thequotient map. The following result generalizes (3.18).

2Here again the convention is the natural one: the i th d-factor appearing on the left-hand-side corresponds tothe i th factor appearing on the right-hand-side.


Theorem 3.2.4 (Fusion product in stages). Suppose X is obtained from X by fusing the latter

along the i th1 , i th

2 , . . . , i thkC1

factors of .D � .d˚ d//n, where k D n �m. Then

(3.19) .Ji1Ji2 � � �JikC1; ��!/ D .Ji1; !/.Ji2; 0/ � � � .JikC1; 0/:

Proof. Without loss of generality the indices can be taken to be i1 D 1, i1 D 2, . . . , ikC1 Dk C 1. The proof proceeds by induction on k. The base case k D 1 is essentially (3.18). Nowsuppose the statement holds for k � l < n � 1; one wishes to show it holds for k D l C 1

as well. Let .X 0; J 0; !0/ be the L0-Hamiltonian space obtained by taking the internal fusionproduct of X along the first l C 1 factors (the subgroup L0 � DmC1 is implicitly defined), let� 0 W X ! X 0 be the quotient map. Then by the induction hypothesis

.J1; !/.J2; 0/ � � � .JlC1; 0/.JlC2; 0/ D .J1J2 � � �JlC1; .�0/�!/.JlC2; 0/

and one is thus left with the task of showing that the right-hand-side of the equation above isequal to the left-hand-side of (3.19). Let !0 2 �2.X 0/ be the pullback of ! to X 0. Accordingto (3.18) again,

.J 01J02;!

0/ D .J 01; !0/.J 02; 0/;

Note that J1 � � �JlC1 D J 01 ı � and JlC2 D J 02 ı � and thus part (a) of Lemma 3.2.2 incombination with the above equation give

.J1 � � �JlC2; .�0/�!0/ D .J1 : : : JlC1; .�

0/�!0/.JlC2; 0/:

Since .� 0/�!0 D ��!, this completes the proof.

Remark 3.2.2. Equation (3.19) appears in [54] (although only the case ! D 0 is considered)under quite different considerations; there it is interpreted as a form of Stokes theorem relatinga flat connection on a polygon to its pullback onto the boundary. More will be said on thismatter in the ultimate chapter.

Example 3.2.4 ([4]). In [4], the double D.D/ and the fused double D.D/ were introduced asfundamental examples of quasi-Hamiltonian spaces. To illustrate the usefulness of Theorem3.2.4, these constructions are recovered by means of internal fusion products of suitable L-Hamiltonian spaces.

Let l � .d˚ d/4 be the Lagrangian subalgebra

(3.20) l D f.�1; �2; �01; �02; �2; �1; �

02; �01/ W �i ; �

0i 2 dg:


Its orbit through the group identity in D4 is

O D f.a; b; a�1; b�1/ W a; b 2 Dg ' D2;

which is a L-Hamiltonian space for .D � .d˚ d//4, where

L D f.d1; d2; d01; d

02; d

�12 ; d�11 ; .d 02/

�1; .d 01/�1/ W di ; d

0i 2 Dg � D

8

is the closed and connected subgroup integrating l. The 2-form !O defined in Example 3.1.1 iseasily seen to be trivial (this is also argued in Proposition 5.1.1 in Chapter 4). Let D.D/ be itsinternal fusion product along the first three factors of .D�.d˚d//4, which is aD��D� ' D

2-Hamiltonian space for D2 � .d ˚ d ˚ d ˚ d/. The space D.D/ is the quotient of O by theaction of the closed and connected subgroup of D8 integrating

l \ f.0; �1; �1; �2; �2; 0; 0; 0/ W �i 2 dg;

which is, in view of (3.20), trivial. Thus D.D/ D O ' D2 as manifolds. Its moment map isJ.a; b/ D .aba�1; b�1/ and the L ' D2-action is

.d1; d2/:.a; b/ D .d1ad�12 ; d2bd

�12 /:

According to Theorem 3.2.4, the 2-form ! 2 �2.D.D// is given by

.aba�1; !/ D .a; 0/.b; 0/.a�1; 0/

D .ab; .a � b/�&/.a�1; 0/

D .aba�1; .a � b/�& C .ab � a�1/�&/

which can then be expanded to

! D .a � b/�& C .ab � a�1/�&

D1

2

�h�L;1; �R;2i C h�L;1; �L;2i � hAdb�1 �

L;1; �L;1i�

Let D.D/ be the internal fusion product ofD.D/; equivalently it is the internal fusion productof O along all four factors of .D � .d˚ d//4. Here once again D.D/ D O as manifolds andthe 2-form ! 2 �2.D.D// is characterized by

.aba�1b�1;!/ D .a; 0/.b; 0/.a�1; 0/.b�1; 0/


D .ab; &/.a�1b�1;1

2h�R;1; �L;2i/

and so

! D1

2.h�L;1; �R;2i C h�R;1; �L;2i C h.ab/��L;1; .a�1b�1/��R;2i:

The spaces D.D/ and D.D/ are called the double and fused double of D, respectively. Theyare first defined in [4]. As shown there, and in this thesis in Chapter 4, when taken as Hamilto-nian spaces forD � .d˚ d/ and .D � .d˚ d//2, respectively, they correspond to spaces liftingthe moduli spaces of flat connections on a cylinder and on a 1-holed torus, respectively.

3.3 Dual, symplectic reduction and shifting trick

3.3.1 Dualization

Let inv W D ! D be the group inversion. The inverse map of the pair groupoid (3.7) is

(3.21) .�1; �2/�1D .�2; �1/;

which is an isometry d˚ d! d˚ d. Viewing d˚ d as the constant sections of D � .d˚ d/,one has

inv�aD.�1; �2/ D inv�.�L2 � �R1 / D �

L1 � �

R2 D aD.�2; �1/;

and thus according to Example 2.3.2 the product of Gr.inv/ and the graph of .�/�1 is a Courantmorphism D � .d ˚ d/ Ü D � .d˚ d/, which will be denoted by Inv. Note that Inv isequivariant with respect to the group homomorphism

(3.22) D �D ! D �D; .a; b/ 7! .b; a/:

Proposition 3.3.1. The morphism Inv W D � .d ˚ d/ Ü D � .d˚ d/ is exact. With the

identifications D � .d˚ d/ ' TD�D and D � .d˚ d/ ' TD��D , the morphism Inv is T inv.

Proof. An element of ran.a�D/ D ker.aD/ is of the form .d;Add �; �/ for some � 2 d. Thiselement is Inv-related to and only to .d�1; �;Add �/ D .d�1;Add�1 � 0; � 0/ where � 0 D Add �.This shows that an element of ran.a�D/ � D � .d˚ d/ is Inv-related to a unique element ofran.a�D/ � D � .d˚ d/, which establishes the exactness of Inv.

For the second part of the claim, it suffices to show that Inv ı ran.jD/ D ran.jD/ (recall


that jD is the splitting (3.4)). This is clear since

.Invjd / ı ran.jD/ D f.d; �;�Add�1 �/ W � 2 dg

D f.d;�Add �; �/ W � 2 dg

D ran.jDjd /:

Given a subspace a � d ˚ d, let a�1 denote the subspace of elements obtained fromelements of a by inversion. If a�1 D a, one will say that a is symmetric. For composableelements �1; �2 � d ˚ d one has .�1 ı �2/�1 D ��12 ı �

�11 and thus for another subspace

b � d˚ d there is a relationship

(3.23) .a ı b/�1 D b�1 ı a�1:

For a Lagrangian subalgebra l � d˚ d, one has a Dirac morphism

Inv W .D � .d˚ d/; E.l//Ü .D � .d˚ d/; E.l�1//:

If l is multiplicative and symmetric then, by Proposition 3.2.1, it is a subset of d˚d of the form

(3.24) l D c� C .c?˚ c?/;

where c � d is a coisotropic subalgebra. For example, the diagonal d� � d˚ d and the directsum g ˚ g, where g � d is a Lagrangian subalgebra of d, are multiplicative and symmetricLagrangian subalgebras.

Proposition 3.3.2. Let l � d ˚ d be a Lagrangian subalgebra integrating to the connected

subgroup L � D �D. Suppose the triple .X; J; !/ is a L-Hamiltonian space forD � .d˚ d/

as per Proposition 3.2.1. Then the triple .X; inv ı J;�!/ is a L�1-Hamiltonian space for

D � .d˚ d/.

Proof. By definition, the function J and the 2-form ! determine the L-equivariant Diracmorphism TJ! W .TX; TX/Ü .TD�D ; E

.l//. The composition T inv ıTJ! D T .inv ı J /!is a Dirac morphism

T .inv ı J /! W .TX; TX/Ü .TD��D ; E.l�1//:

The resulting l�1-action % W l! X.X/ is the l-action on X composed with the isometry (3.21).Since Inv is equivariant with respect to the homomorphism (3.22), it follows that T .inv ı J /!is equivariant with respect to L�1. This shows that .X; inv ı J; !/ is a L�1-Hamiltonian


space for D � .d˚ d/. According to Proposition 3.2.1 one has (a) d! D J �inv��D, (b)ker.!/ \ ker.inv�J�/ D 0 and (c) �%.�1;�2/! D

12J �inv�.h�2; �Li C h�1; �Ri/ for all .�1; �2/ 2

l�1. The conditions may be rewritten (a) d.�!/ D �J ��D, (b) ker.�!/ \ ker.inv�J�/ D 0

and (c) �%.�1;�2/.�!/ D �12J �inv�.h�2; �LiC h�1; �Ri/. These are precisely the conditions for

.X; inv ı J;�!/ to be a L�1-Hamiltonian space according to Proposition 3.2.1 again.

Definition 3.3.1 (Dualization). Given a L-Hamiltonian space .X; J; !/, the L�1-Hamiltonianspace .X; inv ı J;�!/ is called the dual of X and is denoted by X�.

Remark 3.3.1. Note that .X�/� D X . If X is a L-Hamiltonian space for D � .d ˚ d/. Infact, the proof of the preceding proposition makes it clear that in the special case l D d� theq-Hamiltonian space X and its dual X� are isomorphic asD�-space. (The space X� coincideswith the space X� constructed in [4, Prop. 4.4].) It will be seen in the next chapter that thisdoes not hold for l D g˚ g.

The next result is the L-Hamiltonian analogue of the relationship (3.23).

Proposition 3.3.3. Suppose the Lagrangian subalgebra li � d˚ d (i D 1; 2) integrates to a

closed and connected subgroup Li . Let .Xi ; Ji ; !i/ be a Li -Hamiltonian space. Then

(3.25) .X1 ~X2/� ' X�2 ~X�1

as Lie.l�12 ı l�11 /-Hamiltonian spaces.

Proof. The li -action on X�i is obtained by composing the li -action on Xi with the group iso-morphism .d1; d2/ 7! .d2; d1/. It is clear that the .l1 � l2/ \D�.2;3/-action on X1 � X2 andthe .l�12 � l�11 / \ D�.2;3/-action on X�2 � X

�1 are identical under the obvious identification

X1 � X2 D X�2 � X�1 . Since .l1 ı l2/�1 D l�12 ı l

�11 , the spaces X1 � X2 and X�2 � X

�1 are

moreover also equal as L-spaces, where l D Lie..l1 ı l2/�1/ D Lie.l�12 ı l�11 /.

Recall that the maps J1�J2 and J1~J2 are related by the following commutative diagram

(3.26)

X1 �X2 D �D

X1 ~X2 D

J1 � J2

�=Lie..l1 � l2/ \ d�.2;3// multJ1 ~ J2

:

The moment map of .X1 ~X2/� is inv ı .J1 ~ J2/ and, as mult intertwines inv � inv and inv,


one also has the commutative diagram

(3.27)

X1 �X2 D �D

.X1 ~X2/� D

.inv � inv/ ı .J1 � J2/

�=Lie..l1 � l2/ \ d�.2;3// multinv ı .J1 ~ J2/

:

On the other hand, the moment map .inv ı J2/~ .inv ı J1/ of X�2 ~X�1 fits in the diagram

(3.28)

X�2 �X�1 D �D

X�2 ~X�1 D

.inv ı J2/ � .inv ı J1/

�=Lie..l�12 � l�11 / \ d�.2;3// mult

.inv ı J2/~ .inv ı J1/

:

The top, left and right maps in the diagrams (3.27) and (3.28) all agree, and thus the momentmaps of .X1 ~X2/� and X�2 ~X�1 also agree.

It remains to be shown that the 2-forms carried by .X1 ~ X2/� and X�2 ~ X�1 coincide. Inthe same notation as in (3.17), the 2-form corresponding to .X1~X2/� is �.!1~!2/ and onehas

.�=Lie..l � l/ \ d�.2;3///�.�.!1 ~ !2// D �!11 � !

22 � .J1 � J2/

�&:

On the other hand, the 2-form corresponding to X�2 ~X�1 is .�!2/~ .�!1/ and one has

.�=Lie..l � l/ \ d�.2;3///�..�!2/~ .�!1// D �!11 � !

22 C .J1 � J2/

�.inv � inv/�&

D �!11 � !22 � .J1 � J2/

�&:

This completes proof.

3.3.2 Symplectic reduction

Let l � d ˚ d be a Lagrangian subalgebra integrating to a closed and connected subgroupL � D �D. Suppose R W TXÜ D � .d˚ d/ is a L-Hamiltonian space with moment mapJ and corresponding 2-form ! 2 �2.X/ as per Proposition 3.2.1. Let % W X � l! TX be thel-action on X .

Lemma 3.3.1. The 2-form ! sends ker.J�/ to the annihilator3 Ann.ran.%// � T �X of ran.%/

3The subscript indicates the space in whose cotangent bundle the annihilator is taken (one must distinguishbetween the annihilator in T �X and the annihilator in T �S ).


bijectively.

Proof. Recall from the proof of Proposition 3.1.1 that

ker.R/ D fv � �v! W v 2 ker.J�/g:

As R intertwines the generators, it relates the fundamental vector field %.�/ 2 X.X/ and theconstant section � 2 �.D � .d˚ d// for all � 2 l by definition. Thus ran.%/ � ran�.R/ andtaking the orthogonal on either side gives the containment ran.%/? � ker.R/. The differentialform �v!, for v 2 ker.J�/, must thus be in the annihilator Ann.ran.%// � T �X of ran.%/.On the other hand the intersection ker.!/ \ ker.J�/ is trivial by part (b) of Proposition 3.2.1.The upshot is that ! sends ker.J�/ to Ann.ran.%// injectively and it must only be shown thatthe dimensions of ker.J�jm/ and Ann.ran.%/jm/ are equal for all m 2 X . Now the backwardimage

E.l/ ıR D ran.%/C ker.R/

is a Dirac structure of TX (Example 2.3.3). As TX \ ker.R/ D 0 (by the definition of aDirac morphism), the dimension of ran.%/jm is dim.X/ � dim.ker.J�jm//. The dimension ofAnn.ran.%/jm/ is the codimension of ran.%/jm in TmX , which is dim.ker.J�jm//. This is whatneeded to be proven.

Theorem 3.3.1 (Symplectic reduction). Suppose �O W O ,! D is an orbit of L and that J is

transversal to O so that S D J�1.O/ is a submanifold �S W S ,! X . Then XO D S=L is a

symplectic orbifold.

To clarify, it is claimed that there is a closed L-invariant 2-form !0 2 �2.S/ whose kernelconsists of the L-orbit directions. In particular, if XO is a quotient manifold 4 then !0 descendsto a symplectic form !0red 2 �

2.XO/.

Proof. Let !O 2 �2.O/ be the form (3.2). Define a 2-form !0 2 �2.S/ by putting

(3.29) !0 D ��S.! � J�!O/

The 2-form !0 is closed since ��Sd! D ��SJ�d!O D ��

�SJ�� according to part (a) of Proposi-

tion 3.2.1. It is in addition L-invariant since J is L-equivariant and ! and !0 are L-invariant.It must be shown that ker.!0/ D ran.%/jS , which will be argued by double containment.

4Given a Lie group G acting on a manifold M , the quotient M=G will be called a quotient manifold providedM=G has a (necessarily unique) smooth structure compatible with the quotient topology such that the quotientmap M !M=G is a smooth submersion. For more details see [10, 5.9.5].


Since the fundamental vector field %.�/ 2 X.X/ and the constant section � 2 �.D�.d˚d//are R-related for all � 2 l, one has J�.ran.%/jS/ D TO. Consequently

T S D ker.J�jS/C ran.%/jS :

As X and O are L-Hamiltonian spaces with corresponding Courant morphisms R D TJ! andT .�O/!O respectively, the differential forms �%.�/��S! and �%.�/��SJ

�!O are equal by part (c) ofProposition 3.2.1. The 2-form !0 therefore vanishes on ran.%/jS , i.e. ran.%/jS � ker.!0/.

For the other containment, it will be argued that the kernel of the restriction of !0 to ker.J �/is ker.J�jS/\ ran.%/jS . First note that !0 coincides with ��S! on ker.J�/. According to Lemma3.3.1, the 2-form ! sends ker.J�/ to the annihilator5 AnnX.ran.%// � T �X of ran.%/. Onethus has a diagram

ker.J�jS/ AnnX.ran.%/jS/

AnnS.ran.%/jS/:

��!

��!0

��S

where the top map ��! is an isomorphism. The kernel of ��!0 is the preimage of the intersectionAnnX.ker.J�jS// \ AnnX.ran.%/jS/ by ��!. The rank of this last intersection is

dim.X/ � rank.AnnX.ker.J�jS//C AnnX.ran.%/jS// D dim.X/ � .dim.X/C

rank.ker.J�jS/ \ ran.%/jS//

D rank.ker.J�jS/ \ ran.%/jS/;

which shows that the kernel of the restriction of !0 to ker.J�/ is indeed ker.J�jS/ \ ran.%/jS .The claim is thus proven.

Remark 3.3.2. In the context of Theorem 3.3.1, the map J� sends ran.%jm/ surjectively ontoTJ.m/O for m 2 S as J is L-equivariant. Thus J is transversal to O if and only if some, andtherefore every, point of O is a regular value of J . In particular, one may instead of S considerthe preimage J�1.o/ of a point o 2 O. Then XO D J�1.o/=Zo, where Zo is the stabilizer ofthe L-action on O at o.

Remark 3.3.3. For q-Hamiltonian spaces, this theorem is a special case of [4, Thm. 5.1].

Via Poisson structures. The 2-form (3.29) may be characterized by a Poisson structure itinduces, which is the approach taken by Bursztyn-Crainic [13, � 4.4] and Bursztyn-IglesiasPonte-Severa [16, � 3.4]. This will now be explained.

5The subscript indicates the space in whose cotangent bundle the annihilator is taken (one must distinguishbetween the annihilator in T �X and the annihilator in T �S).


Consider the algebra of L-invariant functions C1.X/L. For f 2 C1.X/L, one has�%.�/df D 0 whenever � 2 l and thus df 2 AnnX.ran.%//. Since ! sends ker.J�/ toAnnX.ran.%// bijectively by Lemma 3.3.1 , there exists a unique (rough) vector field vf ofX with values in ker.J�/ such that

(3.30) �vf ! D �df;

i.e. vf is the “Hamiltonian” vector field corresponding to f . Actually, the vector field vf issmooth.

Proposition 3.3.4 ([16, Prop. 3.14]). The vector field vf is smooth.

Proof. Let F � D � .d ˚ d/ be an arbitrary isotropic complement of E.l/. This defines aprojection D � .d˚ d/ ! E.l/. Considering R as a subset of J �.D � .d˚ d// � TX , thereis a projection pr W R ! J �E.l/ � T �X . The map pr is injective. Indeed, if pr.x2; x1/ D 0

for x1 2 TX and x2 2 J �.D � .d˚ d// then x1 2 TX . As x1 �R x2, one has x2 2 J �E.l/

according to the definition of a Dirac morphism, implying that x2 D 0. But the definition of aDirac morphism then also implies that x1 is trivial.

Now the pair .vf ; df / is a rough section of ker.R/ and thus the triple .vf; df; 0/ is a roughsection of R � J �.D � .d˚ d// � TX . The image of this triple under pr is .df; 0/, meaningthat .vf ; df; 0/ is the section pr�1.df; 0/. So .vf ; df; 0/ is in fact smooth and therefore vf issmooth.

Next, define an L-invariant bracket f�; �g on C1.X/L by putting

(3.31) ff; �g D vf :

Proposition 3.3.5 ([16, Lem. 3.12]). The bracket f�; �g is Poisson.

Proof. That f�; �g is R-bilinear and Leibniz is standard. Let f; g; h 2 C1.X/L. Since ker.R/ Dfv� �v! W v 2 ker.J�/g, one has vf Cdf 2 ker.R/. In particular hvf Cdf; vgCdgi vanishesand thus

fg; f g D �vgdf D ��vf dg D �ff; gg;

i.e. f�; �g is skew-symmetric.

Next, observe that since dff; gg D Lvf dg � �vf ddg D Lvf dg one has

Jvf C df; vg C dgK D Œvf ; vg �C Lvf dg � �vgdg D Œvf ; vg �C dff; gg;


where the equality �vgdg D 0, which is equivalent to hvgCdg; vgCdgi D 0, was used. Sinceker.R/ is involutive and !] is injective on ker.J�/, it follows that vff;gg D Œvf ; vg �. PropertyC1 then gives

ff; fg; hgg D fff; gg; hg C fg; ff; hgg;

which is the Jacobi identity.

Let C1.S/L be the space of L-invariant smooth functions on S . If the conditions ofTheorem 3.3.1 are met then the 2-form (3.29), by virtue of descending to a symplectic form onXO D S=L, defines a Poisson bracket f�; �g0 on C1.S/L via

ff; �g0 D wf ;

where wf 2 X.S/ is the unique L-invariant vector field such that

(3.32) �wf !0D �df:

Proposition 3.3.6 ([16, Prop. 3.16]). If the conditions of Theorem 3.3.1 are met, then f�; �g0

is the unique Poisson bracket on C1.S/L such that the restriction map ��S W C1.X/L !

C1.S/L, with C1.X/L equipped with the Poisson bracket f�; �g, is Poisson.

Proof. Let f 2 C1.X/L. Then the vector field vf 2 X.X/ with values in ker.J�/ defined by(3.30) is parallel to S since S D ker.J�/ C ran.%/. Furthermore, as !0 and ��S! coincide onker.J�/, one may substitute !0 for ! in (3.30), i.e �vf jS!

0 D ��Sdf . From this it follows thatvf jS is equal to wf ı�S as defined by (3.32). In particular this means, if g is another function inC1.X/L, that

ff ı �S ; g ı �Sg0D �S ı ff; gg;

meaning that the restriction map ��S is indeed Poisson. On the other hand, the above equationdetermines f�; �g0, hence its uniqueness.

If Xred D X=L and XO D S=L are quotient manifolds, then Proposition 3.3.6 equivalentlysays that XO is a symplectic leaf of Xred.

Shifting trick. The final result of this chapter relates the fusion product, dualization and sym-plectic reduction. Let l � d ˚ d be a multiplicative and symmetric Lagrangian subalgebraintegrating to the closed and connected subgroup L � D �D.

Theorem 3.3.2 (Shifting trick). Let .X; J; !/ be aL-Hamiltonian space forD�.d˚d/. Given

an L-orbit �O W O ,! D, then .X; J; !/ and O meet the conditions of Theorem 3.3.1 if and


only if X ~ O� and the L-orbit �Oe W Oe ,! D through the group identity e 2 D do6 In that

case there is a canonical symplectomorphism

(3.33) XO ' .X ~O�/Oe :

The second part of the statement must be clarified. Let SX and SX~O� be the preimages of Oand Oe under the moment maps of X and X ~O� respectively. Then the claim is that there isa commutative diagram

(3.34)SX SX~O�

XO XOe

f

�=L �=Lfred

where 1. the bottom horizontal arrow fred is a bijection, and 2. the top horizontal arrow f isa smooth map under which the 2-form !0X 2 �

2.SX/ defined in (3.29) is the pullback of itscounterpart !0X~O� 2 �

2.SX~O�/.

Proof. The moment map ofX~O� is J~.invı�O/, which is related to the map J �.invı�O/ WX �O� ! D �D via the diagram

(3.35)X �O� D �D

X ~O� D

J � .inv ı �O/

�=Lie..l � l/ \ d�.2;3// multJ ~ .inv ı �O/

;

Now J ~ .inv ı �O/ is transversal to Oe if and only if the group identity is a regular value for it(Remark 3.3.2), or equivalently (according to the diagram (3.35)) if and only if J � .inv ı �O/is transversal to the anti-diagonal D� D f.d; d

�1/ W d 2 Dg. The preimage of D� byJ � .inv ı �O/ is

f.m; J.m// W m 2 Sg ' S:

For m 2 S, consider the sum of subspaces

(3.36) ran..J � .inv ı �O//�j.m;J.m///C T.J.m/;J.m/�1/D�:

Since T.J.m/;J.m/�1/D� D f.v; inv�v/ W v 2 TJ.m/g, the dimension of the intersection of thetwo summands in (3.36) is equal to that of the intersection ran.J�jm/\TJ.m/O. The dimension

6Note that in general Oe is not a singleton like in other symplectic reduction theorems.


of (3.36) is therefore equal to

rank.J /C dim.O/C dim.D/ � dim.ran.J�jm/ \ TJ.m/O/:

So the sum (3.36) is equal to T.J.m/;J.m/�1/D�D if and only if the sum ran.J�jm/CTOJ.m/ isequal to TJ.m/D, that is to say J � .inv ı �O/ is transversal toD� if and only if J is transversalto O. This establishes the first part of the statement.

Suppose now that X and O, and hence X ~O� and Oe, satisfy the conditions of Theorem3.3.1. LetbSX~O� be the lift of SX~O� to X �O�. Explicitly,

(3.37) bSX~O� D f.m; o/ 2 X �O� W J.m/o�1 2 Oeg:

In particular there is a natural embedding of S into bSX~O� sending m 2 SX to .m; J.m//.Define f W SX ! SX~O� to be the composition this embedding and the quotient map �=..L �L/ \D�.2;3//. It is now claimed that

(3.38) !0X D f�!0X~O�

or equivalently that !0X D .Id � J /�b!0X~O� , where b!0X~O� 2 �2.bSX~O�/ is the pullback of

!0X~O� to bSX~O� . First, recall Corollary 3.2.1: the 2-form (3.2) corresponding to Oe is trivialsince l is multiplicative. In particular the 2-form !0X~O� is simply equal to the pullback toSX~O� of the 2-form of theL-Hamiltonian spaceX~O�, which is !~.�!O/ 2 �

2.X~O�/where !O 2 �

2.O/ is the 2-form (3.2) corresponding to O. On the other hand, the pullback of! ~ .�!O/ to X �O� is given by (3.17). The upshot is that

b!0X~O� D ��bSX~O�

.!1 � !2O C .J � inv/�&/:

Now,

.Id � J /�b!0X~O� D ��SX! � �

�SXJ

�!O C1

2hJ ��L; J �inv��Ri

D ��SX! � ��SXJ

�!O ��

��1

2hJ ��L; J ��Li

D ��SX .! � J�!O/

D !0X ;

proving the equality (3.38).

Finally, it will be shown that the map f W SX ! SX~O�; m 7! Œm; J.m/� descends toa bijection XO ! XOe . First notice that .d1; d2/:m, where .d1; d2/ 2 L, is sent by f to


Œ.d1; d2/:m; .d2; d1/:J.m/� (recall that the L-action on O� is .d1; d2/:o D d2od�11 ). Since

.d1; d2/ ı .d2; d1/ D .d1; d1/, one has

Œ.d1; d2/:m; .d2; d1/:J.m/� D .d1; d1/:Œm; J.m/�:

according to the definition of the L-action on X ~ O�. This shows that f descends to a mapfred W XO ! XX~O� as in the diagram (3.34).

Suppose now f .m/ D .d1; d2/:f .m0/ for m;m0 2 SO and .d1; d2/ 2 L. Then

f .m/ D .d1; d2/:f .m0/ H) Œm; J.m/� D Œ.d1; d /:m

0; .d; d2/:m0�

H) m D .e; d 0/.d1; d /m0

where d; d 0 2 D are elements such that .d1; d /; .d; d2/; .e; d 0/ 2 L. This shows that Œm� DŒm0� in XO and therefore that fred is injective.

For surjectivity, suppose an element Œm; o� 2 SX~O� is given. One has J.m/o�1 2 Oe andthus J.m/o�1 D d1d�12 for some .d1; d2/ 2 L. Now

(3.39) d�11 J.m/d2.d�12 od2/

�1D d�11 J.m/o�1d2 D e:

The pairs .d�11 ; d�12 / and .d�12 ; d�12 / are in L since the first is the inverse of .d1; d2/ andthe second is equal to .d�12 ; d�11 / ı .d�11 ; d�12 /. One may thus rewrite (3.39) in terms of theL-actions on X and O� as

J..d�11 ; d�12 /:m/..d�12 ; d�12 /:o/�1 D e ” .d�12 ; d�12 /:o D J..d�11 ; d�12 /:m/:

This leads to

.d�11 ; d�12 /Œm; o� D Œ.d�11 ; d�12 /:m; .d�12 ; d�12 /:o� D f .m0/

where m0 D .d�11 ; d�12 /:m. So f sends m0 to an element of SX~O� that is L-equivalent toŒm; o� and fred is therefore surjective. The proof is now complete.

Chapter 4

D=G-valued moment maps

The results proven for L-Hamiltonian spaces in the preceding chapter can now be broughtto bear on the theory of D=G-valued moment maps. Of particular interest, fusion and dualitybehave differently in this category than in other familiar settings: the fusion product is not com-mutative and duals are not isomorphic as G-spaces. Explicit counterexamples are provided.

4.1 Hamiltonian spaces for action Courant algebroids

Suppose d is a quadratic Lie algebra and g � d is a Lagrangian subalgebra. Let h � d bea Lagrangian complement of g, which is guaranteed to exist by Proposition 2.1.1. The triple.d; g; h/ is called a Manin quasi-triple. If h is a Lagrangian subalgebra of d, the triple .d; g; h/is simply called a Manin triple.

Example 4.1.1. Let g be a real semisimple Lie algebra and consider its complexification gC

viewed as a real Lie algebra. The imaginary part of the Killing form of gC is an Ad-invariantmetric under which g is a Lagrangian subalgebra of gC. The triple .gC; g;

p�1g/ is an ex-

ample of a Manin quasi-triple. According to results by P. Delorme [22], there is at least oneLagrangian subalgebra h � gC complementary to g. If g is compact then there is a canonical(up to conjugation) such complementary Lagrangian subalgebra; it is the solvable Lie algebrab D a˚ n appearing in the Iwasawa decomposition gC D k˚ a˚ n with k D g [45].

Let .d; g; h/ be a Manin quasi-triple. Suppose S is a manifold on which d acts with coiso-tropic stabilizers thus defining an action Courant algebroid S � d. The anchor will be denotedby aS W S � d! TS . Consider a Hamiltonian space

R W .TX; TX/Ü .S � d; E.g//

63

CHAPTER 4. D=G-VALUED MOMENT MAPS 64

with moment map J W X ! S . The same considerations as in the case of L-Hamiltonianspaces reveal that the Lie algebroid action (3.1) is a Lie algebra action

% W g! X1.X/:

The extension of % to a morphism of exterior algebrasV� g ! X�.X/ will also be denoted

by %. Recall Propositions 2.1.2 and 2.1.3, and the intervening discussion: the backward imageE.h/ ı R is the graph Gr.��h/ of a bivector field ��h 2 X2.X/. Moreover, R induces to abundle map

%�h W Gr.��h/ ' T �X ! J �E.h/

dual to % (taken as a bundle map J �E.g/ ! TX ).

Recall Lemmas 2.3.1 and 2.4.3: the Courant tensor ‡Gr.��h/ 2 �.V3

TX/ and the Diractensor ƒGr.��h/ 2 g� � �.

V2E/ defined in Chapter 2 are related to the Courant and Dirac

tensors of E.h/ via

‡Gr.��h/D %.J �‡E

.h/

/ D %.‡h/

andƒGr.��h/

D .Idg � %/.J�ƒE

.h/

/ D .Idg � %/.ƒh/

where % has been extended to mapV�

E.g/ !V�

TX in the second terms of either equation.The tensors

‡E.h/

2 �.^3

E.g// D �.X �^3

g/

ƒE.h/

2 �.g� �^2

E.g// D �.X � g� �^2

g/

are of course the Courant and Dirac tensors of E.h/, respectively.

According to Examples 2.2.8 and 2.4.3, the tensors ‡Gr.��h/ and ƒGr.��h/ are 12Œ�h; �h�

and L%. /�h respectively. The upshot is that

(4.1)1

2Œ�h; �h� D %.‡h/; L%. /�h

D �%.ƒh. //:

These conditions are precisely the ones defining a quasi-Poisson action of g on a manifold Xfound in [2] equipped with a bivector field �h in [2].

Definition 4.1.1 ([2]). Let X be a manifold equipped with a bivector field �h 2 X2.X/. Aquasi-Poisson action of g on X is a g-action % W g ! X.X/ such that the conditions (4.1) aresatisfied. In this case X is called a quasi-Poisson g-space (G-space if the g-action integrates to


a G-action). The quasi-Poisson g-space X is called non-degenerate if

ran.%/C ran..�h/]/ D TX:

As shown in [2], the choice of Lagrangian complement h in the previous definition is immater-ial; the prescription

�h0D �h

� %.t/;

where t 2V2 g is the twist defined by

t ] W h ' g� ! g; � 7! prh0 � � �

(Proposition 2.1.3) shows how to modify �h given another choice of Lagrangian complementh0 � d˚ d of g. The next proposition is essentially due to Bursztyn and Crainic [14]. Therethe corresponding results are presented as a categorical equivalence between two moment maptheories, one stated in terms of Dirac geometry and the other in terms of quasi-Poisson actions.

Proposition 4.1.1. A Hamiltonian space for .S � d; E.g// is equivalently a quasi-Poisson g-

space X together with a g-equivariant map J W X ! S such that the moment map condition

(4.2) .�h/] ı J � D % ı .aS jE .h//�;

where % W g ! X is the g-action and �h 2 X2.X/ is the corresponding l-Hamiltonian quasi-

Poisson bivector field, is satisfied1. If S � d is exact then X is an exact Hamiltonian space if

and only if it is non-degenerate as a quasi-Poisson g-space.

Proof. In one direction, suppose X is a Hamiltonian space for .S �d; E.g//. It was seen abovethat h determines a bivector field �h 2 X2.X/ satisfying (4.1). One knows that

.aS � a/�.Gr.J �// � R

and thus for � 2 T �X and ˛ 2 T �S

h��hC �; J �˛i D h%�h.�/; a

�S˛i

D h�; % ı prE .g/ ı a�S.˛/i

D h�; % ı .aS jE .h//�.˛/i;

where the duality of %�h and % was used. Since �h is anti-symmetric, the above equation is

1It is emphasized that this characterization depends on h.


precisely (4.2).

In the other direction, suppose a bivector field �h 2 X2.X/ satisfying (4.1) and (4.2) isgiven. One wishes to recover a Dirac morphism

R W .TX; TX/Ü .S � d; E.g//

with base map J and corresponding bivector field �h. Since R must define a bundle map%�h W T

�X ! J �E.h/ dual to % W J �E.g/ ! TX , one is forced to write

(4.3) R D ff . / W 2 gg C fg.�/ W � 2 T �Xg

wheref W g! S � d � TX; 7! . ; %. //

andg W T �X ! S � d � TX; � 7! .%�h.�/;��

hC �/:

The two summands on the right-hand-side of (4.3) intersect trivially and a dimension countestablishes thatR is indeed Lagrangian in S �d�TX . Dualizing the g-equivariance conditionJ� ı % D aS jE .g/ , one has

%�h ı J�D .aS jE .g//� D prE .h/ ı a�S :

For ˛ 2 T �S , let D .aS jE .h//�.˛/. Then

%�h.J�˛/C D .aS jE .g//�.˛/C .aS jE .h//�.˛/ D a�S.˛/:

Moreover (4.2) gives �J�˛�h D %�h. / and thus

.J �˛; ˛/ D f . /C g.J �˛/;

and the containment .aS � a/�.Gr.J �// � R, equivalently .aS � a/.R/ � Gr.J�/, follows.The Courant tensor ‡R 2 �.

V3R�/ of R is therefore defined and one must show that it

vanishes to verify that R is a Dirac structure supported on Gr.J /. Consider sections �i ; �j 2�.E.h/ � Gr.��h// (i D 1; 2; 3 and j D 1; 2) such that �i restricts to g.�i/ on Gr.J / for adifferential form �i 2 �

1.X/ and �j restricts to f . i/ on Gr.J / for some i 2 g. Then form 2 X one has

‡R.g.�1/; g.�2/; g.�3//j.J.m/;m/ D hJ�1; �2K; �3ij.J.m/;m/


D ‡E.h/�Gr.��h/.�1; �2; �3/j.J.m/;m/

D ‡E.h/

.%�h.�1/; %�h.�2/; %

�h.�3//jJ.m/ �

‡Gr.��h/.�1; �2; �3/jm

D %.‡h/.�1; �2; �3/jm �1

2Œ�h; �h�.�1; �2; �3/jm

D 0:

Also

‡R.f . 1/; g.�1/; g.�2//j.J.m/;m/ D hJ�1; �1K; �2ij.J.m/;m/

D ‡E.h/�Gr.��h/.. 1; %. 1//; �1; �2/j.J.m/;m/

D ƒE.h/�Gr.��h/. 1; �1; �2/j.J.m/;m/

D ƒE.h/

. 1; %�h.�1/; %

�h.�2//jJ.m/ �

ƒGr.��h/. 1; �1; �2/jm

D %.ƒh/. 1; �1; �2/jm C L 1�h.�1; �2/jm

D 0:

Finally for any section � 2 �.R/ one has

‡R.f . 1/; f . 2/; �/ D hJ�1; �2KjGr.J /; �i

D hf .Œ 1; 2�/; �i

D 0:

These calculations show that ‡R vanishes identically and thereby prove the first part of thestatement.

The second part follows immediately from the fact that the Courant morphismR W TXÜS � d is exact if and only if

a.ran�.R// D ran.%/C ran..�h/]/ D TX:

A quasi-Poisson g-space X is called a Hamiltonian quasi-Poisson g-space (with moment

map valued in S ) if a moment map J W X ! S satisfying the moment map condition 4.2has been chosen. In this case the bivector field �h 2 X2.X/ will be called a g-Hamiltonian

bivector field for X .


4.1.1 Fusion product of L-Hamiltonian spaces

TheL-Hamiltonian spaces considered in Chapter 3 are special cases of the kind of Hamiltonianspaces considered above. One could expect the bivector fields associated to the fusion productof two L-Hamiltonian spaces to be related to their respective associated bivector fields in someway. This section culminates in Theorem 4.1.1, which gives this relation and generalizes aquasi-Poisson geometric characterization of the fusion product of quasi-Hamiltonian spacesappearing in [3]. Specifically, given Lagrangian subalgebras li � d˚ d (i D 1; 2) integratingto closed and connected subgroups Li � D and Li -Hamiltonian spaces

Ri W TXiÜ D � .d˚ d/;

Theorem 4.1.1 exhibits a l1 ı l2-Hamiltonian bivector field for the fusion product X1 ~ X2in terms of li -Hamiltonian bivector fields for Xi . This result relies on the existence of adLi -invariant Lagrangian complements of the Lagrangian subalgebras li � d ˚ d. Such comple-ments do not always exist (see Examples 4.4.1 and 4.4.2), but presumably do in most casesencountered in practice according the following proposition.

Proposition 4.1.2. Suppose the Lagrangian subalgebra l � d˚ d integrates to the closed and

connected subgroup L � D �D. Then l admits an AdL-invariant Lagrangian complement if

it is semisimple or if L is compact.

Proof. If l is semisimple then any real finite-dimensional representation of it is completelyreducible (this is the real analogue of Weyl’s theorem on complete reducibility, see e.g. [32,Appx. B]). Applying this to the adjoint representation of l in d˚ d, it follows that l � d˚ d

admits an ad-invariant complement. On the other hand, if L is compact then one can define anAdL-invariant positive definite metric on d˚ d. So l admits an AdL-invariant complement ind ˚ d in either case. One can thus start with such a complement in the proof of Proposition2.1.1; the constructions derived from this complement given there are also AdL-invariant andtherefore produce an AdL-invariant Lagrangian complement of l in the end.

Suppose now the Lagrangian subalgebras li � d˚d admit AdLi -invariant Lagrangian com-plements ki � d˚ d. Choose a (not necessarily AdLie.l1ıl2/-invariant) Lagrangian complementk of l1 ı l2. Denote the moment maps of the Li -Hamiltonian spaces Xi by Ji W Xi ! D andthe corresponding li -actions by %i W li 7! X.Xi/. Let

C? D .l1 ˚ l2/ \ d�.2;3/

where d�.2;3/ D 0�d��0. Denote by .D�.d˚d//red the coisotropic reduction ofD�.d˚d/


by E.C/, i.e.

.D � .d˚ d//red DE.C/

E.C?/

ıLie..l1 � l2/ \ d�.2;3//:

Let R D R1 �R2 and J D J1 � J2 and Jred D J1~ J2 and % D %1 � %2. Since C? � d�.2;3/,part (a) of Theorem 2.4.2 gives a commutative diagram

TX1 � TX2 D2 � .d˚ d˚ d˚ d/

T .X1 ~X2/ .D � .d˚ d//red D � .d˚ d/

q

R

q0Mult

Rred Multred

where q and q0 (and Mult) are the reduction morphisms and Multred descends from the identityautomorphism of D2 � .d˚ d˚ d˚ d/. According to Proposition 2.4.1, the Dirac structureE.l1˚l2/ � D2 � .d˚ d˚ d˚ d/ descends to a Dirac structure E.l1˚l2/red of .D � .d˚ d//red.Moreover, (b) of Theorem 2.4.2 implies that Rred is a Dirac morphism

Rred W .T .X1 ~X2/; T .X1 ~X2//Ü ..D � .d˚ d//red; E.l1�l2/red /:

Denote by %red the bundle map J �redE.l1�l2/red ! T .X1 ~X2/ that Rred induces.

Recall from the previous section that the backward image of E.ki / by Ri is the graph ofa bivector field ��ki . Since ki is AdLi -invariant, the bivector field ��ki is Li -invariant. Inparticular the bivector field ��k1˚k2 D .��k1/ � .��k2/ is L1 � L2-invariant and its forwardimage under the reduction morphism q is a Lagrangian subbundle of T .X1 ~ X2/. On theother hand, the Lagrangian subbundle E.k1˚k2/ descends to a Lagrangian subbundle E.k1˚k2/red .

Lemma 4.1.1. The forward image of the graph of ��k1˚k2 under q coincides backward image

Rred ıE.k1˚k2/red . In particular it is the graph of a bivector field ��k1˚k2

red 2 X2.X1 ~X2/.

Proof. Given an element x 2 q ı .Gr.��k1˚k2//, choose x0 2 Gr.��k1˚k2/ \ ran.%/? suchthat x0 �q x. Then, as Gr.��k1˚k2/ is the backward image of E.k1˚k2/ by R, there exists anelement y 0 2 E.k1˚k2/ such that x0 �R y 0. Dualizing the containment J �E.C

?/ � R ı ran.%/gives

J �.R ı ran.%/?/ � E.C/;

and thus y 0 2 E.C/. So there is an element y 2 E.k1˚k2/red such that y 0 �q0 y. This implies thatx �Rred y and consequently

q ı .Gr.��k1˚k2// � E.k1˚k2/red ıRred:

As both sides are Lagrangian, equality ensues. The second part of the claim follows from


the fact that E.k1˚k2/red ı Rred, being a Lagrangian complement of T .X1 ~ X2/ according toProposition 2.1.2, is the graph of a bivector field.

Finally, consider the backward image E.k/ ı Multred of E.k/ by Multred. As both it andE.k1˚l2/red are Lagrangian complements of E.l1˚l2/red one can define a twist t 2

V2E.l1˚l2/red by

puttingt ] W E

.k1˚k2/red ' .E

.l1˚l2/red /� ! E

.l1˚l2/red ; x 7! prE .k/ıMultred

x � x:

Theorem 4.1.1. The bivector field

(4.4) �k1˚k2red � %red.t/

is a l1 ı l2-Hamiltonian bivector field for X1 ~X2.

Proof. The graph of��k1˚k2red is the backward image ofE.k1˚k2/red as seen in the previous lemma.

The claim then follows immediately from Proposition 2.1.3.

4.2 D=G-valued moment maps

4.2.1 Background

Let .d; g/ be a Manin pair and suppose D is a Lie group integrating d such that its connectedsubgroup G integrating g is closed. The pair .D;G/ is called a group pair for the Manin pair.d; g/. There is a natural action of the group D on the homogeneous space D=G

(4.5) d:d 0G D dd 0G:

The restriction of this action to the acting group G is called the dressing action. The stabilizerof the action (4.5) at a coset dG 2 D=G is Add G and so the stabilizer algebras of the corres-ponding d-action are all Lagrangian. In particular it defines a Courant algebroid D=G � d. Asa consequence of Proposition 4.1.1, a Hamiltonian space for .D=G � d;D=G � g/ is equival-ently a quasi-Poisson g-space (G-space if the induced g-action integrates) with a moment mapvalued in D=G as defined by Alekseev and Kosmann-Schwarzbach [2].

The following observation is central to this chapter.

Theorem 4.2.1. There is a one-to-one correspondence between non-degenerate quasi-Poisson

G-spaces with moment maps valued in D=G and L-Hamiltonian spaces where L D G �G.

Proof. One may regard D=G � d as the Courant algebroid reduced from D � .d ˚ d/ withC? D E.0˚g/. Since the corresponding G-action g:d D dg�1 is free and proper, Proposition


2.4.2 ensures that D=G � d is exact. According to Proposition 4.1.1 (in this special casedue Bursztyn-Crainic [14] building on work of Bursztyn-Severa-Iglesias Ponte [16]), a non-degenerate quasi-Poisson G-space X is equivalently an exact Hamiltonian space

R W .TX; TX/Ü .D=G � d;D=G � g/;

with base map J W X ! D=G, such that the induced g-action % W g ! X.X/ integrates to aG-action. Let Xlift ,! X �D be the pullback of J W X ! D=G and the principal G-bundleD ! D=G so that one has a commutative diagram

Xlift D

X D=G

Jlift

�=G �=G

J

;

Note that Xlift is a G � G-space. Denote the corresponding g ˚ g-action by %lift W g ˚ gÜX.Xlift/. The projection onto the D-factor is a G � G-equivariant map Jlift W Xlift ! D. SinceX D Xlift=.e � G/ and J is the map to which Jlift descends, part (c) and (d) of Theorem 2.4.2imply that R lifts to a G-equivariant exact Courant morphism

Rlift W TXlift Ü D � .d˚ d/

intertwining the generators of the e �G-actions and with base map Jlift such that the diagram

(4.6)TXlift D � .d˚ d/

TX D=G � d

Rlift

q1 q2

R

commutes, where q1 and q2 are the reduction morphisms. It will be shown that Rlift is a Diracmorphism with respect to TXlift and E.g˚g/. The element .d; 1; 2/ 2 E.g˚g/ is q2-relatedto .dG; 1/, which in turn is R-related to a unique element of TX . It follows that .d; 1; 2/is Rlift-related to an element of TXlift. This shows existence. Now if v 2 TXlift \ ker.Rlift/

then Œv� belongs to TX \ ker.R/ D 0 and so v D %lift.0; / for some 2 g. Since v �R 0and .Jlift/�v D .0; / by equivariance, it follows that D 0 and thus v D 0. This establishesuniqueness. Note that the g˚ g-action on Xlift induced by Rlift coincides with %lift.

Conversely, if the G � G-Hamiltonian space Xlift was the starting point then coisotropicreduction gives the diagram (4.6). The two constructions being inverse to each other, the claim


is proven.

4.2.2 Fusion product and dual

Suppose .d; g/ is a fixed Manin pair admitting the group pair .D;G/. Since g ˚ g � d ˚ d

is multiplicative and invertible, Theorem 4.2.1 gives a way to bring over the notions of fusionproduct and duality developed in the context of L-Hamiltonian spaces in Chapter 3 as well astheir attendant results to the category of non-degenerate quasi-Poisson G-spaces with momentmaps valued in D=G. The notation introduced for the category of L-Hamiltonian spaces isretained.

The picture for the fusion product in this category is as follows. Given two non-degeneratequasi-Poisson G-spaces

Ri W .TXi ; TXi/Ü .D=G � d;D=G � g/ .i D 1; 2/

with moment map Ji W Xi ! D=G, form the product

J1 � J2 W X1;lift �X2;lift ! D2� .d˚ d˚ d˚ d/:

As a manifold, the fusion product X1 ~X2 is equal to the quotient

X1;lift �X2

e �G�

and one has a commutative diagram

.T .X1;lift �X2;lift/; T .X1;lift �X2;lift// .D2 � .d˚ d˚ d˚ d/; E.g˚g//

.T .X1 ~X2/; T .X1 ~X2// .D=G � d; E.g//

R1;lift �R2;lift

R1 ~R2

where the vertical arrows are the reduction morphisms corresponding to the appropriate coiso-tropic reductions.

On the other hand, the dual X� of a Hamiltonian space X in this category is obtained byquotient the liftXlift ofX by the leftG-action as opposed to the rightG-action. In other words,the spaces X�lift and Xlift are equal as manifolds and the G �G-action of the former is obtainedfrom the G �G-action of the latter by trading the two G-factors.

Example 4.2.1. To illustrate the novelty of the notion of fusion product just given, an exampleof a target homogeneous spaceD=G admitting no Lie group structure is given. Let d D sl2.C/


and g D sl2.R/ with D D SL.2;C/ and G D SL.2;R/. One can identify the homogeneousspace D=G D SL.2;C/=SL.2;R/ with the 3-dimensional de Sitter space dS3, i.e. the hyper-boloid of one sheet

dS3 D f.t; x1; x2; x3/ 2 R1;3W t2 � x21 � x

22 � x

23 D �1g;

where R1;3 is Minkowski spacetime. One may identify R1;3 with the space of 2 � 2 complexHermitian matrices with the assignment

.t; x1; x2; x3/ 7!

t C x3 x1 �

p�1x2

x1 Cp�1x2 t � x3

!;

whereby the metric of R1;3 becomes the determinant. Then dS3 is identified with the setof Hermitian matrices of determinant �1 and SL.2;C/ acts transitively on dS3 via C:H DCHC � where C 2 SL.2;C/ and H is a Hermitian matrix. The stabilizer at

�0p�1

p�1 0

�is

SL.2;R/ and therefore SL.2;C/=SL.2;R/ D dS3. Now since dS3 is topologically the productR � S2 of the real line and the 2-sphere, it does not admit a Lie group structure. Indeed, thesecond homotopy group of a Lie group must vanish [11] whereas �2.dS3/ D Z.

Non-isomorphic dual and non-commutativity. In the context of Example 4.2.1, let O ,!

D=G through dG 2 D=G, where d D�p�1 0

0 �p�1

�. Then its dual O� D O�lift=.e �G/ is the

G-orbit of d�1G in D=G. Consider now the matrices

a D

1 1

0 1

!; b D

1 �1

0 1

!:

Note that dbd�1 D a, from which it follows that a 2 Stab.xG/ and b 2 Stab.d�1G/. Thematrices a and b are not conjugate in SL.2;R/: any matrix conjugate to a in SL.2; R/ hasa non-negative upper-right entry, which is seen by a simple computation. So Stab.dG/ andStab.d�1G/ are not conjugate subgroups of G and consequently O and O� are of differentorbit types [26, Lemma 2.6.2], i.e. they are not isomorphic as G-spaces.

Now by definition

O~O� D Olift �G O�; O� ~O D O�lift �G O:

Consider the element Œ.d; d�1G/� 2 O~O�. It is clear that a 2 Stab.Œ.d; d�1G/�/ since

a:Œ.d; d�1G/� D Œ.ad; d�1G/� D Œ.db; d�1G/� D Œ.d; b�1d�1G/� D Œ.d; d�1G/�:


Now consider the element Œ.d�1; dG/� 2 O� ~ O. As similar argument shows that b 2Stab.Œ.d�1; dG/�/. One thus concludes as before that O ~ O� and O� ~ O are not iso-morphic as G-spaces. Therefore the fusion product in the category of non-degeneratequasi-Poisson G-spaces with moment maps in D=G is in general non-commutative andduals are generally not isomorphic as G-spaces.

4.3 Comparison with Poisson g-spaces

4.3.1 Poisson g-spaces

Suppose .D;G/ is a Lie group pair for .d; g/ and that .d; g; h/ is a Manin triple. Let H � Dbe the connected subgroup integrating h. In J.-H. Lu’s moment map theory [44], moment mapstake value in H (which may be substituted by its universal cover with minor adjustments),which she denotes by G�. Lu views G� as the Lie group dual to G; the Manin triple .d; g; h/defines a multiplicative Poisson structure on G [25] and the linearization of such a Poissonstructure at the group identity gives a Lie bracket on g� which coincides with the Lie algebrastructure of h from the point view just adopted. Consider the mapsG�H ! D andH �G !D given by multiplication. Since they are local diffeomorphisms, one can find neighbourhoodof the group identities UG � G and UH � H on whose products UG � UH and UH � UG themultiplication map is diffeomorphic with UGUH D UHUG . For this reason, the decompositionD D GH D HG will be assumed to be global in the sequel with the understanding that thediscussion can be adapted to the general case by considering suitable neighbourhoods of thegroup identities in the groups D, G and H .

Now one can define an action of D on H extending multiplication on the left in H byputting, for d 2 D and h 2 H ,

(4.7) dh D .d:h/g;

where d:h 2 UH and g 2 UG are the unique elements such that (4.7) holds. This actioncoincides with the action (4.5), and likewise its restriction to the acting group G is called theleft dressing action on H . The induced d-action is

(4.8) � Lh prh Adh�1 �

for � 2 d. Likewise, one can define a right D-action on H extending right multiplication in H


by putting

(4.9) hd D g.d:h/

where g 2 G and d:h 2 H are the unique elements such that this equality holds. Naturally, itsrestriction to the acting group G is called the right dressing action on H .

Similarly, there are D-actions on G with their restrictions to the acting group H called theleft and right dressing actions on G.

Definition 4.3.1 ([44]). A Poisson g-space (G-space) is a g- (resp. G)-manifold P togetherwith a 2-form ! 2 �2.P / and a g-equivariant map J W P ! H such that

(a) ! is closed,


(c) �%. /! D �J �h�R; i, where % W g! P is the g-action.

Remark 4.3.1. As a consequence of (4.1) and this last proposition, a Poisson g-space carries aPoisson structure, hence the terminology.

The kernel at h 2 H of the Courant algebroidH � d D D=G � d is Adh g and there is thusa natural isotropic splitting

(4.10) jH .v/ D ��v�R:

Since ran.jH / D E.h/ is a Dirac structure of H � d, the corresponding 3-form is trivial and soH � d ' TH .

Proposition 4.3.1. A Poisson g-space is equivalently an exact Hamiltonian space for .H �

d; E.g//.

Proof. Identifying H � d ' TH via the splitting jH , (a) and (c) follow directly from Pro-position 3.1.1. Since (b) here is stronger than its counterpart in Proposition 3.1.1, it must onlybe only that if TJ! W .TP; TP /Ü .H � d; E.g// is an exact Hamiltonian space then ! isnon-degenerate. Recall that Gr.�!/ D ran.jH / ı TJ! . However, since ran.jH / D E.h/ istransversal to E.g/, its backward image ran.jH / ı TJ! is also the graph of a bivector field onP . It follows that ! is non-degenerate.

Example 4.3.1 (Classical moment map theory). Given an arbitrary Lie algebra g integrating toa connected Lie group G, let d D h Ì g be the semi-direct product of the Lie algebra g and theabelian Lie algebra h D g� where the underlying map g! Der.h/ is 7! ad� . As a metrized


vector space, the semi-direct product h Ì g is equal to g� ˚ g. As subalgebras of d, the Liealgebras g and h are Lagrangian and .d; g; h/ is a Manin triple. The Lie group D D h Ì Gintegrates d and its connected subgroups integrating g and h are respectively G and H D h.The left dressing action on H is simply the coadjoint representation. As h is G-invariant inthis case, the above proposition then specializes as follows: an exact Hamiltonian space for.h � d; h � g/ is equivalently a g-manifold P together with a g-equivariant map J W P ! g�

and a 2-form ! such that

(a) ! is closed,


(c) �%. /! D �d h ; J i for all 2 g, where % W g! P is the g-action.

This is the classical definition of a moment map valued in the dual of a Lie algebra.

Let j 0D W TD ! D � .d ˚ d/ be the isotropic splitting induced by jH as in Proposition2.3.2, in other the isotropic splitting with image E.h˚g/. It identifies D � .d˚ d/ ' TD andis given explicitly by

(4.11) j 0D.v/ D .�Adh prh �HG;Adg�1 prg �

HG/;

where �HG is defined by �v�HG D Lh�1Rg�1v for v 2 ThgD where h 2 H and g 2 G.The 2-form corresponding to the change of isotropic splitting TD ' TD�D is (according to(2.10))

(4.12) $ D1

2h�HG; prg �

HGi:

and the 2-form corresponding to the morphism MultD W D2� .d˚ d˚ d˚ d/Ü D� .d˚ d/

is

(4.13) & 0 D & C$1C$2

�mult�$:

The group G (or H ) is itself a G � G-Hamiltonian space since it is the orbit of E.g˚g/

through the identity. In terms of the identification D � d ˚ d ' TD, the correspondingCourant morphism is T �G where �G W G ,! D is the inclusion, according to Corollary 3.2.1.Consequently, the group G carries a natural non-trivial Poisson structure �h˚h

G 2 X2.G/ cor-responding to the Lagrangian complement h˚ h of g˚ g in d˚ d. Now

v C � �T �G .�1; �2/ ” �2 D prh.Adg�1 �1/; v D �Lg prg Adg�1 �1; � D h�1; �Li


for v 2 TgG and � 2 T �gG. Since Gr.��h˚hG / is the backward image of E.h˚h/ under T �G ,

the above expression shows that the symplectic leaves of �h˚hG are precisely the orbits of the

dressing actions on G.

4.3.2 Fusion product

Recall Example 2.3.4: The generalized tangent bundle TH carries a natural Lie groupoidstructure TH � h� ' g where the source and target map are respectively the left and righttrivializations and multiplication is the exact morphism TmultH W TH � TH Ü TH .Through the identification H � d ' TH , the source and target maps become

(4.14) s.h; �/ D prg Adh�1 �; t.h; �/ D prg �;

and multiplication becomes

(4.15) .h1; �1/ ı .h2; �2/ D .h; �/ ” h D h1h2; ��aH .�1/�RC Adh1 �2 D �

for composable pairs .h1; �1/ and .h2; �2/.

Proposition 4.3.2. The morphism MultH is a Dirac morphism

MultH W .H � d �H � d; E.g˚g//Ü .H � d; E.g//:

Proof. Suppose h1; h2 2 H and say 2 g. Writing h D h1h2, one wishes to solve

.h1; 1/ ı .h2; 2/ D .h; /:

By (4.14), one is forced to take 1 D and 2 D prg Adh�11 . Then

��aH . 1/�RC Adh1 2 D Adh1 prh Adh�11 C Adh1 prg Adh�11 D ;

i.e. this choice of 1 and 2 satisfies (4.15). This proves the claim.

Remark 4.3.2. Note thatg:.h1h2/ D .g:h1/..h1:g/:h2/

whereH acts onG by the right dressing action. Putting g D exp.�t / for 2 G and differen-tiating .g:h1; .h1:g/:h2/ at t D 0, one obtains .aH . /jh1; aH .prg Adh�11 /jh2/ in accordancewith the proof above.


The upshot of Proposition 4.3.2 is that a fusion product in the category of Poisson g-spacescan be defined: Given Poisson g-spaces

Ri W TP1Ü H � d .i D 1; 2/;

their fusion product isP1 ~ P2 D P1 � P2

with corresponding morphismMultH ı .R1 �R2/:

In terms of Definition 4.3.1, the corresponding 2-form and moment map are, respectively, thesum and product of those of P1 and P2, and the g-action on P1 ~ P2 is given by

(4.16) %. /j.p1;p2/ D .%1. /jp1; %2.prg AdJ1.p1/�1 /jp2/;

where %i W g! X.Pi/ (i D 1; 2) is the g-action on Pi and Ji W Pi ! H the moment map. Thisdefinition of the fusion product of Poisson g-spaces agrees with that derived by H. Flaschkaand T. Ratiu [27, Lem. 2.19]. Following Remark 4.3.2, if the g-actions on P1 and P2 integrateto G-actions, then the g-action on P1 ~ P2 integrates to the G-action

(4.17) g:.p1; p2/ D .g:p1; .J.p1/:g/:p2/:

Now if Pi (i D 1; 2) are Poisson G-spaces with moment maps Ji W P ! H then they areexact Hamiltonian spaces for

.H � d; E.g// D .D=G � d; E.g//:

There are therefore a priori two ways of taking the fusion product P1~P2: one in the categoryof Poisson g-spaces and one in the category of non-degenerate quasi-Poisson G-spaces. Thenext results show that these two fusion products agree. In the following, the subscript H willbe appended to the fusion product symbol ~ to indicate it is interpreted in the category ofPoisson g-space. Otherwise it is interpreted in the category of quasi-Poisson G-spaces.

The group G is the orbit �G W G ,! D through the group identity e 2 D of the Diracstructure E.g˚g/ and is therefore a G � G-Hamiltonian space. According to Corollary 3.2.1,its 2-form with respect to the identification D � .d˚ d/ ' TD�D is trivial.

Lemma 4.3.1. Let X be a G �G-Hamiltonian space. Then G ~X D X .

Proof. Let J be the moment map of X and ! its 2-form with respect to the identification


D � .d˚ d/ ' TD. Its 2-form with respect to the identification D � .d˚ d/ ' TD�D on theother hand is !C J �$ and, as the latter is G �G-invariant and J ı .g; e/ D Lg ı J (the map.g; e/ here stands for the action by .g; e/ 2 G �G on X ), one has

.g; e/�! D ! C J �$ � J �L�g$:

Furthermore, by Proposition 3.2.1, one has

�%. ;0/! D �1

2J �h�1; �

Ri C J �� R$:

As manifolds, one has G ~X D X with the identification descending from the action map

A W G �X ! X

of the G � e-action on X . As such, the G � G-action on G ~ X coincides with that on Xand the moment maps of those spaces are equal. Now, the 2-form of G ~ X with respect toD � .d˚ d/ ' TD is

! C .��G � J�/.� 0/ D ! C .��G � J

�/.� C$1C$2

�mult�$/;

which must be shown to be equal to A�! to complete the proof. Note that, since ��G$ D 0,the term $1 may be omitted in the above expression. For i 2 g and vi 2 X.X/, one has

A�!.��G R1 C v1; ��G R2 C v2/ D !..g; e/�v1 � %. 1; 0/; .g; e/�v2 � %. 2; 0//

D .g; e/�!.v1; v2/ � !.%. 1; 0/; .g; e/�v2/C

!.%. 2; 0/; .g; e/�v1/ C !.%. 1; 0/; %. 2; 0//

D .! C J �$/.v1; v2/C1

2h 1; �

RLgJ�v2i �

1

2h 2; �

RLgJ�v1i � J�Lg$.v1; v2/ �$.

R1 ; LgJ�v2/C

$. R2 ; LgJ�v1/ �$. R1 ;

R2 /

D !.��G R1 C v1; �

�G

R2 C v2/C

.Id � J /�.� C$2�mult�$/.��G

R1 C v1; �

�G

R2 C v2/;

as claimed.

Theorem 4.3.1. LetRi W TPiÜ H �d (i D 1; 2) be Poisson g-spaces with moment maps Ji .

Suppose the actions %i integrate to G-actions. Then the fusion product P1 ~ P2 interpreted in


the category of Poisson g-spaces agrees with its interpretation in the category of quasi-Poisson

G-spaces with moment maps valued in D=G .

Proof. In the notation of Theorem 4.2.1,

Pi;lift D f.p; Ji.p/g/ W p 2 Pi ; g 2 Gg D Pi �G

andRi;lift D TmultD ı.Ri�Id/ in terms of the identificationsH �d ' TH andD�.d˚d/ '

TD. Likewise, .P1 ~H P2/lift D P1 � P2;lift and its corresponding Courant morphism isTmultD ı .R1 �R2;lift/. Furthermore, as manifolds,

P1;lift ~ P2;lift D P1 � .G ~ P2;lift/ D .P1 ~H P2/lift:

Now denote by multHG , multHD and multGD the restrictions of the multiplication mapmultD to H �G, H �D and G �D respectively. Consider the diagram

.TP1 � TG/ � TP2;lift TH � TG � TD T .D �D/

T .P1 ~H P2/lift TH � TD TD

q

R1 � Id �R2;lift TmultHG � Id

Id � T .multGD/.�G�Id/�& 0 MultDR1 �R2;lift TmultHD

where q is the morphism labelled q1 in (4.6). This diagram is commutative; commutativity inthe left square follows from Lemma 4.3.1 and commutativity in the right square amounts to theequality

.multHG � Id/�& 0 D ..�G � Id/�& 0/2;

which can be verified by direct computation. Composing the top horizontal morphisms givesR1;lift �R2;lift, which therefore descends to TmultD ı .R1 �R2;lift/ in the diagram (3.14). Thatis to say P1;lift ~ P2;lift D .P1 ~H P2/lift as G �G-Hamiltonian spaces, which is what neededto be argued.

4.4 Some examples

Let .d; g/ be a Manin pair. In the case where it can be completed to a Manin triple .d; g; h/,the D=G-valued moment map theory might be argued to be redundant since one already hasLu’s G�-valued moment map theory. Likewise, in the case where g admits an adg-invariantcomplement then the theory reduces to the classical theory near the coset of the group identity(Appendix 4.A). Following are examples of Manin pairs .d; g/ where no such complementh � d may found. In the case of the first example, the subalgebra g is an ideal of d and


therefore D=G is a Lie group for any group pair .D;G/ and thus an objection may still beraised. The second example does away with any such objection: the subalgebra g given theredoes not admit a Lagrangian complement that is a subalgebra or adg-invariant and there is agroup pair .D;G/ for .d; g/ for which D=G cannot be made into a Lie group.

Example 4.4.1. Let h be the Heisenberg algebra, i.e. the 3-dimensional Lie algebra admittingthe presentation

hA;B;E j ŒA; B� D Ei :

Let A, B , E be a basis as above and let ˛, ˇ, " be the basis dual, with the obvious pairingconvention. Let � D ˛ ^ ˇ ^ " and define the following bracket on h˚ h�:

Œx C '; y C � D Œx; y�h C adx � ady � C �.x; y/:

This bracket is Lie and moreover the natural metric on h ˚ h� is ad-invariant with respect toit; see [9, Ex. 4.2]. Denote this metrized Lie algebra by T �

�h. The non-zero commutation

relations in T ��h with respect to these bases are

(4.18) ŒA; B� D E C "; ŒA;E� D �ˇ; ŒB;E� D ˛; ŒA; "� D �ˇ; ŒB; "� D ˛:

It is readily seen from the above that T ��h is nilpotent with nilindex 3. Clearly h�, identified

with the second factor, is a Lagrangian subalgebra (in fact an abelian ideal) of T ��h.

Now Suppose that there were a 3-dimensional subalgebra g � T ��h such that T �

�h D

h� C g. Since the only nilpotent Lie algebras of dimension 3 are the abelian and HeisenbergLie algebras [21], the subalgebra g would be nilpotent of nilindex 1 or 2. Note that ŒT �

�h; h��

is contained in the center of T ��h. Therefore given p; q; r 2 T �

�h, one has Œp; Œq; r�� D 0 if any

of p, q, or r is in T ��h. It follows that

ŒT �� h; ŒT�� h; T

�� h�� Œg; Œg; g�� D 0;

meaning T ��h has nilindex 2, a contradiction.

Note that h� does not admit an adh�-invariant Lagrangian complement since that wouldcontradict the commutations (4.18). Therefore .T �

�h; h�/ is an example of a Manin pair

whose Lagrangian subalgebra h� admits no Lie algebra complement nor any adh�-invariantcomplement. Since h� is an ideal of T �

�h, the connected subgroup integrating the former is

closed in the simply connected Lie group integrating the latter, meaning .T ��h; h/ admits a

group pair.


Example 4.4.2. Retaining the notation in the previous example, consider the Manin pair

.T �� h; h�/˚ .sl2.C/; sl2.R// D .T

�� h˚ sl2.C/; h

�˚ sl2.R//:

Suppose g � T ��h ˚ sl2.C/ were an Lagrangian complement of h ˚ sl2.R/. Let g1 and g2

be the images of g in the first and second factors respectively. Then h� C g1 D T ��h and

sl2.R/ C g2 D sl2.C/, hence dim.g1/ � 4 and dim.g2/ � 3. Let k1 D .g1 ˚ 0/ \ g andk2 D .0˚g2/\g. According to Goursat’s lemma, the image of g in g1=k1˚g2=k2 is the graphof an isomorphism of vector spaces

(4.19) g1=k1 ' g2=k2:

Note that dim.gi/ � 3 and dim.ki/ � 3 and

(4.20) dim.g1/C dim.k2/ D dim.g2/C dim.k1/ D 6:

Case where g is a Lie algebra. In this case g1 and g2 are Lie algebras with k1 and k2 re-spectively as ideals. Furthermore (4.19) is an isomorphism of Lie algebras. From the previousexample it is known that dim.g1/ � 4. Suppose first that dim.g1/ D 4. Arguing similarly as inthe previous example, one has ŒT �

�h; ŒT �

�h; T �

�h�� Œg1; Œg1; g1�� and the nilindex of g1 must

therefore be 3. There is only one 4-dimensional nilpotent Lie algebra with nilindex 3 [21], itadmits the presentation

hX1; X2; X3; X4 j ŒX1; X2� D X3; ŒX1; X3� D X4i :

Since h� is an abelian ideal in T ��h, h� \ g1 is a 1-dimensional ideal of T �

�h. It is readily seen

that the only 1-dimensional ideals of T ��h contained in h� are spanned by linear combinations

of the basis elements ˛ and ˇ. Therefore h� \ g1 is in the center of T ��h and a fortiori in that

of g1. Taking a basis Xi as in the above presentation, it is clear that the center of g1 is the spanof X4. Thus X4 D a˛ C bˇ for a; b 2 R not both 0. Thus

ŒT �� h; T�� h� D h˛; ˇ;E C "i

but also h˛; ˇ;X3i. So without loss of generality one has X3 D E C " C k˛ C lˇ for somek; l 2 R. But then

ker adX3 D hX; Y;Z; zi D T�� hC hX3i ;

which does not include X2, a contradiction. One concludes that dim.g1/ � 5.


Now consider g2. There are no 5-dimensional Lie subalgebras of sl2.C/2. Up to con-jugation there is only one 4-dimensional Lie subalgebra, the Borel subalgebra, which is infact a complex subalgebra of complex dimension 2 with real form B, the Borel subalgebra ofsl2.R/. Suppose then that g2 D B C

p�1B. As it is a centerless complex Lie algebra, it

does not admit a real 1-dimensional ideal. Also dim.k2/ ¤ 0 since B Cp�1B is not nilpo-

tent. Thus dim.k2/ � 2 and dim.g1/C dim.k2/ � 7, contradicting (4.20). One concludes thatdim.g2/ D 3.

There are up to conjugation five non-isomorphic 3-dimensional subalgebras of sl2.C/, noneof which are abelian or nilpotent. So dim.k2/ ¤ 0, forcing dim.k2/ D 1. Since the quotientalgebra g2=k2 is nilpotent of dimension 2, it is abelian. It follows that the derived algebraŒg2; g2� is just k2 and in particular 1-dimensional. Now the real forms of sl2.C/ are up toconjugation su.2/ and sl2.R/, both of which are semisimple, excluding them as possibilitiesfor g2. Therefore the complex Lie algebra g2 C

p�1g2 has complex dimension 2 and the

complex Lie algebra g2 \p�1g2 has complex dimension 1. As g2 \

p�1g2 is abelian but g2

is not, it follows that dim.Œg2 \p�1g2; g2�/ > 0. But that means Œg2 \

p�1g2; g2� is a non-

trivial complex vector space contained in the real 1-dimensional space Œg2; g2�, a contradiction.

Case where g is adh˚sl2.R/-invariant. In this case g1 and k1 are adh-invariant and g2 and k2 areadsl2.R/-invariant. As was argued in the previous example, the subalgebra h does not admit anadh-invariant Lagrangian complement and thus dim.k1/ � 2. As sl2.R/ is a simple Lie algebra,the adsl2.R/-invariant subspace g2 \ sl2.R/ must be trivial and thus dim.g2/ D 3. However,(4.20) then forces dim.k1/ D 3, a contradiction.

Therefore .T ��h ˚ sl2.C/; h� ˚ sl2.R// is an example of a Manin pair that does not

admit a Lie algebra complement nor an adh˚sl2.R/-invariant Lagrangian complement,and whose Lagrangian subalgebra is not an ideal. Furthermore it admits a group pair .D;G/obtained as the componentwise product of .SL.2;C/;SL.2;R// and a group pair .K;H/ for.T ��h; h�/. Since K=H is a Lie group, the second homotopy group of the quotient D=G is

isomorphic to that of dS3 D SL.2;C/=SL.2;R/which is non-trivial as seen above. ThereforeD=G does not admit a Lie group structure.

2For a list of the subalgebras of sl2.C/, identified as the Lie algebra of the Lorentz group SO.1; 3/, see [29,Chapter 6].


4.A Appendix: Classicism near the coset of the group iden-tity

Suppose .d; g/ is a Manin pair admitting the group pair .D;G/. In some circumstances, forinstance if g is semisimple or if G is compact according to Proposition 4.1.2, an AdG-invariantLagrangian complement h � d of g can be found. In that case the exponential map exp W d!D sends a small neighbourhood of 0 2 h diffeomorphically onto a neighbourhood of the cosetŒe� 2 D=G of the group identity e 2 D. The exponential map furthermore intertwines thecoadjoint representation of G (identifying h ' g�) and the dressing action on D=G.

On the other hand, in the classical theory of g-Hamiltonian spaces, moment maps are valuedin the dual h ' g� of the Lie algebra g and intertwine a G-action on their domain and thecoadjoint representation of G. It would therefore seem that the D=G-valued moment maptheory might converge towards the classical moment map theory near the coset Œe�. This indeedwill be argued here. There is however an important caveat: the discussion requires a passagethrough infinite-dimensional manifolds. These are handled loosely so as to not get boggeddown in auxiliary questions of regularity that would interfere with the geometric aspects of thearguments given. License has been taken in extending results presented in Chapter 2 to thiscontext. This, however, is not entirely unjustified; the results in question can also be derived inthe formalism of Hilbert manifolds [17], barring additional topological closure conditions. Seealso [4, �8.1, �9.1].

4.A.1 Gauge-theoretic preliminaries

Let PD be the group of paths W Œ0; 1� ! D into the Lie group D and let P d be the Liealgebra of paths Œ0; 1�! d in the Lie algebra d. Consider the space of connections

(4.21) P d� D �1.D � Œ0; 1�; d/D

of the trivial principal bundle3 Œ0; 1��D. A connection A 2 P d� is completely determined byits pullback onto Œ0; 1�, embedded in Œ0; 1��D in the natural way. Alternatively, it is determinedby its corresponding horizontal path P 2 PD with initial condition P.0/ D e. The space ofconnections of Œ0; 1��D may thus be regarded as the group of paths inD starting at the groupidentity e.

Consider the Lie algebra P d� Ì P d of the semi-direct product P d� Ì PD where PD actson P d� via d:A D Add A. It carries a natural Ad-invariant metric given by the pairing

3Contrary to the usual convention, the group D acts on Œ0; 1� � D on the left via d:.t; d 0/ D .t; dd 0/. Thespace P d� consists of d-valued 1-forms ! that are equivariant in the sense that d�! D Add .


(4.22)Z 1

0

hA; �i;

for A 2 P d� and � 2 P d. Consider then the natural action of P d� Ì PD on P d�. As thisaction is transitive, the stabilizer algebras of the action of P d� Ì Pd on P d� are translationsof the stabilizer algebra at the identity, which is the Lagrangian subalgebra P d of P d� Ì P d.These considerations determine an action Courant algebroid

(4.23) Pd D P d� � .P d� Ì PD/:

Its anchor will be denoted by aPd� . It admits the natural P d�ÌPD-invariant isotropic splitting

(4.24) jPd�.A/ D .A; 0/;

with corresponding identification Pd ' TP d�.

Consider next the action of PD on P d� by gauge transformations

(4.25) d:A D Add A � d��R:

Identifying A 2 P d� with its corresponding horizontal path P 2 PD with initial conditionP.0/ D e, the action (4.25) admits the alternative description

(4.26) .d:P /.t/ D d.0/�1P.t/d.t/;

from which one sees that the action (4.25) by gauge transformations is transitive. Define theholonomy map hol W P d� ! D as follows: if A D d:0 then hol.A/ D d.0/d.1/�1. This mapis well-defined since if d is in the stabilizer of 0 then so is d�1 and therefore d.0/d.1/�1 D eaccording to the description (4.26). Note that

(4.27) hol.d:A/ D d.0/holt.A/d.1/�1:

The infinitesimal counterpart of the action (4.25) by gauge transformations is the covariantderivative ²A� D d� C adA � . For a subspace s � d˚ d, define

(4.28) E .s/ D f.d�;��/ 2 P d� Ì P d W � 2 P d�; .�.0/; �.1// 2 sg;

which is a subspace of P d� Ì P d and will also be regarded as a (trivial) subbundle of Pd. For


a subgroup K � D �D define

(4.29) GK D f.�d��R; d / 2 P d� Ì PD W d 2 PD; .d.0/; d.1// 2 Kg

which is a subgroup of P d�ÌPD. If s � d˚d is a subalgebra, then in fact E .s/ is a subalgebraof P d�ÌP d (see Proposition 4.A.1 below) integrating to the subgroup GS , where S � D�Dis the connected subgroup integrating s. For the special case s D d ˚ d and S D D � D,the supercript in (4.28) and the subscript in (4.29) will be omitted and these will be calledrespectively the gauge Lie algebra and the gauge group. The gauge group G implements theaction (4.25), hence the terminology.

Proposition 4.A.1. Let s � d˚ d be a subspace. The subbunbdle E .s/, seen as a subbundle of

Pd, is closed in Pd and its orthogonal is E .s?/. Furthermore E .s/ is,

(a) isotropic if s is isotropic,

(b) coisotropic if s is coisotropic,

(c) involutive if s is a Lie subalgebra.

Proof. For .d�1;��1/; .d�2;��2/ 2 E .d˚d/, observe that

h.d�1;��1/; .d�2;��2/i D �

Z 1

0

hd�1; �2i �

Z 1

0

h�1; d�2i

D h�1.0/�2.0/i � h�1.1/�2.1/i;

(4.30)

where integration by parts was used. It is immediate from (4.30) that .E .s//? D E .s?/. Inparticular .E .s?//? D E .s/ and it follows that E.s/ is closed. If s is (co)isotropic in d˚ d thenit also follows from (4.30) that E .s/ is (co)isotropic in Pd. Finally, if s is a subalgebra of d˚ d

then, regarding E .s/ as a subspace of P d� Ì P d, one has

Œ.�d�1; �1/; .�d�2; �2/� D .ad�2 d�1 � ad�1 d�2; Œ�1; �2�/ D .�dŒ�1; �2�; Œ�1; �2�/

and thus E .s/ is a subalgebra of P d� Ì P d. Arguing as in Example 2.2.5 then, one concludesthat E .s/ is an involutive subbundle of Pd.

The gauge Lie algebra E is coisotropic with isotropic complement E .0/. From the descrip-tion (4.26), it follows that the action of Ge on P d� is free and, as P d�=G0 D D is finite-dimensional, it is also proper. This defines a Courant algebroid

(4.31)E .d˚d/

E .0/=G0


obtained by coisotropic reduction. As ŒE .0/; E � � E .0/, according to Example 2.4.4 the reducedCourant algebroid (4.31) is an action Courant algebroid with base manifold D and acting Liealgebra

E .d˚d/

E .0/D d˚ d;

where E .d˚d/ and E .0/ are interpreted as subspaces of P d� Ì P d this time. The quotient mapwill identify two connections A1; A2 2 P d� if only if there is an element d 2 PD withd.0/ D d.1/ D e such that A2 D d:A1. According to (4.26) and (4.27), this means that A1and A2 are identified if and only if hol.A1/ D hol.A2/. One can thus take the view that thequotient map is the holonomy map hol W P d� ! D and, by (4.27) again, the reduced Courantalgebroid (4.31) is non other than the familiar Courant algebroid D � .d˚ d/. The reductionmorphism PdÜ D � .d˚ d/ will be denoted by

Hol W PdÜ D � .d˚ d/:

Note that Hol is equivariant with respect to the quotient of Lie groups �=G0 W G ! D �D andintertwines the generators.

Proposition 4.A.2. The reduction morphism Hol W PdÜ D � .d˚ d/ is exact.

Proof. Since ŒE ; E .0/� � E .0/, it follows from C5 that the subbundle E .0/ � Pd is invariantunder the action of the gauge group G on Pd. Now the kernel of aPd� at A D 0 is f.0; �/ W� 2 P dg, which intersects E .0/jAD0 trivially. As the action of G on P d� is transitive and Holis equivariant with respect to the quotient �=G0 W G ! D �D, it follows that the intersectionE .0/ \ ker.aPd�/ is everywhere trivial and the claim then follows from Proposition 2.4.2.

Let j 0Pd� be the isotropic splitting of Pd defined by the isotropic splitting jD of D � .d˚d/ as in Proposition 2.3.2. Note that j 0Pd� is G-invariant. It is therefore determined by itsrestriction j 0Pd�j0. Explicitly,

j 0Pd�.A/j0 D .A;��A/

where �A 2 P d solves d�A D A with initial condition .d.0/; d.1// 2 d� (recall that d� is theanti-diagonal of d˚ d). In fact,

�A.t/ D

Z t

0

A �1

2

Z 1

0

A:

For B 2 P d� and let dB W Œ0; 1� ! D be the (necessarily unique) solution to the differentialequation �d��R D B with initial condition d.0/ D e (if B is a constant connection then


d is an integral curve of the fundamental vector field ��R 2 X.D/). Then according to thedescription (4.25), one has d:0 D �. In particular

j 0Pd�jB.A/ D j0Pd�j0.Ad�1.B;dB/A/ D j

0Pd�j0.Ad�1dB A/ D .d�Ad�1

dBA;��Ad�1

dBA/:

Let ! 2 �2.P d�/ be the 2-form corresponding to the Courant morphism Hol under the iden-tifications D � .d ˚ d/ ' TD and Pd ' Td. This 2-form is G-invariant and according to(2.10), it is given by

(4.32) !.A1; A2/jB D hjPd�.A1/; j0Pd�.A2/ijB D �

Z 1

0

hA1; �Ad�1dBA2i:

4.A.2 Classicism of D=G-valued moment maps

Consider the action Courant algebroid

Ah D g� � .g� Ì g/ ' h � .h Ì g/

introduced in Example 4.3.1. Denote its anchor by ah. It is exact, admitting the isotropicsplitting jh.A/ D .A; 0/ thereby identifying it with Th. (This is completely analogous to theconstruction of Pd above.) The inclusion map �h W h ! P d�, where h is identified with theconstant h-valued connections, thus defines a Courant morphism

T �h W Ah ' ThÜ TP d� ' Pd�:

Note that the inclusion map �h is equivariant with respect to the inclusion

(4.33) G ,! GG�G; g 7! .0; g/:

and thus the Courant morphism T �h is also equivariant with respect to (4.33). Now the coiso-tropic reduction of D � .d˚ d/ by E.0˚g/ is D=G � d. The upshot is the diagram

(4.34)

Ah Pd

D � .d˚ d/

D=G � d

T �h

eHol Hol

q


where q is the reduction morphism and eHol D q ıHolıT �h. As q is equivariant with respect tothe projection D � G ! D, the morphism eHol is G-equivariant. The holonomy of a constantconnection � 2 d � P d� is the exponential exp.tA/ and the base map of the Courant morphismeHol is therefore

� 7! Œexp.�/�;

and for this reason it will be denoted by exph W h! D=G. The base map exph is thus invertiblenear 0 in the sense that there is an open neighbourhood U � h of 0 sent diffeomorphically ontoan open neighbourhood V of Œe� 2 D=G. Let .D=G�d/jV denote the restriction ofD=G�d tothe base manifold V , made into a Courant algebroid in the obvious way, similarly for .Ah/jU .Retain eHol to denote the restriction of eHol W AhÜ D=G � d to .V � U/ \ Gr.exph/.

Theorem 4.A.1 (Classicism near the coset of the group identity). The Courant morphism eHol W.Ah/jU Ü .D=G � d/jV is a Dirac morphism

..Ah/jU ; .h � g/jU /Ü ..D=G � d/jV ; E.g/jV /:

Proof. Since eHol W .Ah/jU Ü .D=G � d/jV is a Courant isomorphism, it suffices to showthat

eHol ı .h � g/jU � E.g/jV :

Actually, as eHol W Ah ! D=G � d is G-equivariant, it suffices to verify that

(4.35) eHol ı .h � g/j0 � E.g/jŒe�:

Now E.g/j0 D g is the kernel of the anchor of Ah at 0. Its forward image by T �h is thereforethe kernel of aPd� at 0, which is .P d� � P d�/j0 D P d�. On the other hand, the intersectionof .P d� � P d�/j0 and E .d˚d/ (seen as a subbdundle of Pd) is .P �d � d/j0 D d, where d isinterpreted as the space of constant connections. The upshot is that

prg.�/ �T �h � �Hol .�; �/ 2 .D � .d˚ d//je

for any � 2 d. In particular, for 2 g one has

�T �h �Hol . ; / �q 2 .D=G � d/jŒe�:

This establishes the containment (4.35) and the proof is therefore complete.

Theorem 4.A.1 is significant in that it shows that if X is a Hamiltonian space for .D=G �d; E.g//, in other words a quasi-Poisson g-space with a moment, say J , valued in D=G, then


X jJ�1.V / is a g-Hamiltonian space in the classical sense. Thus, if g admits an AdG-invariantLagrangian complement h � d, then the theory of quasi-Poisson g-space with D=G-valuedmoment maps becomes the classical moment map theory near the coset of the group identity.It should be noted however that their respective notions of the fusion product are different.

The morphism eHol is exact since it is the composition of exact morphisms. Consider thesubbundle E.h/ of D=G � d. Its intersection with the kernel of the anchor of D=G � d at Œe�,which is Œe� � g D g, is trivial. Thus, making U (and V ) smaller if need be, the subbundleE.h/jV is a Lagrangian complement of the kernel of the anchor ofD=G�d and thus defines anisotropic splitting jD=G W T .D=G/jV ! .D=G � d/jV . Let �D=G 2 �3.V / be its correspond-ing 3-form. Now the quotient morphism q present in the diagram (4.34) restricts to a Courantmorphism .D � .d ˚ d//jexp.U /G Ü .D=G � d/jV where exp.U /G is the flow out by thefeg�G-action of exp.U / � D. Let j 0D W TDjexp.U /G ! .D� .d˚d//jexp.U /G be the isotropicsplitting induced by jD=G as per Proposition 2.3.2, let �0D 2 �

3.exp.U /G/3 be its correspond-ing curvature tensor and let $ be the 2-form corresponding to the change of isotropic splittingTD�0D

' TD�D , where �D is the Cartan 3-form defined in (3.5). The calculations carried outto obtain the expressions (4.11) and (4.12), substituting exp.U / for G, are still valid and thus

j 0D.v/ D .�Adexp.�/ prh �exp.U /G;Adg�1 prg �

exp.U /G/;

and

(4.36) $ D1

2h� exp.U /G; prg �

exp.U /Gi;

where �v� exp.U /G D Lexp.�/�1Rg�1v for v 2 Texp.�/g and � 2 U . The 2-form $ can also beinterpreted as the 2-form corresponding to quotient morphism q W .D � .d˚ d//jexp.U /G Ü.D=G � d/jV .

Finally, let !h 2 �2.U / be the 2-form such that eHol D T!h

. Putting (4.36) and (4.32)together gives

(4.37) !h D ��h.hol�$ C !/:

Theorem 4.A.1 may be recast as follows.

Theorem 4.A.2. Let !h 2 �2.U / defined by be the 2-form defined by (4.37). Then

1. d!h D � exp�h �D=G ,

2. !h is G-invariant4,

4Here G-invariance is considered only insofar the G-action on h ' g� preserves V .


3. and �ah. /! D �d h ; �i C exp�h j�D=G

for 2 g.

Proof. Since eHol is equal to T .exph/!hunder the identifications afforded by the splitting jh

and jD=G , the first property follows. As jh and jD=G are G-invariant and eHol is G-equivariant,the second property follows as well. Finally, by Theorem 4.A.1 one has �eHol which thenimplies

aD=G. / D .exph/�ah. /; j�h . /C �ah. /! D exp�h j

�D=G :

Since j �h . / D d h ; �i, the claim follows.

Chapter 5

Moduli space examples

Moduli spaces of flat principal bundles over surfaces carrying certain decorations provide arich class of examples of L-Hamiltonian spaces. The coloured surfaces introduced belowgeneralize those of Severa [54] and it is indeed with an eye towards his work that this chapterproceeds. In particular, a class of examples of D=G-valued moment maps is provided.

5.1 Moduli spaces of flat D-bundles on coloured surfaces

5.1.1 Coloured surfaces

Let † be a compact oriented 2-manifold with corners. It is assumed that † has at least oneboundary component and that each boundary component has at least one corner (the case ofa single corner corresponds to a boundary circle with a base point). Let d be a quadratic Liealgebra integrating to a connected group D. A boundary segment a of † may be decoratedwith a Lagrangian subalgebra ga � d integrating to a closed and connected subgroupGa � D,in which case a is called coloured, or left undecorated in which case a is called free. Let V andE denote respectively the set of vertices (corners) and the set of boundary segments of †. Thesubsets of coloured and free boundary segments will be denoted by Ecol; Efree � E respectively.It is required that there be at least one free boundary segment and that no two free boundarysegments be adjacent1. With these decorations, the surface † is called a coloured surface.

1The second requirement is to simplify the characterization of the subalgebra l.†/ given below; it can be liftedat the cost of somewhat complicating the exposition.

92

CHAPTER 5. MODULI SPACE EXAMPLES 93

†

Figure 5.1: A coloured surface with 1-sided, 2-sided and 3-sided boundary components. Free boundarysegments are indicated by dotted lines. Colours correspond to Lagrangian subalgebras of d.

Consider the restricted fundamental groupoid of †

…1.†;V/� V

consisting of all homotopy classes of paths with end points in V . Choose an orientation on† so that each boundary segment is now directed. One will be interested in the space ofhomomorphisms of groupoids

M† WD fF 2 Hom.…1.†;V/;D/ W F.Œa�/ 2 Ga for a 2 Ecolg:

There is an action of Map.V;D/ on Hom.…1.†;V/;D/ given by

(5.1) .d:�/.Œp�/ D dp.0/�.Œp�/d�1p.1/

for � 2 Hom.…1.†;V/;D/ and Œp� 2 …1.†;V/. The acting group is restricted so that thisaction preserves M†. Namely let

Mapcol.V;D/ � Map.V;D/

be the subgroup defined by the following rules on its elements f 2 Mapcol.V;D/:

Rules 5.1.1. For v 2 V , let a; b 2 E be the (possibly identical) boundary segments incident tov.

1. If a 2 Ecol and b 2 Efree then f .v/ 2 Ga;

2. if a; b 2 Ecol then f .v/ 2 Lie.ga \ gb/;

3. if a; b 2 Efree then f .v/ is unconstrained.


Let Mapcol;0.V;D/ be the subgroup of Mapcol.V;D/ of elements f 2 Mapcol.V;D/ such thatf .v/ D e whenever v is incident to a free boundary segment. Consider the quotient

(5.2) M† WDM†=Mapcol;0.V;D/:

The quotient M† is interpreted as the space of flat D-bundles over † (determined by theirholonomies) up to a suitable notion of gauge transformation. It inherits an action by the group

L.†/ WDMapcol.V;D/

Mapcol;0.V;D/

which can also be seen as the subgroup of elements f 2 Mapcol.V;D/ such that f .v/ D e

whenever v is not incident to a free boundary segment. Let

(5.3) J WM† ! Map.Efree;D/

be the pullback map. There is a natural action of L.†/ on Efree and J is Mapcol.V;D/-equivariant. Consider the action Courant algebroid

(5.4) A D Map.Efree;D � .d˚ d//

defined analogously to .D � .d˚ d//r where r D jEfreej. The group L.†/ may be regarded asthe closed and connected subgroup of Map.Efree;D�D/ integrating the Lagrangian subalgebra

(5.5) l.†/ � Map.Efree; d˚ d/

defined by the following rules on its elements � 2 l.†/:

Rules 5.1.2. For a 2 Efree, let b; c 2 E be the (possibly identical) boundary segments adjacentto a in the order dictated by the orientation of † (i.e. the boundary segments appear in theorder b � a � c ) .

1. If b; c 2 Ecol then �.a/ 2 .gb; gc/,

2. if b D c so that a is a boundary circle, then �.a/ 2 d�.

The central result of this chapter can now be stated. Its proof is discussed peacemeal in whatfollows it.

Theorem 5.1.1. The space M† possesses a canonical smooth structure and is canonically a

L.†/-Hamiltonian space for the Courant algebroid A.


Smooth structure of M†. To endow M† with a smooth structure, start by cutting the surface† until a glueing polygon P is obtained. Specifically choose a vertex v incident to a freeboundary segment in one of the boundary polygons of†. Then, capping off all boundary com-ponents of †, introduce loops at v corresponding to a fundamental polygon for the resultingclosed orientable surface in the usual way, i.e. with loops going around and loops going along“handles”. Next, for each boundary polygon other the one on which v lies choose a vertex andjoin it to v via a path. This is done in such a way that no two loops or paths intersect otherthan at v. The surface † is then cut along the paths and loops introduced in this way to give apolygon P .

P†

A

A

B

B

C

D

C

D

Figure 5.2: The coloured surface † is cut into a glueing polygon P .

By convention, it will be assumed that the orientation on† corresponds to the CCW orientationof P . Denote the set of edges of P by OE . Its elements will be listed as, in CCW order,

(5.6) a0; : : : ; aj OEj�1

where a0 is by convention a free boundary segment (now an edge of P ) of† The free boundarysegments of † will be listed as af0 D a0; : : : ; afr�1 in the order in which they appear in (5.6).

The edges of P , together with the identifications corresponding to the cutting of†, may beregarded as a set of generators for M† and consequently there is an inclusion

M† ,! Map. OE ;D/:

As such, the space M† inherits a smooth structure as an embedded submanifold. To show


the canonicity of this smooth structure, suppose P 0 is another polygon obtained by cutting †and denote its edges by OE 0. Then the respective embeddings of M† into Map. OE ;D/ and intoMap. OE 0;D/ both lift to the same embedding

M† ,! Map. OE [ OE 0;D/

end must therefore induce the same smooth structure. Since M† is the quotient of M† by afree and proper Lie group action, it follows that M† possesses a canonical smooth structure aswell.

Hamiltonian space structure. Having cut† into a polygon P , it will be shown how to endow† with the structure of a L.†/-Hamiltonian space for A. Consider the quadratic Lie algebra

(5.7) Map. OE n fa0g; d˚ d/ �Map.Efree n fa0g; d˚ d/:

Define a Lagrangian subalgebra l.P / of (5.7) according to the following rules on its elements.�; / 2 l.P /, where � 2 Map. OE n fa0g; d˚ d/ and 2 Map.Efree n fa0g; d˚ d/:

Rules 5.1.3.

1. If a 2 Efree then2 �.a/ D .a/�1;

2. if a; a0 2 OE are edges corresponding (as a pair) to a cut introduced on † then �.a/ D�.a0/�1;

3. if a 2 Ecol then �.a/ 2 .ga; ga/.

Consider now the Courant algebroid

(5.8) OA D Map. OE n fa0g;D � .d˚ d// �Map.Efree n fa0g;D � .d˚ d//:

By way of its definition, the orbit O.P / of l.P / through the group identity of the group

(5.9) Map. OE n fa0g;D/ �Map.Efree n fa0g;D/

can be identified withM†; under this identification, regarding the edges of P as generators forM†, the inclusion of M† into the group (5.9) becomes the map

(5.10) J W F 7! F.Œa1��1/; F .Œa2�

�1/; : : : ; F .Œaj OEj�1�

�1/; F .Œaf0�/; : : : ; F .Œafr�1�/:

for F 2M†. Let L.P / denote the closed subgroup of the group (5.9) integrating l.P /. As per2Recall that in the pair groupoid d˚ d� d, inversion consists in substituting .ˇ; ˛/ for .˛; ˇ/.


Proposition 3.2.1, there is a 2-form ! 2 �2.M†/ for which .M†; J; !/ is aL.†/-Hamiltonianspace for OA.

Proposition 5.1.1. The two form ! 2 �2.M†/ is trivial.

Proof. The Lagrangian subalgebra l.P / is a product of Lagrangian subalgebras of d˚ d of theform ga ˚ ga and a number of copies of the Lagrangian subalgebra

d�.1;4/ C d�.2;3/ D f.˛; ˇ; ˇ; ˛/ W ˛; ˇ 2 dg:

of .d ˚ d/2. The l.P /-orbit O.P / is therefore the product of the orbits through the groupidentity ofD (orD2) of those various Lagrangian subalgebras of d˚d (resp. d˚d˚d˚d) and!is the product of the 2-forms thereof. As the subalgebras ga˚ga are multiplicative, the 2-formsthey contribute are trivial according to Corollary 3.2.1. It must thus only be shown that the 2-form associated to the orbit through the group identity e 2 D2 of the Lagrangian subalgebrad�.1;4/ C d�.2;3/ is trivial. This is a matter of direct computation; it suffices to compute this2-form at the group identity and for elements .˛; ˇ; ˇ; ˛/; .˛0; ˇ0; ˇ0; ˛0/ of .d˚ d/2 one has

hj �D2.˛; ˇ; ˇ; ˛/; aD2.˛0; ˇ0; ˇ0; ˛0/i D h.˛; ˇ; ˇ; ˛/; jD2 aD2.˛0; ˇ0; ˇ0; ˛0/i

D1

2h.˛; ˇ; ˇ; ˛/; .˛0 � ˇ0; ˇ0 � ˛0; ˇ0 � ˛0; ˛0 � ˇ0/i

D 0;

which completes the proof.

Take fusion product of M† is taken along the factors

Map.aj OEj�1;D � .d˚ d//;Map.a

j OEj�2;D � .d˚ d//; : : : ;Map.a1;D � .d˚ d//;

in that order. Let C be the coisotropic subalgebra of (5.7) defined by the condition3 on itselements .�;�/ 2 C .

s.prMap.ai ;d˚d/ �/ D t.prMap.aiC1;d˚d/ �/ for i D 1; : : : ; j OE j � 2:

At the level of the Courant algebroids TM† and OA, the fusion product just described amountsto the coisotropic reductions by %.C?/? and C respectively, where % W l.P /! X.M†/ is thel.P /-action. In fact, the reduction of TM† is precisely TM† and the reduction of OA is the

3Here s and t are the source and target maps in the pair groupoid d˚ d� d, respectively.


Courant algebroid A. Since one has

F.Œa0�/ D F.Œaj OEj�1��1/ � F.Œa

j OEj�2��1/ � � �F.Œa1�/

�1

for F 2M†, the reduced moment map of M† is the map J defined earlier. On the other handthe reduced Lagrangian subalgebra

l.P / \ C

l.P / \ C?

coincides with the Lagrangian subalgebra l.†/. Denoting the 2-form reduced from ! by ! 2

�2.M†/, the upshot is that .M†;J ;!/ is a L.†/-Hamiltonian space for A. According toTheorem 3.2.4 this 2-form is characterized by

(5.11) .F.Œa0�/; ��!/ D .F.Œa

j OEj�1��1/; 0/ � .F.Œa

j OEj�2��1/; 0/ � � � .F.Œa1�

�1/; 0/;

where � WM† !M† is the quotient map.

Canonicity of the 2-form. Next, it is argued that the 2-form ! 2 �2.M†/ does not depend onthe cutting of †. The case where every boundary component is a boundary circle correspondsto that considered by M. Atiyah [7] in the spirit of his famous paper with R. Bott on the Yang-Mills equations on Riemann surfaces [8]; see also [4]. The more general case considered herecan be reduced to the preceding one; this is now explained.

To simplify matters, it is assumed that every boundary segment of † is free. Although thiscontradicts the requirement that no two free segments be adjacent, there is actually no harmin doing so as far as the foregoing constructions are concerned. The Lie algebra l.†/ maythen be understood as follows. For any boundary component, suppose ai0; ai1; : : : ; ail�1 are itssegments, listed in CCW order. Then for � 2 l.†/,

t.prMap.aij ;d˚d/�/ D s.prMap.aijC1 ;d˚d/

�/ for j D 0; : : : ; l � 1;

where the subscript j is interpreted modulo l . One has M† D Hom.…1.†;V/;D/. If † isan arbitrary coloured surface on the other hand, it may be substituted by a surface †0 whoseboundary segments are all free and as such M† � M†0 . A cutting of † is also a cutting of†0 and in view of part (a) of Lemma 3.2.2, the formula (5.11) for † is the “pullback” of itscounterpart for †0, meaning it suffices to deal with †0.

Let Q† be the surface obtained by contracting all but one boundary segment in each boundarycomponent of † so that Q† is a surface where all boundary components are boundary circleswith the same genus and number of boundary components as †. Denote by B � E the set of


boundary segments of † chosen for contraction. There is a natural embedding of

(5.12) � WM Q† ,!M†

whereby the segments in B map identically to the group identity. In terms of (5.12), one has

(5.13) M† D Map.B;D/ �M Q†:

Now suppose P1 and P2 are two polygons obtained by cutting the surface † in possiblydifferent ways. Their respective edges a0 may be a priori different. However, it follows im-mediately from part (b) of Lemma 3.2.2 that the right-hand-side of (5.11) augmented with thefactor .F.Œa0�; 0/ is invariant under cyclic permutations of its factors, i.e. under admissiblerotations of the polygon P . As a consequence, one can assume that a0 is the same in eithercase. Subscripts will be added to the objects defined in the previous subsections in function ofwhether they correspond to P1 or to P2 when there is a need to distinguish them. Each edgeof P2 can be expressed as a concatenation of edges of P1 (and their inverses). According toTheorem 2.4.2, there is an exact Courant morphism R W OA! OA lifting the identity morphismAÜ A whose base map sends Map.a;D/, where a 2 OE2 n fa0g, to its expression in terms ofthe edges in of P1 in OE2 n fa0g. The upshot is a commutative diagram

TM†OA

TM†OA

TM† A

TM† A

TJ2

RTJ1

TJ!2

TJ!1

;

where the vertical arrows are reduction morphisms. The composition of R and TJ2 is a Cour-ant morphism

TJ1;! W TM†Ü OA

for some 2-form ! 2 �2.M†/. To complete the argument, it must be argued that ! is trivial.By the very fact that TJ1 is a Dirac morphism .TM†; TM†/Ü . OA; E.l.P1/// one can con-clude that

E.l.P1// ı TJ1;! D Gr.�!/:


An element of E.l.P2// where all the entries corresponding to edges in OE2 n E (i.e. cuts) are setto 0 is also an element of E.l.P1// where all the entries corresponding to cuts are set to 0. Infact, such elements must beR-related to themselves. In particular, in terms of the identification(5.13), one has �v! D 0 whenever v 2 T Map.B;D/.

Finally, a cutting of† can be interpreted as a cutting of Q† in the obvious way. Let QPi denotethe polygon for Q† under the cutting of † that gives the polygon Pi ; the polygon QPi is obtainedfrom Pi by contracting some of its edges. Provided a0 … B , the inclusion O. QPi/ ,! O.Pi/ isthe inclusion (5.12) after the identifications

M† ' O.Pi/; M Q† ' O. QPi/:

As a consequence, the pullback of ! to M Q† is precisely its counterpart for the two cuttingsof † interpreted as cuttings for Q†. Since Q† is a surface whose boundary components are allboundary circles, this last 2-form is known to be trivial. The upshot is that ! is indeed trivialand the proof that the 2-form ! 2 �2.M†/ is independent of the cutting of† is thus complete.

5.1.2 Fusion of coloured surfaces

Let†1 and†2 be two coloured surfacers each having at least one boundary circle. Define theirfusion product †1 ~†2 to be the coloured surface obtained by joining the two surfaces alongboundary circles by a “pair of pants”, i.e. a 3-holed sphere (Figure 5.3).

Let Pi be glueing polygons for the coloured surfaces †i . Denote the edges of P1 bya0; : : : ; am1 and those of P2 by b0; : : : ; bm2 , with the usual convention: a0 and b0 are freeboundary segments and the edges are listed in CCW order. Form a new polygon P1 ~ P2by joining a0 and b0 along a triangle (Figure 5.4). Then P1 ~ P2 is a polygon for †1 ~ †2corresponding to a certain cutting of the latter.

Let !1 2 �2.M†1/ and !2 2 �

2.M†2/ be the 2-forms constructed above. Then as aconsequence of formula (5.11) one has

Theorem 5.1.2. The L.†1 ~ †2/-Hamiltonian space M†1~†2 is the fusion product M†1 ~M†2 along the respective first factors of Map.E1;free;D�.d˚d// and Map.E2;free;D�.d˚d//.

Remark 5.1.1. Since any coloured surface can be obtained by joining via pairs of pants numberof 1-holed tori equal to the genus of the surface and cylinders each with one boundary circleand one boundary having possibly multiple segments, the computation of the 2-form ! reducesto computing the corresponding 2-forms for a number of “building blocks”, analogously to [4,Thm. 9.3].


†1 †2

†1 ~†2

Figure 5.3: Illustration of the fusion of two coloured surfaces.


P1 P2

P1 ~ P2

A

A

B

C

B

C A

A B

C

B

C

A

A

B

A

BC

D

D

E

F

E

F

Figure 5.4: The polygon P1 ~ P2.


5.1.3 Duality

The dual of the L.†/-Hamiltonian structure on M .†/ corresponds to the Hamiltonian struc-ture obtained via the opposite orientation of†. Indeed, according to (5.11) one has forF 2M†

.F.Œa�10 �/;��!/ D .F.Œa1�/; 0/ � .F.Œa2�; 0/; // � � � .F.Œaj OEj�1�/; 0/:

This is formula (5.11) for the glueing polygon P oriented CW; equivalently it is formula (5.11)for the horizontal reflection of P , which is a glueing polygon for † with the opposite orienta-tion.

5.1.4 Symplectic structure

Let B ,! Map.Efree;D/ be an L.†/-orbit. Then, if J is transversal to B, the quotient spaceJ�1.B/=L.†/ inherits a symplectic form according to Theorem 3.3.1. Define

M† WDM†=L.†/:

The quotient M† is interpreted as the space of flat D-bundles with prescribed holonomiesalong the free boundary segments, again up to a suitable notion of gauge transformation.

The connected components of the preimages J�1.Bi/, where i indexes the family of L-orbits in Map.Efree;D/, foliate M†. The quotient map �=L.†/ W M† !M† sends the mem-bers of this foliation bijectively onto the connected components of M†. Thus each connectedcomponent of the orbifold M† carries a symplectic form.

5.2 Severa’s coloured surfaces

The surfaces Severa considers correspond to the coloured surfaces discussed above with theexception that there are no free boundary segment [54] and if a and b are adjacent edgesthen the Lagrangian subalgebras decorating them must intersect trivial. Such a surfaces canbe brought into the fold of the coloured surfaces considered here by removing a disc therebycreating a free boundary segment. The resulting surface will be denoted by †0. The modulispace of interest will then correspond to that of coloured flat connections whose holonomyalong the free boundary segment (i.e. the boundary of the disc removed) is trivial.


† †0

Figure 5.5: As before, colours correspond to Lagrangian subalgebras of d.

Since the only free boundary segment is a boundary circle, the space M 0† is a quasi-Hamiltonian

space forD�.d˚d/ according to Theorem 5.1.1. In fact since adjacent coloured edges are dec-orated with transverse Lagrangian subalgebras, it follows that M†0 D M†0 . One is interestedin the symplectic reduction

M† WD J�1.e/=D�:

Now cut the coloured surface †0 into a glueing polygon P 0 as before (Figure 5.6). Call itsedges a0; : : : ; am�1 with the same convention as before, i.e. a0 is the free boundary circle of†0 and the edges are listed in CCW order. In terms of the identification

O.P 0/ 'M†0 DM†0;

the space M† admits a section

(5.14) M† ,! J�1.e/ � O.P 0/ 'M†0

whereby an element of M† is sent to the element of O.P / ' M†0 whose holonomy alongthe free boundary segment as well as the holonomies along its two adjacent edges are trivial.The symplectic form !symp 2M† descends from the pullback of the 2-form ! 2 �2.M†0/ toJ�1.e/ according to the proof of Theorem 3.3.1. In terms of the section (5.14), the symplecticform !symp is the pullback of ! to M† ,! J�1.e/. According to part (a) of Lemma 3.2.2 andthe equation (5.11), one has

(5.15) .e; !symp/ D((((((.F.Œa1�/; 0/.F.Œa2�/; 0/ � � � .F.Œam�2�/; 0/((((

((((.F.Œam�1�/; 0/:

In this way [54, Thm. 3.1] is rederived.


P†0

A

A

B

B

Figure 5.6: Cutting the coloured surface †0 to obtain a coloured polygon P .

5.3 D=G-valued moment maps examples

Let g � d be a Lagrangian subalgebra integrating to the closed and connected subgroup G �D. Consider a coloured surface † whose coloured boundary segments are decorated with g

exclusively and which does not possess a boundary circle (i.e. each free segment is adjacent toa coloured one). Since the boundary segments adjacent to free segments are decorated with g,the Lagrangian subalgebra l.†/ � Map.Efree; d˚ d/ is

l.†/ D Map.Efree; g˚ g/:

In particular, the space M† is aG�G-Hamiltonian space (or power thereof) and thus provides,after quotienting by the e�G-actions, an example of a Hamiltonian space for .D=G�d; E.g//(or power thereof). Any such surface may be obtained by joining a number equal to the genusof † of 1-holed tori and as many cylinders as † has boundary components.

Figure 5.7: Surface where all free boundary segments are adjacent to coloured segments, which aredecorated with the same Lagrangian subalgebra.

Suppose some given boundary component of † has m segments. The decorations present


on the boundary can be encoded by a string of length m

f.s0; s1; : : : ; sm�1/ 2 f0; 1g�W .si ; siC1/ ¤ .1; 1/ for i D 0; : : : ; m � 1g;

where the subscripts are interpreted modulo m. Entries with 0 correspond to segments decor-ated by g and entries with 1 to free segments. In general, the notation

.†rg ; s1; s2; : : : ; sr/

will be used to denote the coloured surface of genus g and r boundary components with dec-orations corresponding to the maps s1; : : : ; sr . Denote by D.D/ and Ds.D/ respectively thespace (5.2) for the 1-holed torus and the cylinder with a free boundary circle and the otherboundary component decorated in accordance to the map s (cf. Example 3.2.4). With thisnotation, the foregoing discussion amounts to the following corollary of Theorems 5.1.1 and5.1.2:

Theorem 5.3.1. Let .†rg ; s1; : : : ; sr/ be a coloured surface of the kind considered above. Then

the space (5.2) for this surface is a Map.Efree; G �G/-Hamiltonian space and as such is equal

to the fusion product

D.D/~ � � �~D.D/„ ƒ‚ …g times

~Ds1.D/~Ds2.D/~ � � �~Dsr .D/;

where the fusion product is always taken along factors corresponding to boundary circles.


Figure 5.8: Joining 1-holed tori and cylinders to obtain .†22; 01; 0101/.

Bibliography

[1] A. Alekseev, H. Bursztyn, and E. Meinrenken. Pure spinors on Lie groups. Asterisque,(327):131–199 (2010), 2009.

[2] A. Alekseev and Y. Kosmann-Schwarzbach. Manin pairs and moment maps. J. Differen-

tial Geom., 56(1):133–165, 2000.

[3] A. Alekseev, Y. Kosmann-Schwarzbach, and E. Meinrenken. Quasi-Poisson manifolds.Canad. J. Math., 54(1):3–29, 2002.

[4] A. Alekseev, A. Malkin, and E. Meinrenken. Lie group valued moment maps. J. Differ-

ential Geom., 48(3):445–495, 1998.

[5] A. Alekseev and E. Meinrenken. The non-commutative Weil algebra. Invent. Math.,139(1):135–172, 2000.

[6] A. Alekseev, E. Meinrenken, and C. Woodward. Duistermaat-Heckman measures andmoduli spaces of flat bundles over surfaces. Geom. Funct. Anal., 12(1):1–31, 2002.

[7] M. Atiyah. The geometry and physics of knots. Cambridge University Press, CambridgeNew York, 1990.

[8] M. F. Atiyah and R. Bott. The Yang-Mills equations over Riemann surfaces. Philos.

Trans. Roy. Soc. London Ser. A, 308(1505):523–615, 1983.

[9] M. Bordemann. Nondegenerate invariant bilinear forms on nonassociative algebras. Acta

Math. Univ. Comenian. (N.S.), 66(2):151–201, 1997.

[10] N. Bourbaki. Varietes differentielles et analytiques : fascicule de resultats. Springer,Berlin New York, 2007.

[11] W. Browder. Torsion in H -spaces. Ann. of Math. (2), 74:24–51, 1961.

[12] H. Bursztyn, G. R. Cavalcanti, and M. Gualtieri. Reduction of Courant algebroids andgeneralized complex structures. Adv. Math., 211(2):726–765, 2007.

108

BIBLIOGRAPHY 109

[13] H. Bursztyn and M. Crainic. Dirac structures, momentum maps, and quasi-Poisson man-ifolds. In The breadth of symplectic and Poisson geometry, volume 232 of Progr. Math.,pages 1–40. Birkhauser Boston, Boston, MA, 2005.

[14] H. Bursztyn and M. Crainic. Dirac geometry, quasi-Poisson actions and D=G-valuedmoment maps. J. Differential Geom., 82(3):501–566, 2009.

[15] H. Bursztyn, M. Crainic, and P. Severa. Quasi-Poisson structures as Dirac structures.In Travaux mathematiques. Fasc. XVI, Trav. Math., XVI, pages 41–52. Univ. Luxemb.,Luxembourg, 2005.

[16] H. Bursztyn, D. Iglesias Ponte, and P. Severa. Courant morphisms and moment maps.Math. Res. Lett., 16(2):215–232, 2009.

[17] A. Cabrera, M. Gualtieri, and E. Meinrenken. Dirac geometry of the holonomy fibration,2015.

[18] T. Courant and A. Weinstein. Beyond Poisson structures. In Action hamiltoniennes de

groupes. Troisieme theoreme de Lie (Lyon, 1986), volume 27 of Travaux en Cours, pages39–49. Hermann, Paris, 1988.

[19] T. Courant and A. Weinstein. Beyond Poisson structures. In Action hamiltoniennes de

groupes. Troisieme theoreme de Lie (Lyon, 1986), volume 27 of Travaux en Cours, pages39–49. Hermann, Paris, 1988.

[20] T. J. Courant. Dirac manifolds. Trans. Amer. Math. Soc., 319(2):631–661, 1990.

[21] W. A. de Graaf. Classification of 6-dimensional nilpotent Lie algebras over fields ofcharacteristic not 2. J. Algebra, 309(2):640–653, 2007.

[22] P. Delorme. Classification des triples de Manin pour les algebres de Lie reductives com-plexes. J. Algebra, 246(1):97–174, 2001. With an appendix by Guillaume Macey.

[23] S. K. Donaldson. Boundary value problems for Yang-Mills fields. J. Geom. Phys., 8(1-4):89–122, 1992.

[24] I. Dorfman. Dirac structures and integrability of nonlinear evolution equations. Nonlin-ear Science: Theory and Applications. John Wiley & Sons, Ltd., Chichester, 1993.

[25] V. G. Drinfeld. Hamiltonian structures on Lie groups, Lie bialgebras and the geometricmeaning of classical Yang-Baxter equations. Dokl. Akad. Nauk SSSR, 268(2):285–287,1983.

BIBLIOGRAPHY 110

[26] J. J. Duistermaat and J. A. C. Kolk. Lie groups. Springer, Berlin New York, 2000.

[27] H. Flaschka and T. Ratiu. A convexity theorem for Poisson actions of compact Lie groups.Ann. Sci. Ecole Norm. Sup. (4), 29(6):787–809, 1996.

[28] J. Grabowski and M. a. Rotkiewicz. Higher vector bundles and multi-graded symplecticmanifolds. J. Geom. Phys., 59(9):1285–1305, 2009.

[29] G. S. Hall. Symmetries and Curvature Structure in General Relativity. World Scientific,2004.

[30] L. Hormander. The analysis of linear partial differential operators. Springer, Berlin NewYork, 2003.

[31] D. Iglesias-Ponte, C. Laurent-Gengoux, and P. Xu. Universal lifting theorem and quasi-Poisson groupoids. J. Eur. Math. Soc. (JEMS), 14(3):681–731, 2012.

[32] A. Knapp. Lie groups beyond an introduction. Birkhauser, Boston, 2002.

[33] Y. Kosmann-Schwarzbach. Quasi-bigebres de Lie et groupes de Lie quasi-Poisson. C. R.

Acad. Sci. Paris Ser. I Math., 312(5):391–394, 1991.

[34] Y. Kosmann-Schwarzbach. Jacobian quasi-bialgebras and quasi-Poisson Lie groups. InMathematical aspects of classical field theory (Seattle, WA, 1991), volume 132 of Con-

temp. Math., pages 459–489. Amer. Math. Soc., Providence, RI, 1992.

[35] Y. Kosmann-Schwarzbach. Lie bialgebras, Poisson Lie groups and dressing transforma-tions. In Integrability of nonlinear systems (Pondicherry, 1996), volume 495 of Lecture

Notes in Phys., pages 104–170. Springer, Berlin, 1997.

[36] C. Laurent-Gengoux, M. Stienon, and P. Xu. Lectures on Poisson groupoids. In Lectures

on Poisson geometry, volume 17 of Geom. Topol. Monogr., pages 473–502. Geom. Topol.Publ., Coventry, 2011.

[37] D. Li-Bland and E. Meinrenken. Courant algebroids and Poisson geometry. Int. Math.

Res. Not. IMRN, (11):2106–2145, 2009.

[38] D. Li-Bland and E. Meinrenken. Dirac Lie groups. Asian J. Math., 18(5):779–815, 2014.

[39] D. Li-Bland and E. Meinrenken. Dirac Lie groups. Asian J. Math., 18(5):779–815, 2014.

[40] D. Li-Bland and P. Severa. Quasi-Hamiltonian groupoids and multiplicative Manin pairs.Int. Math. Res. Not. IMRN, (10):2295–2350, 2011.

BIBLIOGRAPHY 111

[41] Z.-J. Liu, A. Weinstein, and P. Xu. Manin triples for Lie bialgebroids. J. Differential

Geom., 45(3):547–574, 1997.

[42] Z.-J. Liu, A. Weinstein, and P. Xu. Dirac structures and Poisson homogeneous spaces.Comm. Math. Phys., 192(1):121–144, 1998.

[43] J.-H. Lu. Multiplicative and affine Poisson structures on Lie groups. ProQuest LLC, AnnArbor, MI, 1990. Thesis (Ph.D.)–University of California, Berkeley.

[44] J.-H. Lu. Momentum mappings and reduction of Poisson actions. In Symplectic geometry,

groupoids, and integrable systems (Berkeley, CA, 1989), volume 20 of Math. Sci. Res.

Inst. Publ., pages 209–226. Springer, New York, 1991.

[45] J.-H. Lu and A. Weinstein. Poisson Lie groups, dressing transformations, and Bruhatdecompositions. J. Differential Geom., 31(2):501–526, 1990.

[46] K. Mackenzie. General theory of lie groupoids and lie algebroids. Cambridge UniversityPress, Cambridge England New York, 2005.

[47] E. Meinrenken. Poisson geometry from a dirac perspective, 2016.

[48] G. P. Ovando. Lie algebras with ad-invariant metrics: A survey-guide. Rend. Semin. Mat.

Univ. Politec. Torino, 74(1):243–268, 2016.

[49] H. Samelson. Topology of Lie groups. Bull. Amer. Math. Soc., 58:2–37, 1952.

[50] M. A. Semenov-Tian-Shansky. Dressing transformations and Poisson group actions.Publ. Res. Inst. Math. Sci., 21(6):1237–1260, 1985.

[51] P. Severa. Letters to alan weinstein about courant algebroids, 2017.

[52] K. Uchino. Remarks on the definition of a Courant algebroid. Lett. Math. Phys.,60(2):171–175, 2002.

[53] P. Severa. Some title containing the words “homotopy” and “symplectic”, e.g. this one.In Travaux mathematiques. Fasc. XVI, volume 16 of Trav. Math., pages 121–137. Univ.Luxemb., Luxembourg, 2005.

[54] P. Severa. Moduli spaces of flat connections and Morita equivalence of quantum tori.Doc. Math., 17:607–625, 2012.

[55] P. Severa. Left and right centers in quasi-Poisson geometry of moduli spaces. Adv. Math.,279:263–290, 2015.

BIBLIOGRAPHY 112

[56] P. Severa and A. Weinstein. Poisson geometry with a 3-form background. Number 144,pages 145–154. 2001. Noncommutative geometry and string theory (Yokohama, 2001).

[57] J. Vysoky. Hitchhiker’s guide to Courant algebroid relations. J. Geom. Phys.,151:103635, 33, 2020.

[58] A. Weinstein. The local structure of Poisson manifolds. J. Differential Geom., 18(3):523–557, 1983.

[59] P. Xu and A. Alekseev. Derived brackets and courant algebroids. unpublished manuscript,2002.

[60] M. Zambon. A construction for coisotropic subalgebras of Lie bialgebras. J. Pure Appl.

Algebra, 215(4):411–419, 2011.

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by Selim Elias Tawﬁkblog.math.toronto.edu/GraduateBlog/files/2020/12/main.pdfSelim Elias Tawﬁk Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2020