Lie 2-Algebras as Homotopy Algebras Over a Quadratic Operad · 2012-11-03 · homotopy algebras over a quadratic operad. Algebras over an operad have a rich theory. 3 which includes

Lie 2-Algebras as Homotopy Algebras Over a QuadraticOperad

by

Travis Squires

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

University of Toronto

Copyright c� 2011 by Travis Squires

Abstract

Lie 2-Algebras as Homotopy Algebras Over a Quadratic Operad

Travis Squires

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2011

We begin by discussing motivation for our consideration of a structure called a Lie 2-

algebra, in particular an important class of Lie 2-algebras are the Courant Algebroids

introduced in 1990 by Courant. We wish to attach some natural definitions from operad

theory, mainly the notion of a module over an algebra, to Lie 2-algebras and hence to

Courant algebroids. To this end our goal is to show that Lie 2-algebras can be described

as what are called homotopy algebras over an operad. Describing Lie 2-algebras using

operads also solves the problem of showing that the equations defining a Lie 2-algebra

are consistent.

Our technical discussion begins by introducing some notions from operad theory,

which is a generalization of the theory of operations on a set and their compositions. We

define the idea of a quadratic operad and a homotopy algebra over a quadratic operad.

We then proceed to describe Lie 2-algebras as homotopy algebras over a given quadratic

operad using a theorem of Ginzburg and Kapranov.

Next we briefly discuss the structure of a braided monoidal category. Following this,

motivated by our discussion of braided monoidal categories, a new structure is introduced,

which we call a commutative 2-algebra. As with the Lie 2-algebra case we show how a

commutative 2-algebra can be seen as a homotopy algebra over a particular quadratic

operad.

Finally some technical results used in previous theorems are mentioned. In discussing

ii

these technical results we apply some ideas about distributive laws and Koszul operads.

iii

Contents

Introduction 1

1 k-Linear Operads 5

1.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Free Algebras and Algebras Over and Operad . . . . . . . . . . . . . . . 8

1.3 Koszul and Cobar Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Homotopy Algebras Over an Operad . . . . . . . . . . . . . . . . . . . . 13

2 Lie 2-Algebras 15

2.1 Definition of a Lie 2-algebra . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 2-Term EL∞-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Lie 2-Algebras as Homotopy Algebras Over a Quadratic Operad . . . . . 21

2.4 Examples of Lie 2-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Commutative 2-Algebras 39

3.1 Monoidal Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Commutative 2-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Commutative 2-Algebras as Homotopy Algebras Over a Quadratic Operad 43

4 Distributive Laws and the Koszulness of R 52

4.1 Distributive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 The Spectral Sequence of a Double Complex . . . . . . . . . . . . . . . . 58

iv

4.3 The Koszulness of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Conclusion 63

Bibliography 64

v

Introduction

The idea of a Courant algebroid was first introduced in 1990 as an algebraic structure that

is something like a Lie algebra however the equations on the bracket are relaxed (see [2]

where Courant algebroids are introduced as Dirac manifolds). Like the bracket of a Lie

algebra, the bracket belonging to a Courant algebroid satisfies the Jacobi identity however

the antisymmetry condition is relaxed and is required only to hold up to homotopy. To

be a little more precise this means that [x, y] + [y, x] = D(x, y) for some differential

D. The process of taking algebraic structures and relaxing certain equational laws in

this manner is typically called categorification. Various varieties of Lie 2-algebras will

be defined in this paper, each of them corresponding to the structure we get through

the categorification of a Lie algebra by relaxing some subset of the antisymmetry and

Jacobi equational laws. In this paper the main example of Lie 2-algebras are the Courant

algebroids described above (the fact that a Courant algebroid is a Lie 2-algebra was first

described in [10]). By showing that Lie 2-algebras have natural structures associated

with them we can conclude the same about well know objects like Courant algebroids.

In particular we’d like to obtain a theory of modules over a Courant algebroid.

The notion of a semistrict Lie 2-algebra was first introduced as an algebraic struc-

ture with a Lie bracket that was antisymmetric and satisfied the Jacobi identity up to

homotopy. This says that instead of insisting [x, [y, z]] = [[x, y], z] + [y, [x, z]] we instead

relax this condition and allow that [x, [y, z]] − [[x, y], z] − [y, [x, z]] = d�x, y, z� where d

is some differential and �−,−,−� is a bracket satisfying some coherence conditions. For

1

2

a treatment of semistrict Lie 2-algebras see [1]. In [9] Roytenberg continued this trend

of weakening the Lie algebra axioms by then relaxing the antisymmetry condition on

the Lie bracket. In this paper Roytenberg introduced a binary bracket �−,−� such that

[x, y] + [y, z] = d�x, y� with �−,−� satisfying some coherence conditions. Upon relaxing

the Jacobi identity in this way we get the notion of a Lie 2-algebra. The main goal of

this paper is to use operad theory to describe a Lie 2-algebra as a homotopy algebra over

an explicitly given quadratic operad. This will accomplish two things; it will allow us

to attach some natural structures from operad theory to Lie 2-algebras and it will show

that the equations defining a Lie 2-algebra are consistent.

In the first chapter we introduce some ideas from operad theory. Operads were

introduced by May in [8] as structures that incode different types of operations and their

compositions. In this discussion we quickly narrow in on the idea of a quadratic operad

and algebras over a quadratic operad. Quadratic operads are constructs which describe

generalized algebraic structures built up from binary operations and ternary relations.

Examples of such objects are the Com,Ass, and Lie (commutative, associative and Lie)

operads whose algebras are the usual commutative, associative and Lie algebras. For

a given quadratic operad we discuss the quadratic dual to that operad, which parallels

the usual notion of the quadratic dual to a classical quadratic algebra. Along with the

quadratic dual to an operad we have another notion of dualizing an operad given by

the cobar construction (again this parallels a similar idea for algebras). In general these

two objects are different however we call a quadratic operad Koszul if they are quasi-

isomorphic. It turns out that for Koszul operads one can describe homotopy algebras

over an operad in a reasonable way. A theorem of Ginzburg and Kapranov in [4] allows

us to encode the equations for a homotopy algebra in a certain dg-complex using the

fact that δ2 = 0 for some differential. We use this theorem to extract the equations

describing a Lie 2-algebra and hence show that these structures can be thought of as

homotopy algebras over a quadratic operad. Algebras over an operad have a rich theory

3

which includes modules over an algebra over an operad. In the future we hope to use this

fact to obtain a theory of modules over a Courant algebroid.

In chapter 2 we begin by discussing Lie 2-algebras in some detail. We then proceed to

describe a particular operad R that gives us the equations for a hemistrict Lie 2-algebra.

We would like to arrive at the equations for general Lie 2-algebras using the theory of

homotopy algebras. As noted above, when we relax the conditions on the Jacobi identity

we introduce a new bracket �−,−,−� however this bracket cannot be described using

a quadratic operad. We use the theorem of Ginzburg and Kapranov mentioned above

to show that this bracket and the coherence conditions it satisfies can be arrived at by

defining our differential ∂ as dual to �−,−,−� and realizing the consequences of ∂2 = 0.

The fact that these coherence conditions follow from ∂2 = 0 says that Lie 2-algebras

can be thought of as homotopy algebras over this operad R. This fact also immediately

implies that the equations describing a Lie 2-algebra are consistent.

Chapter 3 is structurally identical to chapter 2 except the story is now being told

for a new structure we call a commutative 2-algebra. We arrive at the relations for a

commutative 2-algebra by considering a braided monoidal category. We do not insist

that our monoidal category is symmetric however we do suppose that symmetry holds

up to homotopy, that is xy−yx = d[x, y] for some new bracket [−,−]. Next we follow the

philosophy of chapter 2 and suppose that the associative law for our monoidal product

only holds up to some triple bracket [−,−,−] that again satisfies some coherence condi-

tions. The structure given by these new brackets along with their coherence conditions

is what we define as a commutative 2-algebra. Following the exact same procedure as

that of chapter 2 we are able to describe commutative 2-algebras as homotopy algebras

over a quadratic operad.

We leave some of the technical details to chapter 4. Here the goal is to prove that the

operad R considered in chapter 2 is Koszul. This fact allows us to use our main theorem

from [4] and also ensures that the equations describing Lie 2-algebras and commutative

4

2-algebras are coherent in the sense of MacLane (see VII.2 in [5]). The operad R is

built up from two brackets [−,−] and �−,−�. Each of these brackets (and their given

relations) generates its own operad which are both known to be Koszul. We use what

is called a distributive law (defined in [7]) to combine these two operads to obtain our

original operad R. It is the use of this process that allows us to conclude that R is

Koszul.

Chapter 1

k-Linear Operads

In this chapter we introduce the tools we need for the remainder of the paper, namely

the theory of quadratic operads and homotopy algebras over an operad.

An operad is an object that allows us to encode information about operations and

their compositions. We will typically denote operads with uppercase script: P ,Q,R,

...etc. A given operad P gives us a collection of operations and tells us how to compose

these operations to get other operations of higher order. For each n ≥ 1, P(n) is a set

of operations thought of as having arity n. Given operations of arities n1, n2, ..., nk and

another operation of arity k we should be able to construct a new operation by placing

the outputs of the k operations of arities n1, ..., nk into the inputs of the operations of

arity k, giving us an operation of arity n1 + n2 + ... + nk. This is precisely what the

operad structure gives us: a map P(k)×P(n1)×P(n2)× . . .P(nk) ��P(n1 + . . . + nk).

Using operads, for example, we can describe the usual commutative, associative, and Lie

algebras. These algebras are of a special type for two reasons: the first is that they are all

defined as having underlying vector spaces and second they are all generated by binary

operations. Operads of this special type are called quadratic operads and are discussed

in detail in this chapter.

We begin with some definitions and examples and then state the main theorem of

5

Chapter 1. k-Linear Operads 6

this chapter, which allows us to describe homotopy algebras over an operad in a useful

way.

1.1 Definitions and examples

Let k be a field of characteristic 0 (we will be using k=R or C).

Definition 1.1.1. A k-linear operad P consists of a collection of k-vector spaces {P(n) :

n ∈ N} together with the following data

• For each n ∈ N an action Σn × P(n) �� P(n)

• Linear maps called compositions

γm1,...,ml: P(l)⊗ P(m1)⊗ · · ·⊗ P(ml) �� P(m1 + · · ·+ ml)

for all m1, ...,ml ≥ 1. We will write µ(ν1, ..., νl) instead of γm1,...,ml(µ⊗ν1⊗ · · ·⊗νl)

• An element 1 ∈ P(1), called the unit, such that µ(1, ..., 1) = µ for any l ∈ N and

any µ ∈ P(l)

these composition maps must also be associative and equivariant with respect to the

group action.

Given an operad P we think of an element of P(n) as an operation with n inputs,

we sometimes refer to elements of P(n) as n-ary operations. Notice that the space

K = P(1) has the natural structure of a k-algebra with unit given by the k-vector space

structure on K and the multiplication γ1,1 : K ⊗ K �� K. In what follows we will

assume that K = k, the underlying field. Given an element c ∈ k and µ ∈ P(l) we have

that µ(c, 1, ..., 1) = µ(1, c, 1, ..., 1) = · · · = µ(1, ..., c) = cµ(1, ..., 1) which says that the

operations we consider do act like multilinear operations. We think of an element of the

symmetric group as acting on the operations by taking a given operation and producing


a new operation obtained by permuting the order of the inputs. For example take an

operation µ ∈ P(3) and suppose upon evaluation at x⊗ y ⊗ z we write µ(x, y, z). Then

the new operation ((123) ∗ µ) would have the value µ(z, x, y) at x⊗ y⊗ z. The action of

the symmetric group allows us to express all of the familiar algebraic relations.

Example 1.1.2. The operad Lie is defined by Lie(�)=k and Lie(�) = �[−,−]�, the

1-dimensional k-vector space generated by [−,−], with the action of S2 given by (12) ∗

[−,−] = −[−,−]. The space Lie(3) is generated by the symbols [−, [−,−]], [[−,−],−],

the action induced by that of Lie(�) and the composition maps subject to the Jacobi

relation

[[−,−],−] = [−, [−,−]] + (12) ∗ [−, [−,−]]

The vector spaces Lie(n) for n > 3 are all generated by Lie(�) and Lie(�) in the obvious

way. To be precise for all n ≥ 1 the vector space Lie(n) is the subspace of the free

Lie algebra on x1, . . . , xn formed by elements of multidegree (1, . . . , 1) 1. Because of the

Jacobi identity these monomials are not all linearly independent. In fact dim(Lie(n)) =

(n− �)!.

Example 1.1.3. In a similar way we define the commutative and associative operads

Com and Ass as the operads describing one binary operation which is commutative

and associative respectively. For each n we can identify Com(n) = k and Ass(n) as the

subspace of polynomials with noncommuting variables x1, .., xn containing the monomials

of multidegree (1,...,1).

Notation: We denote a permutation of the inputs of an operation (the result of

a permutation acting on a given element) by a subscript, the absence of any subscript

indicating no permutation of inputs. For example the Jacobi identity in example 1.1.2

will be denoted

[[−,−],−] = [−, [−,−]] + [−, [−,−]]1,3

1This is because an operad cannot express a relation that involves duplicating an element, e.g. (aa)b =(ab)a.


We end this section by giving another important example of an operad generated

(this notion will be made more precise later) by two binary operations.

Example 1.1.4. The poisson operad, which we denote Poiss, is generated by two ele-

ments µ, ν ∈ Poiss(�) where µ is associative, ν is a Lie bracket (it is antisymmetric and

satisfies the Jacobi identity) and these two operations satisfy the identity

ν(1, µ) = µ(ν, 1) + µ(1, ν)1,2

We can express this in a more familiar way by denoting the operations µ and ν by · and

�−,−� respectively. Then in the obvious way the following equation

�x, y · z� = �x, y� · z + y · �x, z�

expresses the same relationship.

1.2 Free Algebras and Algebras Over and Operad

We first note that for any n the vector space P(n) is a left K-module (recall that K =

P(1)) and a right K⊗n-module given by γ1, n and γ1,...,1 respectively. One can think

of the left K-module action as multiplying the output of a given operation by a fixed

scalar from K and the right K⊗n-module action as multiplying each of the n inputs of

an operation by a different scalar from K. For α, α1, . . . ,αn ∈ K and µ ∈ P(n) the left

and right actions are given by

α · µ = γ1,n(α⊗ µ)

and

µ · (α1 ⊗ α2 ⊗ . . .⊗ αn) = γ1,...,1(µ⊗ α1 ⊗ . . .⊗ αn)

repsectively.

We now define the notion of an algebra over an operad.


Definition 1.2.1. Let P be a k-linear operad. A P-algebra is a k-vector space A and

for each n an Sn-equivariant map

P(n) �� Hom(A⊗n, A)

that take the composition maps into actual compositions of operations on A.

Example 1.2.2. Algebras over Lie, Com and Ass are the usual Lie, commutative, and

associative algebras.

We could generalize the definitions above by replacing vector spaces with differential

graded vector spaces. In such a case we get the notion of a dg-operad and a dg-algebra over

a dg-operad. The composition maps given with an operad represent a single composition

of operations. We can compose these composition maps in many ways to get iterated

operations. For instance all the operations in Com and Ass are iterations of a single

binary operations. These iterated operations are sometimes best thought of as trees.

Definition 1.2.3. Given a directed tree T = (V, E) with source and target maps s, t :

E �� V ∪ {v0} an external edge is an edge e ∈ E with either no source or no target

(an edge of degree 1), any other edge is an internal edge.1 An external edge e such that

s(e) = v0 will be called an input and an external edge such that t(e) = v0 will be called

an output. For each vertex v ∈ V the adjacent edges are of one of two types; we define

In(v) = {e ∈ E : t(e) = v} and Out(v) = {e ∈ E : s(e) = v}.

We now define the notion of an n-tree. These describe the superposition of operations

and allow us to define free operads.

Definition 1.2.4. An n-tree is a tree T with exactly n numbered inputs such that for

every v ∈ V , |Out(v)| = 1 and |In(v)| ≥ 2.

1We add the element v0 to the set of vertices so that s and t are still defined on an external edge (sofor an external edge e either s(e) = v0 or t(e) = v0).


The following is an example of a 3-tree

1

v1��

��

� 2

v1��

��

v1

v2��

��

3

v2��

��

��

��

v2

��

Definition 1.2.5. Given an n-tree T and an operad P we define

P(T ) =�

v∈V (T )

P(In(v)).

The notion of a free operad can now be defined using n-trees.

Definition 1.2.6. Given a dg-operad P the free operad FP generated by P is given by

FP(n) =�

n−trees T

P(T )

the action of Sn is given by renumbering the n-inputs of a tree according to the acting

permutation. The composition maps are given by adjoining trees in the obvious way.

Using the composition maps from P one gets a canonical map of operads

πP : FP �� P .

Using the tree T from above one can think of the picture

1

[a, b]��

��

2

[a, b]��

��

[a, b]

[a, b]

e��

��

3

[a, b]��

��

��

��

[a, b]

��

Then there is a natural map [a, b]⊗T [a, b] �→ [[a.b], c] obtained by collapsing the edge

e above and composing elements in P(2) at the vertices as suggested by the tree T .

As noted earlier letting K = P(1) we see that K is an associative k-algebra and

every P(n) is a left K-module and a right K⊗n-module. This allows us to talk about


free algebras over an operad. Let V be a left K-module. We can then form the graded

vector space

FP(V ) =�

n≥1

�P(n)⊗K⊗n V

⊗n�

Sn

Compositions in P give rise to maps P(n)⊗ FP(V )⊗n �� FP(V )

Definition 1.2.7. Given an operad P and a left K-module V we call FP(V ) the free P

algebra generated by V .

1.3 Koszul and Cobar Duality

Given an operad P we could ask whether for large n the operations in P(n) are given

by compositions of lower order operations. More precisely, is there a number m such

that if µ ∈ P(n) for any n > m then there exists µi ∈ P(mi) with mi ≤ m such that

µ = γm1,...,mkn(µ1 ⊗ . . . ⊗ µkn). For example Lie, Com and Ass are generated by binary

operations only (so m = 2). These later types of operads are in a special class.

Definition 1.3.1. An operad P is called quadratic if for all n ≥ 2 each element of P(n)

is given as a composition of elements from P(2) and if all relations follow from those

given on ternary operations (elements of P(3)). More precisely P is quadratic if

• πP(2) : FP(2)�� P is surjective

• Ker(πP(2)) is generated by Ker(πP(2)(3) : FP(2)(3) �� P(3))

From the above definition it follows that a quadratic operad can be completely given

by a vector space V = P(2) of generators equipped with an S2 action and a subspace of

relations R ⊂ FP(3) invariant under S3. We will write P = �V, R�.

Definition 1.3.2. Given a quadratic operad P = �V, R� the Koszul dual operad is given

by P ! = �V ∗ ⊗ sgn, R⊥�


Theorem 1.3.3. The operads Ass, Com and Lie are quadratic and their Koszul duals

are Ass, Lie and Com respectively.

For the proof of this theorem see [4]

We next describe the cobar construction for a dg-operad. For a vector space V of

dimension n, let Det(V ) = Λn(V ). For a tree T let Ed(T ) be the set of edges of T except

the output edge and let Det(T ) = Det(kEd(T )) where kEd(T ) is the vector space generated

by Ed(T ). For a dg-operad P and for each n we define a complex given below

P(n)∗ ⊗Det(kn) δ ��

n−trees T|T |=1

P(T )∗ ⊗Det(T ) δ �� . . . ��

n−trees T|T |=n−2

P(T )∗ ⊗Det(T )

where the rightmost term is in degree 0. The differential δ is defined in terms of trees as

follows.

δT �,T : P(T �)∗ ⊗Det(T �) �� P(T )∗ ⊗Det(T )

is zero unless T� = T/e, that is T

� is obtained from T by collapsing the internal edge

e. In this case δT �,T = γ∗T,T � ⊗ le where γT,T � : P(T ) �� P(T �) is the map obtained by

composing in P the operations at the source and target vertices of e along with collapsing

the edge e and le(f1 ∧ f2 ∧ . . . ∧ fm) = e ∧ f1 ∧ f2 ∧ . . . ∧ fm. One can see that δ2 = 0

by considering a tree T and a tree T�� = T/{e1, e2}. The tree T

�� can be obtained from

T by collapsing one edge at a time in two different ways, by collapsing e1 first and e2

second and by collapsing e2 first and e1 second. According to the definition above the

result of applying delta twice in both cases will be the same except collapsing e1 then

e2 will give a factor e2 ∧ e1 ∧ f1 ∧ . . . ∧ fm and collapsing e2 then e1 will give a factor

e1 ∧ e2 ∧ f1 ∧ . . . ∧ fm = −e2 ∧ e1 ∧ f1 ∧ . . . ∧ fm. Note that each P∗(T ) above is a

dg - vector space and the differentials anticommute making the above complex a double

complex for each n. We call the total complex obtained from the double complex above

D(P)(n). One can in fact check that D(P) is a dg-operad.


The next theorem defines the important notion of the cobar-dual to a given operad.

The theorem also clarifies in which sense this new structure is actually dual to P .

Theorem 1.3.4. Let P be a dg-operad with P(0) = 0 and P(1) = k. Then the collection

{D(P)(n)} naturally forms a dg-operad D(P) called the cobar dual to P. There is a

canonical isomorphism of dg-operads D(D(P))αP �� P.

Proof. See [4].

We have defined so far two different notions of duality, the Kozsul (or quadratic) dual

and the cobar-dual. In general these structures are different however

Definition 1.3.5. We call an operad P Koszul if the natural map D(P)βP �� P ! is a

quasi-isomorphism.

Proposition 1.3.6. P is Koszul if and only if P ! is Koszul.

Proof. Assume that P is Koszul. Then

D(P)βP �� P !

is a quasi-isomorphism and

DD(P)D(βP )

�� D(P !)

Hence βP ! = αP ◦D(βP)−1 : D(P !) �� P is a quasi-isomorphism.

1.4 Homotopy Algebras Over an Operad

The existence of the natural map D(P !) �� P tells us that every P-algebra is also a

D(P)-algebra. This fact is motivation for making the following definition

Definition 1.4.1. Let P be a Koszul operad and P ! its quadratic dual. A dg-algebra

over D(P !) is called a homotopy P-algebra.


Loosely speaking homotopy algebras can be thought of as quadratic algebras with

relations that hold up to homotopy. The next theorem allows us to encode the structure

of a homotopy P-algebra in a differential on a shifted complex.

Theorem 1.4.2. Let V = ⊕Vn be a graded k-bimodule with finite dimensional com-

ponents. Giving the structure of a homotopy P-algebra on V is the same as giving a

differential ∂ on the free algebra FP !(V ∗[1]) satisfying the conditions

1. ∂2 = 0

2. ∂ is a derivation with respect to any binary operation in P ! so that for any µ ∈ P(2)

and x, y ∈ FP !(V ∗[1])

∂(µ(x, y)) = µ(∂(x), y) + (−1)deg(x)µ(x, ∂(y))

Example 1.4.3. Consider the operad P = Lie. Then D(P !) = D(Com). We know that

Com(n) = k for each n so the complex D(Com)(n) is as follows

�

|T |=0

Det(T ) ��

|T |=1

Det(T ) �� . . . ��

|T |=n−2

Det(T )

We can see then that a D(Com) algebra is just an assignment on an n-ary operation

[x1, . . . , xn] for each n (corresponding to the one basis vector of Com(n)) which is anti-

symmetric (because of the occurance of Det(T ) in each degree of the above complex).

The consequence of the differential in the above complex is that these operations will

obey some generalized Jacobi identities. Theorem 1.4.2 tells us that this structure is

equivalent to giving a vector space V along with a differential on the complex Λ(V ) that

satifies the Leibniz rule (see [11]).

We will use this theorem in the chapters that follow to describe homotopy algebras

over explicitly given operads. We will construct a complex as in the above theorem and

then read off the homotopy algebra relations by taking the dual of the given complex.

Chapter 2

Lie 2-Algebras

In the following chapter we first introduce the notion of a Lie 2-algebra. From here we

introduce a quadratic operad R and show how a Lie 2-algebra can be described as a

homotopy algebra over R.

2.1 Definition of a Lie 2-algebra

Before we can talk about Lie 2-algebras we need some notions from categorical algebra.

First of all we need to define a linear category, which is a category whose hom-sets are

vector spaces and whose composition maps are linear. To be more precise

Definition 2.1.1. A linear category is a category V whose set of objects L0 is a vector

space, for all x, y ∈ L0 the set HomV(x, y) is a vector space and the composition map

◦ : HomV(y, z)× HomV(x, y) �� HomV(x, z)

identity map

1 : L0�� L1

and source and target maps

s, t : V1�� V0

are linear.

15

Chapter 2. Lie 2-Algebras 16

Given an arrow (a : x �� y) ∈ V1 we can take the arrow

−→a = a− 1x : 0 �� y − x

called the arrow part of a. We can then define a notion of composition of arrows as

follows. If a : x �� y and b : y �� z then−−−→a + b = −→

a +−→b and we define

ab =−−−→a + b + 1a+b

It remains to check that with this definition of composition we have a category. This

is done in [1]. It turns out that the conditions that s, t and 1 are linear maps is strong

enough to insist that this is the only composition that could be defined to form a linear

category. Therefore we can forget about the composition in V and just concentrate on

the addition of the arrow parts. This fact is also verified in [1]. This allows us to translate

commutative diagrams into equations. For example, the diagram

x zc��

y

x

��

a

��

��

��

��y

z

b

��

��

��

��

is equivalent to −→a +−→b = −→

c

Definition 2.1.2. A Lie 2-algebra is a linear category L together with a bilinear functor

[·, ·] : L⊗ L �� L

a bilinear natural transformation called the skew-symmetrizor with components

Sx,y : [x, y] �� − [y, x]

and a trilinear natural transformation called the Jacobiator with components

Jx,y,z : [x, [y, z]] �� [[x, y], z] + [y, [x, z]]

such that the following diagrams commute


[[x, [y, z]], w] + [[y, z], [x, w]]+

+ [[x, z], [y, w]] + [z, [x, [y, w]]]

[[[x, y], z], w] + [z, [[x, y], w]]+

+ [y, [[x, z], w]] + [y, [z, [x, w]]]

[x, [[y, z], w]] + [x, [z, [y, w]]]

[[x, [y, z]], w] + [[y, z], [x, w]]+

+ [[x, z], [y, w]] + [z, [x, [y, w]]]

Jx,[y,z],w+Jx,z,[y,w]��

[x, [[y, z], w]] + [x, [z, [y, w]]] [[x, y], [z, w]] + [y, [x, [z, w]]][[x, y], [z, w]] + [y, [x, [z, w]]]

[[[x, y], z], w] + [z, [[x, y], w]]+

+ [y, [[x, z], w]] + [y, [z, [x, w]]]

J[x,y],z,w+[y,Jx,z,w]��

[x, [[y, z], w]] + [x, [z, [y, w]]] [[x, y], [z, w]] + [y, [x, [z, w]]]

[x, [y, [z, w]]]

[x, [[y, z], w]] + [x, [z, [y, w]]]

[x,Jy,z,w]��

��

[x, [y, [z, w]]]

[[x, y], [z, w]] + [y, [x, [z, w]]]

Jx,y,[z,w]

��

��

[[x, [y, z]], w] + [[y, z], [x, w]]+

+ [[x, z], [y, w]] + [z, [x, [y, w]]]

[[[x, y], z], w] + [[y, [x, z]], w] + [[y, z], [x, w]]

+ [[x, z], [y, w]] + [z, [[x, y], w]] + [z, [y, [x, w]]]

[Jx,y,z ,w]+1+1+[z,Jx,y,w]

��

��

[[x, [y, z]], w] + [[y, z], [x, w]]+

+ [[x, z], [y, w]] + [z, [x, [y, w]]]

[[[x, y], z], w] + [z, [[x, y], w]]+

+ [y, [[x, z], w]] + [y, [z, [x, w]]]

[[[x, y], z], w] + [z, [[x, y], w]]+

+ [y, [[x, z], w]] + [y, [z, [x, w]]]

[[[x, y], z], w] + [[y, [x, z]], w] + [[y, z], [x, w]]

+ [[x, z], [y, w]] + [z, [[x, y], w]] + [z, [y, [x, w]]]

1+1+Jy,[x,z],w+Jy,z,[x,w]

��

��

[[x, y], z]

[[x, y], z]− [y, [x, z]]

J−1x,y,z−1[y,[x,z]]

��

��

[[x, y], z] −[[y, x], z][Sx,y ,z]

�� −[[y, x], z]

[[x, y], z]− [y, [x, z]]

J−1y,x,z−1[x,[y,z]]

��

��

[[x, y], z] + [y, [x, z]] −[[x, z], y]− [z, [x, y]]S[x,y],z+Sy,[x,z]

��

[x, [y, z]]

[[x, y], z] + [y, [x, z]]

Jx,y,z

��

[x, [y, z]] −[x, [z, y]][x,Sy,z ]

�� −[x, [z, y]]

−[[x, z], y]− [z, [x, y]]

−Jx,z,y

��

[[x, y], z] + [y, [x, z]] −[[y, z], x]

[x, [y, z]]

[[x, y], z] + [y, [x, z]]

Jx,y,z

��[x, [y, z]]

−[[y, z], x]

Sx,[y,z]

��

−[[y, x], z]− [y, [z, x]] −[y, [z, x]] + [z, [y, x]]1+Sz,[y,x]

��

[[x, y], z] + [y, [x, z]]

−[[y, x], z]− [y, [z, x]]

[Sx,y ,z]+[y,Sx,z ]

��

[[x, y], z] + [y, [x, z]] −[[y, z], x]−[[y, z], x]

−[y, [z, x]] + [z, [y, x]]

−(J−1y,z,x−1[z,[y,x]])

��

If −Sy,xSx,y = 1[x,y] we say that S is symmetric. In this case the pentagon diagram

above follows from the square and triangle identities.


Definition 2.1.3. A Lie 2-algebra L is called

• semistrict if S = 1

• hemistrict if J = 1

• strict if it is semistrict and hemistrict.

Lie 2-algebras form a 2-category 2Lie. For the definitions of morphisms between Lie

2-algebras and 2-morphisms between these see [9].

2.2 2-Term EL∞-algebras

In [1] it was shown that the category of semistrict Lie 2-algebras is equivalent to the

category of 2 term L∞ algebras. Similarly in [9] it is shown that general Lie 2-algebras are

equivalent to another structure called an EL∞ algebra. We describe this correspondence

below and will in the future use the equations coming from the EL∞ description of a Lie

2-algebra.

Let L0 and L1 be the spaces of objects and arrows of a Lie 2-algebra (L, [·, ·], S, J)

respectively and let s, t : L1�� L0 be the source and target maps respectively.

We define the normalized cochain complex N(L) as

N(L)0 = L0

N(L)−1 = ker s

with differential d : N−1 �� N0 given by the restriction of t to N

−1.

If x ∈ N0 and (b : 0 �� y) ∈ N

−1 then [x, b] : 0 �� [x, y] and so d[x, b] = [x, y].

Similarly if (a : 0 �� x) ∈ N−1 then d[a, y] = [x, y]. Hence we get the following two

relations for all x, y ∈ N0 and all a, b ∈ N

−1

d[x, b] = [x, db] (2.1)

d[a, y] = [da, y] (2.2)


We define two new brackets

�−,−� : N0 ⊗N

0 �� N−1

�−,−,−� : N0 ⊗N

0 ⊗N0 �� N

−1

as

�x, y� =−−→Sx,y = 1[x,y] − Sx,y

�x, y, z� =−−→Jx,y,z = 1[x,[y,z]] − Jx,y,z

respectively. Using the various definitions and the naturality of S and J we get the

following relations

d�x, y� = [x, y] + [y, x] (2.3)

�da, y� = [a, y] + [y, a] (2.4)

�x, db� = [x, b] + [b, x] (2.5)

d�x, y, z� = [x, [y, z]]− [[x, y], z]− [y, [x, z]] (2.6)

�da, y, z� = [a, [y, z]]− [[a, y], z]− [y, [a, z]] (2.7)

�x, db, z� = [x, [b, z]]− [[x, b], z]− [b, [x, z]] (2.8)

�x, y, dc� = [x, [y, c]]− [[x, y], c]− [y, [x, c]] (2.9)

[x, �y, z, w�] + �x, [y, z], w�+ �x, z, [y, w]�+ [�x, y, z�, w] + [z, �x, y, w�] (2.10)

= �x, y, [z, w]�+ �[x, y], z, w�+ [y, �x, z, w�] + �y, [x, z], w�+ �y, z, [x, w]�

�x, y, z�+ �y, x, z� = −[�x, y�, z] (2.11)

�x, y, z�+ �x, z, y� = [x, �y, z�]− �[x, y], z� − �y, [x, z]� (2.12)


Remark 2.2.1. If we assume that the Jacobiator is zero equations (2.6)-(2.12) reduce

to

[�x, y�, z] = 0 (2.13)

[x, �y, z�] = �[x, y], z�+ �y, [x, z]� (2.14)

and the Leibniz relations

[x, [y, z]]− [[x, y], z]− [y, [x, z]] = 0 (2.15)

[a, [y, z]]− [[a, y], z]− [y, [a, z]] = 0 (2.16)

[x, [b, z]]− [[x, b], z]− [b, [x, z]] = 0 (2.17)

[x, [y, c]]− [[x, y], c]− [y, [x, c]] = 0 (2.18)

on [−,−]. This gives us the notion of a hemistrict Lie 2-algebra

Definition 2.2.2. A 2-term EL∞-algebra is a 2-term cochain complex C equipped with

• a chain map [·, ·] : C ⊗ C �� C

• a chain homotopy �·, ·� : [·, ·] + [·, ·] ∗ σ �� 0

• a chain homotopy �·, ·, ·� : [·, [·, ·]]− [[·, ·], ·]− [·, [·, ·]] ∗ σ12�� 0

such that the following equations hold:

[x, �y, z, w�] + �x, [y, z], w�+ �x, z, [y, w]�+ [�x, y, z�, w] + [z, �x, y, w�] (2.19)

= �x, y, [z, w]�+ �[x, y], z, w�+ [y, �x, z, w�] + �y, [x, z], w�+ �y, z, [x, w]� (2.20)

�x, y, z�+ �y, x, z� = −[�x, y�, z] (2.21)

�x, y, z�+ �x, z, y� = [x, �y, z�]− �[x, y], z� − �y, [x, z]� (2.22)

�x, y, z� − �y, z, x� = �x, [y, z]�+ �z, [y, x]� − [�x, y�, z]− [y, �x, z�] (2.23)

(2.24)


Remark 2.2.3. If we assume that �−,−� is symmetric then the last equation follows

from the previous two.

EL∞-algebras form a 2-category 2TermEL∞. For the definitions of morphisms and

two morphisms in this 2-category see [9].

It is clear that given a Lie 2-algebra one can define a 2-term L∞-algebra. Conversely,

given a 2-term L∞ algebra one can define a Lie 2-algebra.

Theorem 2.2.4. The Dold-Kan correspondence induces an equivalence of 2-categories

2Lie 2TermEL∞��2Lie 2TermEL∞��

For more details on the above theorem and the Dold-Kan correspondence see [9].

2.3 Lie 2-Algebras as Homotopy Algebras Over a

Quadratic Operad

We will now describe any Lie 2-algebra L as a homotopy algebra over a quadratic operad

R. To describe the desired homotopy R-algebra we use theorem 1.4.2.

Let N = N(L) be a linear category and let [−,−] be a bilinear bracket on L. We

consider a quadratic operad P generated by [−,−] a binary operation of degree 0 and d

a differential subject to the ternary relation

[[−,−],−] = [−, [−,−]] + [−, [−,−]]1,2 (2.25)

The dg-operad P = (P , d) generated by [−,−] is a free Leibniz algebra and it is well

know that the quadratic dual with binary operation {−,−} is given by the so called

Zinbeil relations (see [6])


{−, {−,−}} = {{−,−},−}+ {{−,−},−}1,2 (2.26)

From the discussion of Lie 2-algebras given above it is clear that N(L) is naturally a

dg-algebra over P .

We now take the complex given by FP !(N∗[1]). Letting A = (N−1)∗, B = (N0)∗ and

m = [−,−]∗ we define a differential δ generated by the maps

d :B �� A

m :B �� B ⊗B

A �� B ⊗ A

A �� A⊗B

their extensions as derivations and by taking tensor products. The beginning of the

resulting complex is as follows with the left most vector space in degree one.

B Ad ��B

B ⊗B

m

��

��

��

��A

B ⊗B

A

B ⊗ Am ��

A

A⊗B

m��

B ⊗B

(B ⊗B)⊗B

��

��

��

��B ⊗B

B ⊗ A��B ⊗B

A⊗B��

B ⊗ A

(B ⊗B)⊗ A��

B ⊗ A

(B ⊗ A)⊗B��B ⊗ A

(A⊗B)⊗B

��

A⊗B

(B ⊗B)⊗ A��

A⊗B

(B ⊗ A)⊗B��A⊗B

(A⊗B)⊗B

��

(B ⊗B)⊗B

(B ⊗B)⊗ A��

(B ⊗B)⊗B

(B ⊗ A)⊗B��

(B ⊗B)⊗B

(A⊗B)⊗B��

(B ⊗B)⊗B

((B ⊗B)⊗B)⊗B��

We call this complex W∗ so that the dual to the above complex is W . We refer to

the differential of this complex as δ. The tensor products in these diagrams represent the

tensor product modulo the dual Leibniz (zinbiel) relations.


Remark 2.3.1. Note that we took the free algebra over the quadratic operad P !, which

has a binary operation denoted by {−,−}. Since {−,−} satisfies the Zinbeil relations

any bracketing can be rearranged to be of the form ((...(B1⊗B2)⊗B3)⊗ ...)⊗Bn). Hence

the tensor products above really represent the tensor product modulo the relations given

for {−,−}

In what follows we se that we can read off the axioms of a strict Lie 2-algebra by

realizing the consequences of δ2 = 0 above.

Notation 2.3.2. Note that the above complex has a bi-grading described by coordinates

(m, n) where m is the degree of the vector space and n is its position as chosen above

in the degree m column, the top position is assigned the value of 0. All relations that

follow from δ2 = 0 above follow from a commutative diagram that starts at a vertex in

degree (m, n1) and ends at one in degree (m+2, n2). We will refer to such a commutative

diagram as (m; n1, n2)

Lemma 2.3.3. Equations (2.1) and (2.2) follow from the squares (1;0,0) and (1;0,1).

Proof. First consider the square (1;0,0) pictured below

B B ⊗ A

A

B

��d

�� A

B ⊗ A

m

��

B

B ⊗B

m ��B B ⊗ AB ⊗ A

B ⊗B

��

1⊗d��

Taking the dual of the above diagram gives the following square

N0 N0 ⊗N−1

N−1

N0

d

�� N−1

N0 ⊗N−1

�� [−,−]

��

N0

N0 ⊗N0

��

[−,−] ��N0 N0 ⊗N−1N0 ⊗N−1

N0 ⊗N0

1⊗d��

Now if we take an element x⊗ b ∈ N0⊗N−1 and follow it along the top and the bottom

of the above square we get

d[x, b] = [x, db]


A similar proof shows that for any a⊗ y ∈ N−1 ⊗N0

d[a, y] = [da, y]

Lemma 2.3.4. Equation (2.15) follows from diagram (1; 0, 3)

Proof. The diagram (1;0,3) is the composition of two morphisms

B B ⊗Bm �� B ⊗B (B ⊗B)⊗B��

We write m(b) = m1(b) ⊗ m2(b), ∀b ∈ B and when no confusion can occur we write

m(b) = m1 ⊗ m2. The map B ⊗ B �� ((B ⊗ B) ⊗ B)�

(B ⊗ (B ⊗ B)) is obtained

by extending m to a derivation however we always rearrange brackets using the Zinbeil

relations to have all the opening brackets on the left. Then for all b ∈ B

((m⊗ 1− 1⊗m) ◦m)(b) = (m⊗ 1− 1⊗m)(m1 ⊗m2)

= (m1m1 ⊗m2m1)⊗m2−

−m1 ⊗ (m1m2 ⊗m2m2)

= (m1m1 ⊗m2m1)⊗m2+

− (m1 ⊗m1m2)⊗m2m2+

− (m1m2 ⊗m1)⊗m2m2

= 0

Taking the dual of the above equation we get that for all x, y, z ∈ N0

[[x, y], z]− [x, [y, z]]− [y, [x, z]] = 0


Notation 2.3.5. For any a ∈ A let us write mr(a) = mr,1(a)⊗mr,2(a) when considering

the map m : A �� B ⊗A and write ml(a) = ml,1(a)⊗ml,2(a) when considering the map

m : A �� A⊗B.

Remark 2.3.6. Before we proceed we must calculate some of the arrows appearing in the

complex above. All the arrows shown are generated by d and m however all bracketing is

rearranged using the zinbeil relations. Consider the arrow with domain B ⊗A in degree

(3, 0) and codomain (A⊗B)⊗B in degree (4, 2). If we take an element b⊗ a ∈ B ⊗A,

apply the map 1⊗ml and rearrange brackets according to the zinbiel relations we get

(1⊗ml)(b⊗ a) = b⊗ (ml,1(a)⊗ml,2(a))

= (b⊗ml,1(a))⊗ml,2(a) + (ml,1(a)⊗ b)⊗ml,2(a)

In the latter expression (b⊗ml,1(a))⊗ml,2(a) ∈ (B⊗A)⊗B and (ml,1(a)⊗b)⊗ml,2(a) ∈

(A⊗B)⊗B. Next we apply the map 1⊗mr and get

(1⊗mr)(b⊗ a) = b⊗ (mr,1(a)⊗mr,2(a))

= (b⊗mr,1(a))⊗mr,2(a) + (mr,1(a)⊗ b)⊗mr,2(a)

In the latter expression (b⊗ml,1(a))⊗ml,2(a) ∈ (B⊗B)⊗A and (ml,1(a)⊗b)⊗ml,2(a) ∈

(B ⊗B)⊗ A. Finally we apply the map m⊗ 1 and get

(m⊗ 1)(b⊗ a) = (m1(b)⊗m2(b))⊗ a

which is an element of (B ⊗ B) ⊗ A. The above calculations show that the arrow B ⊗

A �� (A⊗B)⊗B is given by the assignment

b⊗ a �−→ (ml,1(a)⊗ b)⊗ml,2(a)


hence is dual to the map

1⊗ [−,−]1,3 : N−1 ⊗N0 ⊗N0�� N0 ⊗N−1

a⊗ y ⊗ z �−→ y ⊗ [a, z]

We perform a similar calculation and find that the map A⊗B �� (A⊗B)⊗B is given

by the assignment

a⊗ b �−→ (ml,1(a)⊗ml,2(a))⊗ b + (a⊗m1(b))⊗m2(b)

hence is dual to the map

[−,−]⊗ 1 + 1⊗ [−,−] : N−1 ⊗N0 ⊗N0�� N−1 ⊗N0

a⊗ y ⊗ z �−→ [a, y]⊗ z + a⊗ [y, z]

Lemma 2.3.7. The equations (2.16), (2.17), (2.18) follow from the squares (2; 0, 2),

(2; 0, 1) and (2; 0, 0) respectively.

Proof. Consider the square (2;0,2) depicted below

A (A⊗B)⊗B

B ⊗ A

A

��

m

��

��

��

�B ⊗ A

(A⊗B)⊗B

(1⊗[−,−]1,3)∗��

��

��

��

A

A⊗B

m

��

��

��

��A (A⊗B)⊗B(A⊗B)⊗B

A⊗B

��

([−,−]⊗1−1⊗[−,−])∗��

��

��

�

Taking the dual of the above diagram we see that for any a⊗ y⊗ z ∈ N−1⊗N0⊗N0 we

have [[a, y], z]− [a, [y, z]]− [y, [a, z]] = 0. Similarly we get equations (2.17) and (2.18)


Remark 2.3.8. There are more equations we could extract from FP !(N∗[1]) however

these all follow from the equations already derived. For example the diagram (1; 1, 1)

gives us the relation

x⊗ d[b, z] + db⊗ [x, z] + d[x, b]⊗ z = x⊗ [db, z] + [x, db]⊗ z + db⊗ [x, z]

which follows from equation (2.1).

Let us define a new operad Q generated by the symmetric binary operation �−,−� of

degree -1 with Q(n) = 0 for n ≥ 3. We next consider the operad generated by P and Q

subject to the relations

[−, �−,−�] = �[−,−],−�+ �−, [−,−]�1,3 (2.27)

[�−,−�,−] = 0 (2.28)

We also extend the differential d by defining d�x, y� = [x, y] + [y, x]. Call this operad

R = (R, d)

If we take the dual to this operad we get an operad R! generated by the operations

{−,−} = [−,−]∗ and � = �−,−�∗ which satisfies the following relations

{−,− �−} = (− � {−,−})1,3 (2.29)

{−,− �−} = ({−,−} �−) (2.30)

Remark 2.3.9. Recall that Q(n) = 0 for n ≥ 3. Then under the inner product described

in the proof of the next theorem we have RQ! = Q(3)⊥ = 0, that is � has no n-ary relations

for n ≥ 3. As stated earlier the operation {−,−} satisfies the so called Zinbeil relations.

Remark 2.3.10. The fact that [�−,−�,−] = 0 is equivalent to the fact that there is no

way of rewriting expressions of the form {x � y, z}.


Proposition 2.3.11. The operad described above is dual to the operad R

Proof. For the purposes of computing the inner product we identify R(3) with R(3)∗.

The inner product is given as follows

�[xσ(1), �xσ(2), xσ(3)�], {xσ(1), xσ(2) � xσ(3)}

�= sgn(σ)

��xσ(1), [xσ(2), xσ(3)]�, xσ(1) � {xσ(2), xσ(3)}

�= sgn(σ)

and all other inner products are zero. The relevant ternary relations (elements of RR

and RR!) are generated by

[x, �y, z�]− �[x, y], z� − �y, [x, z]�

[y, �x, z�]− �[y, x], z� − �x, [y, z]�

[z, �x, y�]− �[z, x], y� − �x, [z, y]�

[�x, y�, z]

[�y, z�, x]

[�z, x�, y]

and

{x, y � z}− y � {x, z}

{x, y � z}+ {x, y} � z

{y, x � z}− x � {y, z}

{y, x � z}+ {y, x} � z

{z, y � x}− y � {z, x}

{z, y � x}+ {z, y} � x


We need to show these spaces are orthogonal to each other using the inner product

described above.

�[x, �y, z�]−�[x, y], z� − �y, [x, z]�, {x, y � z}− y � {x, z}

�

=�[x, �y, z�], {x, y � z}

�+

��y, [x, z]�, y � {x, z}

�

= sgn(e) + sgn(12)

= 0

and

�[x, �y, z�]−�[x, y], z� − �y, [x, z]�, {x, y � z}− {x, y} � z

�

=�[x, �y, z�], {x, y � z}

�+

�− �[x, y], z�, {x, y} � z

�

= 1− 1

= 0

Also we have that�[�x, y�, z], {x, y � z}− y � {x, z}

�= 0

and�[�x, y�, z], {x, y � z}+ {x, y} � z

�= 0

Calculating the other inner products is done in a similar fashion. This completes the

proof.

We take the complex (W �)∗ = FQ!(W [1]) depicted below


B Ad ��B

B ⊗B

m

��

��

��

��A

B ⊗BB ⊗B B|Bτ ��B ⊗B

(B ⊗B)⊗B

��

��

��

��B|B

(B ⊗B)⊗B

A

B|B

n

��

��

��

��A

B ⊗ Am ��

A

A⊗B

m��

B ⊗B

B ⊗ A��B ⊗B

A⊗B��

B ⊗ A A|B��B ⊗ A

B|(B ⊗B)��B ⊗ A

(B ⊗B)|B��

A⊗B

A|B��A⊗B B|(B ⊗B)��A⊗B

(B ⊗B)|B��

B|B

B|(B ⊗B)��B|B

(B ⊗B)|B��

(B ⊗B)⊗B

(B ⊗B)|B��

(B ⊗B)⊗B

B|(B ⊗B)��

(B ⊗B)⊗B ((B ⊗B)⊗B)⊗B��(B ⊗B)⊗B ((B ⊗B)⊗B)⊗B��

B ⊗ A

((B ⊗B)⊗ A))⊕ ((B ⊗ A)⊗B)⊕ ((A⊗B)⊗B)��

Where the left hand side is in degree 2. Again we will call the differential in the above

complex δ. The bar product here represents the tensor product modulo the relation

b1 ⊗ b2 = −b2 ⊗ b1 (the relations dual to the relations satisfied by �−,−� .)

In what follows we see that we can read off the relations for a hemistrict Lie 2-algebra

by realizing the consequences of δ2 = 0 from above.

Remark 2.3.12. In chapter 4 we see that R = P ◦ Q where ◦ is some operation on

operads involving what is called a distributive law. We have the following property for

this operation

FP1(FP2(V )) = FP1◦P2(V )

for any dg-vector space V (see [3]). Then in our case FQ!(W ∗[1]) = FQ!◦P !(N∗[1]) =

F(P◦Q)!(N∗[1]) = FR!(N∗[1]).


Proof. Diagram (2;0,3) is

B B|B

A

B

��d

�� A

B|B

n

��

B

B ⊗B

m ��B B|BB|B

B ⊗B

��

τ��


Dualizing the above diagram we get that d�x, y� = [x, y] + [y, x]

Lemma 2.3.14. Equations (2.4), (2.5) follow from diagram (3; 0, 1).

Proof. The proof is the same as that of the lemma above.

Remark 2.3.15. Next we want to obtain equations (2.13) and (2.14) by taking the

dual of diagrams coming from FQ!(W ∗[1]). We cannot get the relations we want directly

from this diagram since the correct vector spaces don’t appear there, for example we

would need (N0 ⊗ (N0|N0))∗ = B ⊗ (B|B) to describe relation (2.14). However we

can use relations 2.29 and 2.30 to interchange ⊗ and |. First we consider the map

1|m : B|B �� B|(B ⊗ B). Applying this map to an element b1|b2 ∈ B|B and using the

relations on �−,−� we get

(1|m)(b1|b2) = b1|(m1(b2)⊗m2(b2)) = m1 ⊗ (b1|m2)

the latter being an element of B ⊗ (B|B). We we could also apply the map m|1 :

B|B �� (B ⊗B)|B which gives

(m|1)(b1|b2) = (m1(b1)⊗m2(b1))|b2 = m1 ⊗ (m2|b2)

Note that the relations (2.29) and (2.30) ensure that both of the maps above have

codomain B ⊗ (B|B). Hence we have a map B|B �� B ⊗ (B|B) which is dual to the

map x⊗ (y|z) �→ [x, y]|z + y|[x, z] and there are no nontrivial maps B|B �� (B|B)⊗B

Lemma 2.3.16. The algebra coming from the complex FQ!(W ∗[1]) satisfies the relations

(2.13) and (2.14).

Proof. In light of the comments preceding the lemma we have the following commutative

square

A B ⊗ (B|B)

B ⊗ A

A

��m

�� B ⊗ A

B ⊗ (B|B)

1⊗n

��

A

B|Bn

��A B ⊗ (B|B)B ⊗ (B|B)

B|B

��([−,−]|1+1|[−,−]1,3)∗

��

��


The dual to this diagram says that for all x, y, z ∈ N0

[x, �y, z�] = �[x, y], z�+ �y, [x, z]�

To prove the second equation notice that the following maps compose to give the zero

map

A A⊗Bm �� A⊗B (B|B)⊗B

n⊗1 ��

and taking the dual gives

[�x, y�, z] = 0

for all x, y, z ∈ N0

Theorem 2.3.17. The complex FQ!(W ∗[1]) described above gives the structure of a

hemistrict Lie 2-algebra.

Proof. By lemmas (2.3.3)-(2.3.16) this complex gives us the relations of a hemistrict Lie

2-algebra.

We wish to weaken the axioms to obtain a general Lie 2-algebra as a homotopy algebra

over the quadratic operad R. To do this we will relax the condition that [−,−] satisfies

the Jacobi identity. To this end we take the complex FR!(W ∗[1]) described above and

add the differential ∂ (refer back to theorem 1.4.2). This component should represent

the correction to the Jacobi identity on �−,−� and so should be dual to a new ternary

bracket

�−,−,−� : N0 ⊗N0 ⊗N0�� N−1

We let p = �−,−,−�∗ : A �� B ⊗ B ⊗ B and add this in as a new generator for our

differential giving us a new differential ∂. We show that the consequences of adding this

extra generator and insisting that ∂2 = 0 gives us the relations of a Lie 2-algebra.


Adding this extra generator to the differential will give us the complex

B Ad∗ ��B

B ⊗B

m

��

��

��

��A

B ⊗BB ⊗B B|Bτ∗ ��B ⊗B

(B ⊗B)⊗B

��

��

��

��B|B

(B ⊗B)⊗B

A

B|B

n

��

��

��

��A

B ⊗ Am ��

A

A⊗B

m��

B ⊗ A

((B ⊗B)⊗ A))⊕ ((B ⊗ A)⊗B)⊕ ((A⊗B)⊗B)��

B ⊗ A A|B��B ⊗ A

B|(B ⊗B)��B ⊗ A

(B ⊗B)|B��

A⊗B


(B ⊗B)|B��

B|B

B|(B ⊗B)��B|B

(B ⊗B)|B��

(B ⊗B)⊗B

(B ⊗B)|B��

(B ⊗B)⊗B

B|(B ⊗B)��

(B ⊗B)⊗B ((B ⊗B)⊗B)⊗B��(B ⊗B)⊗B ((B ⊗B)⊗B)⊗B��

B ⊗ A

((B ⊗B)⊗B)⊗B

��

��

��

��

��

��

��

��

��

��

�

A⊗B

((B ⊗B)⊗B)⊗B

��

��

��

��

��

��

��

��

�

A

(B ⊗B)⊗B

��

��

��

��

��

��

call this complex (W �)∗, which looks like the complex we considered previously with the

addition of new arrows, which are the dashed arrows shown above. As a result any

diagram we obtain from the above complex will look like a diagram (m, n1, n2) from W∗

plus one extra map. Any new relations we obtain will only involve iterated operations

involving three or more arguments so we only consider those relevant diagrams. All other

relations are as before.

Lemma 2.3.18. Equation (2.6) follows from the diagram (1; 0, 3)

Proof. We have the map given by the composition

B Ad �� A (B ⊗B)⊗B

p ��

The dual to this map is given by x⊗ y⊗ z �→ d�x, y, z�. The fact that ∂2 = 0 and lemma

2.3.4 give the result.

Lemma 2.3.19. The equations (2.7)-(2.9) follow from the squares (2; 0, 2), (2; 0, 1) and

(2; 0, 0) respectively.

Proof. We have the map given by the composition

A (B ⊗B)⊗Bp �� (B ⊗B)⊗B (A⊗B)⊗B

d⊗1⊗1 ��


the dual to this map is given by a⊗ y ⊗ z �→ �da, y, z�. The fact that ∂2 = 0 and lemma

2.3.7 give the result.

Lemma 2.3.20. Equations (2.11) and (2.12) hold for the homotopy algebra coming from

the complex FR!(W ∗[1])

Proof. Remark 2.3.15 preceding the proof of lemma 2.3.16 immediately gives us the

following commutative diagrams

A (B|B)⊗B

A⊗B

A

��m

�� A⊗B

(B|B)⊗B

n⊗1

��

A

B ⊗B ⊗B

p ��A (B|B)⊗B(B|B)⊗B

B ⊗B ⊗B

��

τ⊗1��

A

B ⊗ A

m

��A B|Bn ��A

B ⊗B ⊗B

p

��

B ⊗ A

B ⊗ (B|B)

1⊗n

��

B|B B ⊗ (B|B)([−,−]|1+1|[−,−]1,3)∗

��

B ⊗B ⊗B

B ⊗ (B|B)

1⊗τ

��

Equations (2.11) and (2.12) follow from taking the duals of the above diagrams

Lemma 2.3.21. Equations (2.7), (2.8) and (2.9) follow from diagrams (3; 0, 0), (3; 0, 1)

and (3; 0, 2).

Proof. We have arrows from (B⊗B)⊗B to (A⊗B)⊗B, (B⊗A)⊗B and (B⊗B)⊗A

given by d⊗ 1⊗ 1, 1⊗ d⊗ 1 and 1⊗ 1⊗ d respectively. The equations follow by taking

the duals of the indicated diagrams and lemma 2.3.7


Proof. First we describe explicitly some of the maps occurring in diagram (3; 0, 4). We

have the map B⊗B⊗B �� ((B⊗B)⊗B)⊗B which is generated by m as a derivation.


Evaluating this map at an element b1⊗ b2⊗ b3 ∈ B⊗B⊗B and rearranging the brackets

using the zinbeil relations we get

m(b1 ⊗ b2 ⊗ b3) = ((m1 ⊗m2)⊗ b2)⊗ b3 − (b1 ⊗ (m1 ⊗m2))⊗ b3

+ (b1 ⊗ b2)⊗ (m1 ⊗m2)

= ((m1 ⊗m2)⊗ b2)⊗ b3 − ((b1 ⊗m1)⊗m2)⊗ b3

+ ((m1 ⊗ b1)⊗m2)⊗ b3 + ((b1 ⊗ b2)⊗m1)⊗m2

− ((b1 ⊗m1)⊗ b2)⊗m2 + ((m1 ⊗ b1)⊗ b2)⊗m2

which is dual to the map

x⊗ y ⊗ z ⊗ w �−→

[x, y]⊗ z⊗w−x⊗ [y, z]⊗w +y⊗ [x, z]⊗w +x⊗y⊗ [z, w]−x⊗ z⊗ [y, w]+y⊗ z⊗ [x, w]

We next describe the map B ⊗A �� ((B ⊗B)⊗B)⊗B by evaluating it at an element

b⊗ a ∈ B ⊗ A and rearranging brackets using the Zinbeil relations

(1⊗ p)(b⊗ a) = b⊗ ((p1 ⊗ p2)⊗ p3)

= (b⊗ (p1 ⊗ p2))⊗ p3 + ((p1 ⊗ p2)⊗ b)⊗ p3

= ((b⊗ p1)⊗ p2)⊗ p3 + ((p1 ⊗ p2)⊗ b)⊗ p3 − ((p1 ⊗ b)⊗ p2)⊗ p3

which is dual to the map

x⊗ y ⊗ z ⊗ w �→

x⊗ �y, z, w�+ z ⊗ �x, y, w� − y ⊗ �x, z, w�

The map A⊗B �� ((B ⊗B)⊗B)⊗B is simply dual to the map

x⊗ y ⊗ z ⊗ w �−→


�x, y, z� ⊗ w

With these descriptions we have that the dual to the diagram

A A⊗Bm��

B ⊗ A

A

��

m

��

��

��

�B ⊗ A

A⊗BA⊗B ⊗4B��

B ⊗ A

A⊗B

B ⊗ A

⊗4B

��

��

��

�

A A⊗BA

(B ⊗B)⊗B

p

��

��

��

��A⊗B

(B ⊗B)⊗B

A⊗B ⊗4BA⊗B

(B ⊗B)⊗B

⊗4B

(B ⊗B)⊗B

��

��

��

��

�

gives equation (2.10).

Lemma 2.3.23. The complex FR!(W ∗[1]) along with the differential ∂ described above

gives the structure of a Lie 2-algebra.

Proof. This follows from theorem 2.3.17 and lemmas 2.3.18-2.3.22.

Theorem 2.3.24. A Lie 2-algebra is a homotopy algebra over the quadratic operad R.

Proof. Lemma 2.3.22 tells us the complex (FR!(W ∗[1]), ∂�) gives the axioms of a Lie 2-

algebra. If we apply theorem 1.4.2 we see that this data is equivalent to the structure of

a homotopy algebra over R. To apply theorem 1.4.2 we need the fact that the operad R

in question is Koszul. We defer this somewhat involved proof to chapter 4.

2.4 Examples of Lie 2-Algebras

Example 2.4.1. Let (g, [−,−]) be a Leibniz algebra. Let C0 = g and let C−1 = g ⊕ g

where g = span({[x, x] : x ∈ g}). With d : C−1 �→ C0 the inclusion map we get a Lie

2-algebra as follows. We have a vector space of objects L0 = C0 and a vector space of

arrows L−1 = C−1. We define the source, target and identity maps as s(x, [y, y]) = x,

t(x, [y, y]) = x + [y, y] and i(x) = (x, 0). Now take Sx,y = ([x, y], [x, y] + [y, x]) (Note:

[x, y] + [y, x] = [x + y, x + y]− ([x, x] + [y, y]) ∈ g) so that �x, y� = �Sx,y = [x, y] + [y, x].

With J = 0 one readily checks that we have defined a hemistrict Lie 2-algebra.


For the next class of examples we need the notion of an L∞-algebra

Definition 2.4.2. An L∞−algerba (or a homotopy Lie algebra) structure on a graded

vector space V =�

n∈Z V is a collection of skew-symmetric linear maps ln : V⊗n �� V

of degree 2− n satisfying the generalized Jacobi identities

�

i+j=n+1

�

σ

�(σ)sgn(σ)(−1)i(j−1)lj(li(xσ(1), . . . , xσ(i)), xσ(i+1), . . . , xσ(n)) = 0

where the permutations σ vary over all (i, n− i)- unshuffles for i ≥ 1.

Example 2.4.3. Suppose we have a 1-term L∞-algebra V = V0. Then l1 is of degree 1

hence l1 = 0 and for n ≥ 3, ln is of negative degree. Hence ln = 0 for n �= 2. We have

that l2 : V ⊗ V �� V and by the properties of an L∞-algebra listed above (V, l2) is a Lie

algebra in the usual sense.

Next suppose we have a two term L∞ algebra V = V−1 ⊕ V0. Then we have a

differential d = l1 : V−1�� V0 along with a skew symmetric bracket of degree 0

[−,−] = l2 :V0 ⊗ V0�� V0

V−1 ⊗ V0�� V−1

V0 ⊗ V−1�� V−1

We also have a triple bracket of degree -1 given by l3

�−,−,−� = l3 : V0 ⊗ V0 ⊗ V0�� V−1

From the generalized Jacobi identity given above we see that now the Jacobi identity

isn’t satisfied exactly but has the correction term dl3. For example we have that for all

x, y, z ∈ V0

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = d�x, y, z�

We see that a 2-term L∞ algebra is a semistrict Lie 2-algebra.


Example 2.4.4. Let g be a finite dimensional Lie algebra over k with a symmetric

bilinear form �−,−� such that for any x, y, z ∈ g, �x, [y, z]� = �[x, y], z� (for example the

killing form). If we define d : k �� g to be zero and �x, y, z� = ��x, [y, z]� for some � ∈ k

we get a semistrict Lie 2-algebra. When �−,−� is the killing form this lie 2-algebra is

denoted g�

Example 2.4.5. In [12] it is shown that under certain conditions the tangent space at

the identity of a 2-group has a Lie 2-algebra structure. The author defines the notion of a

quasi-coherent sheaf of abelian categories C over a quasi-compact and seperated scheme S.

From here the notation of the 2-group of auto-equivalences of an abelian category GL(C)

is given and this 2-group is shown to satisfy the conditions necessary for the existence of

a Lie 2-algebra at the identity element. In this example, unlike the examples above, a

Lie 2-algebra is given without referring first to a 2-term L∞ algebra.

Our main examples of Lie 2-algebras are Courant algebroids

Definition 2.4.6. A Courant algebroid over a manifold M is a vector bundle E �� M

along with a bracket [−,−] : Γ(E) × Γ(E) �� Γ(E), an inner product �−,−� : E ×

E �� C∞(M) and an anchor map ρ : E �� TM satifying the following conditions

• [x, [y, z]] = [[x, y], z] + [y, [x, z]]

• ρ(x)�y, z� = �[x, y], z�+ �y, [x, z]�

• [x, y] + [y, x] = ρ∗(∂�x, y�)

where ∂ is the de Rham differential (we identify E with E∗ via the inner product).

If we define d = ρ∗∂ : C

∞(M) �� E, [x, b] = ρ(x)b, [a, y] = 0 for a, b ∈ C∞(M) and

x, y ∈ Γ(E) we get a hemistrict Lie 2-algebra.

Chapter 3

Commutative 2-Algebras

In this chapter we introduce the notion of a commutative 2-algebra. The operations

and relations defining a commutative 2-algebra are motivated in the next section by a

discussion on braided monoidal categories. From here this chapter very closely resembles

that of chapter 2; we show that a commutative 2-algebra can be seen as a homotopy

algebra over a quadratic operad.

Note that section 3.3 parallels, almost exactly, section 2.3. In this section some proofs

are omitted since they can be reconstructed using the corresponding proofs in section

2.3.

We begin with a short description of braided monoidal categories.

3.1 Monoidal Categories

Lets start by recalling the definition of a symmetric monoidal category.

Definition 3.1.1. A monoidial category is a category C together with a functor

⊗ : C × C �� C

an identity object e ∈ |C| and three isomorphisms

39

Chapter 3. Commutative 2-Algebras 40

α : ⊗ ◦ (⊗× 1C) �� ⊗ ◦(1C ×⊗)

λ : e⊗ 1C �� 1C

ρ : 1C ⊗ e �� 1C

satisfying the coherence conditions expressed by the following three diagrams for any

x, y, z, w ∈ |C|

Figure 3.1.2.

(x⊗ y)⊗ (z ⊗ w) (x⊗ (y ⊗ z))⊗ w

((x⊗ y)⊗ z)⊗ w

(x⊗ y)⊗ (z ⊗ w)

αx⊗y,z,w

��((x⊗ y)⊗ z)⊗ w

(x⊗ (y ⊗ z))⊗ w

αx,y,z⊗w

��

x⊗ (y ⊗ (z ⊗ w)) x⊗ ((y ⊗ z)⊗ w)x⊗αy,z,w

��

(x⊗ y)⊗ (z ⊗ w)

x⊗ (y ⊗ (z ⊗ w))

αx,y,z⊗w

��

(x⊗ y)⊗ (z ⊗ w) (x⊗ (y ⊗ z))⊗ w(x⊗ (y ⊗ z))⊗ w

x⊗ ((y ⊗ z)⊗ w)

αx,y⊗z,w

��

(x⊗ e)⊗ y

x⊗ y

ρx⊗y

��

��

��

��(x⊗ e)⊗ y x⊗ (e⊗ y)

αx,e,y �� x⊗ (e⊗ y)

x⊗ y

x⊗λy

��

��

��

�

Definition 3.1.3. A monoidal category C is called braided if there is a natural transfor-

mation called a braiding with components γx,y : x ⊗ y �� y ⊗ x such that the following

two hexagons commute for every x, y, z ∈ |C|

Figure 3.1.4.

x⊗ (z ⊗ y) (z ⊗ y)⊗ xγx,z⊗y��

(x⊗ y)⊗ z

x⊗ (z ⊗ y)

(x⊗ y)⊗ z z ⊗ (x⊗ y)γx⊗y,z �� z ⊗ (x⊗ y)

(z ⊗ y)⊗ x

x⊗ (y ⊗ z)

x⊗ (z ⊗ y)

x⊗γy,z

��

��

��

�

(x⊗ y)⊗ z

x⊗ (y ⊗ z)

αx,y,z

��

��

��

(x⊗ y)⊗ z

x⊗ (z ⊗ y)

z ⊗ (y ⊗ x)

(z ⊗ y)⊗ x

α−1z,y,x

��

��

��

z ⊗ (x⊗ y)

z ⊗ (y ⊗ x)

z⊗γx,y

��

��

��

�z ⊗ (x⊗ y)

(z ⊗ y)⊗ x


(y ⊗ x)⊗ z z ⊗ (y ⊗ x)γy⊗x,z��

x⊗ (y ⊗ z)

(y ⊗ x)⊗ z

x⊗ (y ⊗ z) (y ⊗ z)⊗ xγx,y⊗z �� (y ⊗ z)⊗ x

z ⊗ (y ⊗ x)

(x⊗ y)⊗ z

(y ⊗ x)⊗ z

γx⊗y⊗z

��

��

��

�

x⊗ (y ⊗ z)

(x⊗ y)⊗ z

α−1x,y,z

��

��

��

x⊗ (y ⊗ z)

(y ⊗ x)⊗ z

(z ⊗ y)⊗ x

z ⊗ (y ⊗ x)

αz,y,x

��

��

��

(y ⊗ z)⊗ x

(z ⊗ y)⊗ x

γy,z⊗x

��

��

��

�(y ⊗ z)⊗ x

z ⊗ (y ⊗ x)

Definition 3.1.5. A braided monoidal category C that also satisfies the identity

x⊗ y x⊗ y1x⊗y

��

y ⊗ x

x⊗ y

γx,y

��

��

��

�y ⊗ x

x⊗ y

γy,x

��

��

��

��

for all x, y ∈ |C| is called a symmetric monoidal category.

3.2 Commutative 2-Algebras

Next we give the axioms for a commutative 2-algebra. We consider a symmetric monoidal

category V = (V0, V−1,⊗) where V0 = |V| is the vector space of objects and the vector

space V−1 is the collection of arrows in V with domain at 0. The monoidal structure is

denoted by

· : V0 × V0�� V0

x⊗ y �→ xy

and we assume for the moment that the associator α is the identity natural transforma-

tion. We define a complex V

V = (V−1d �� V0)

(0 �� y) �→ y


where V−1 is in degree -1 and V0 is in degree 0. We immediately get the relations, for all

x, y ∈ V0 and a, b ∈ V−1

d(xb) = x(db) (3.1)

d(ay) = (da)y (3.2)

We now define a bracket [−,−] : V0⊗V0��V−1 of degree -1 as follows. Given x, y ∈ V0

we have the arrow γx,y : xy ��yx. We define [x, y] to be the arrow part of this morphism

(see the beginning of chapter 2)

[x, y] = −→γx,y = 1xy − γx,y.

Clearly we have the relation

d([x, y]) = xy − yx (3.3)

Since we assume the monoidal product is associative the hexagon identities collapse

to give the two triangles shown below

yxz yzxy⊗γxz

��

xyz

yxz

γx,y⊗z

��

xyz

yzx

γx,yz

��

��

��

��

xzy zxyγx,z⊗y

��

xyz

xzy

x⊗γy,z

��

xyz

zxy

γxy,z

��

��

��

��

which tell us two relations on the bracket [−,−]

[x, yz] = [x, y]z + y[x, z] (3.4)

[xy, z] = x[y, z] + [x, z]y (3.5)

Definition 3.2.1. A 2-term complex V = (V−1d �� V0) with two operations

· : V0 ⊗ V0�� V0

[−,−] : V0 ⊗ V0�� V−1

satisfying relations (3.3), (3.4) and (3.5) is a hemistrict commutative 2-algebra


We now relax the assumption that the associator is an isomorphism in the monodial

structure V = (V1, V0,⊗). We can then define a new a new operation

[−,−,−] : V0 ⊗ V0 ⊗ V0�� V−1

as follows. Take any x⊗y⊗z ∈ (V ⊗V )⊗V and consider the morphism in the category V

given by the associator αx,y,z. We can now define [x, y, z] = −−−→αx,y,z = 1(x⊗y)⊗z−αx,y,z ∈ V−1.

Figure (3.1.2) and figure (3.1.4) now give us the following relations

[xy, z, w] + [x, y, zw] + x[y, z, w] = [x.y, z]w + [x, yz, w] (3.6)

[x, y, z] + x[y, z] + [x, zy] + [z, y, x] = [xy, z] + z[x, y] (3.7)

[x, y, z] + [x, yz] + [y, z]x + [z, y, x] = [x, y]z + [yx, z] (3.8)

Definition 3.2.2. A hemistrict commutative 2-algebra V with an operation

[−,−,−] : V0 ⊗ V0 ⊗ V0�� V−1

that satisfies equations (3.6), (3.7) and (3.8) is called a commutative 2-algebra. A com-

mutative 2-algebra with γx,y = 1xy is called a semistrict commutative 2-algebra.

3.3 Commutative 2-Algebras as Homotopy Algebras

Over a Quadratic Operad

In this section we define the notion of a commutative 2-algebra as a homotopy algebra

over an operad. What follows will resemble the procedure in chapter 2.

We take the operad P generated by the binary operation − · − which satisfies the

associativity relation. We will denote the dual to this operation by {−,−}. It is well

known that Ass is self dual (see [4]) so that {−,−} also satisfies the associativity relation.

From the discussion in the previous section it is clear that a commutative 2-algebra is

naturally a dg-algebra over − · −. Let V be a symmetric monoidal category. We next


take the complex given by FP !(V ∗[1]). Letting A = (V−1)∗, B = (N0)∗, and m = (− ·−)∗

we define a differential generated by the maps

d :B �� A

m :B �� B ⊗B

A �� B ⊗ A

A �� A⊗B

and their extensions as derivations and by taking tensor products. The beginning of the

resulting complex is as follows with the left most in degree one

B Ad ��B

B ⊗B

m

��

��

��

��A

B ⊗B

A

B ⊗ Am ��

A

A⊗B

m��

B ⊗B

(B ⊗B)⊗B

��

��

��

��B ⊗B

B ⊗ A��B ⊗B

A⊗B��

B ⊗ A

(B ⊗B)⊗ A��

B ⊗ A

(B ⊗ A)⊗B��B ⊗ A

(A⊗B)⊗B

��

A⊗B

(B ⊗B)⊗ A��

A⊗B

(B ⊗ A)⊗B��A⊗B

(A⊗B)⊗B

��

(B ⊗B)⊗B

(B ⊗B)⊗ A��

(B ⊗B)⊗B

(B ⊗ A)⊗B��

(B ⊗B)⊗B

(A⊗B)⊗B��

(B ⊗B)⊗B

((B ⊗B)⊗B)⊗B��

We will call this complex W∗ so that the dual to the above complex is W . We will call

the differential in the above complex δ.

Remark 3.3.1. The same as remark 2.3.1.

Notation 3.3.2. We follow the same notation as that described following remark 2.3.1

Lemma 3.3.3. Equations (3.1) and (3.2) follow from diagrams (1; 0, 0) and (1; 0, 1)

respectively.

Proof. The proof is identical to the one given for lemma 2.3.3


Lemma 3.3.4. The associativity of the product − · − follows from diagrams (1; 0, 3),

(2; 0, 2), (2; 0, 1) and (2; 0, 0) respectively.

Proof. Diagram (1; 0, 3) looks like

B B ⊗Bm �� B ⊗B (B ⊗B)⊗B��

We will use the same notation as that given in the proof of lemma 2.3.4. For any b ∈ B

we have

((m⊗ 1 + 1⊗m) ◦m)(b) = (m⊗ 1 + 1⊗m)(m1(b)⊗m2(b))

= (m1(m1)⊗m2(m1))⊗m2 + m1 ⊗ (m1(m2)⊗m2(m2))

= (m1(m1)⊗m2(m1))⊗m2 + (m1 ⊗m1(m2))⊗m2(m2)

= 0

The dual to this map is

(x⊗ y)⊗ z �−→ (xy)z + x(yz).

This gives the associativity relation. The associativity relations on (V−1 ⊗ V0) ⊗ V0,

(V0 ⊗ V−1)⊗ V0 and (V0 ⊗ V0)⊗ V−1 follow in a similar manner.

We now define a new operad Q generated by the symmetric binary operation [−,−] of

degree -1 and suppose that Q(n) = 0 for all n ≥ 3. We suppose that this new operation

satisfies the following relations

[−,− ·−] = ([−,−],−) + (− · [−,−])1,3 (3.9)

[− ·−,−] = (− · [−,−]) + ([−,−] ·−)1,3 (3.10)

(3.11)

Proposition 3.3.5. Let the quadratic dual to the operation [−,−] be labelled �. Then

the operations {−,−} and � also satisfy the relations (3.9), (3.10).


Proof. The relations (3.9), (3.10) are the Poisson relations and it is well known that these

are self dual. The proof of this fact is the same as that of proposition 2.3.11.

We now take the complex (W �)∗ = FQ!(W ∗[1]) depicted below. Let n = [−,−]∗

B Ad ��B

B ⊗B

m

��

��

��

��A

B ⊗BB ⊗B B|Bτ ��B ⊗B

(B ⊗B)⊗B

��

��

��

��B|B

(B ⊗B)⊗B

A

B|B

n

��

��

��

��A

B ⊗ Am ��

A

A⊗B

m��

B ⊗B

B ⊗ A��B ⊗B

A⊗B��

B ⊗ A A|B��B ⊗ A

B|(B ⊗B)��B ⊗ A

(B ⊗B)|B��

A⊗B


(B ⊗B)|B��

B|B

B|(B ⊗B)��B|B

(B ⊗B)|B��

(B ⊗B)⊗B

(B ⊗B)|B��

(B ⊗B)⊗B

B|(B ⊗B)��

(B ⊗B)⊗B ((B ⊗B)⊗B)⊗B��(B ⊗B)⊗B ((B ⊗B)⊗B)⊗B��

B ⊗ A

((B ⊗B)⊗ A))⊕ ((B ⊗ A)⊗B)⊕ ((A⊗B)⊗B)��

Where the left hand column is in degree 2. Again we call the differential in the above

complex δ. The bar product here represents the tensor product modulo the relations

b1 ⊗ b2 = b2 ⊗ b1 (the relations dual to the relations satisfied by [−,−]).

We now obtain the equations defining a hemistrict commutative 2-algebra from the

complex W�.

Lemma 3.3.6. Equation (3.3) follows from diagram (2; 0, 3).

Proof. The proof is the same as that of lemma 2.3.4


respectively.

Proof. We begin by describing the maps B⊗A �� B|(B⊗B) and A⊗B �� B|(B⊗B)

explicitly. To describe the former we take an element b ⊗ a, apply the map 1 ⊗ n, and


rearrange the brackets using relations (3.9) and (3.10). We get the following

(1⊗ n)(b⊗ a) = b⊗ (n1(a)|n2(a))

= (b⊗ n1)|n2 + n1|(b⊗ n2)

The second term in the sum above is an element of B|(B ⊗ B) and when we take the

dual we get the mapping

x|(y ⊗ z) �−→ y ⊗ [x, z]

We calculate the map A ⊗ B �� B|(B ⊗ B) the same way giving us diagram (3; 0, 3)

depicted below

A B|Bn �� B|B B|(B ⊗B)1|m

��

B ⊗ A

A

��

m

��B ⊗ A

B|B

B ⊗ A

B|(B ⊗B)

(1⊗(−·−)1,3)∗��

��

A B|Bn ��A

A⊗B

m

�� B|B B|(B ⊗B)1|m

��B|B

A⊗B

B|(B ⊗B)

A⊗B

��

((−·−)1,2⊗1)∗��

��

Taking the dual of this square gives us equation (3.4). To get equation (3.5) we follow

the same procedure to get the following commutative square

A B|Bn �� B|B (B ⊗B)|Bm|1��

B ⊗ A

A

��

m

��B ⊗ A

B|B

B ⊗ A

(B ⊗B)|B

(1⊗(−·−)2,3)∗��

��

A B|Bn ��A

A⊗B

m

�� B|B (B ⊗B)|Bm|1��B|B

A⊗B

(B ⊗B)|B

A⊗B

��

((−·−)1,3⊗1)∗��

��

giving equation (3.5).

In the last chapter some of the diagrams obtained from our complex gave us equa-

tions that already followed from previously obtained relations, for example the diagram


corresponding to diagram (3; 1, 3) of chapter 2. This need not be the case here and in-

deed diagram (3; 1, 3) gives us a component to the differential that has yet to have been

described. We describe this component in what follows.

Take diagram (3; 1, 3) depicted below

B ⊗B B|Bτ �� B|B B|(B ⊗B)1|m

��

B ⊗ A

B ⊗B

��

1⊗d

��B ⊗ A

B|B

B ⊗ A

B|(B ⊗B)

(1⊗�1,3)∗

��

B ⊗B B|Bτ ��B ⊗B

A⊗B

d⊗1

�� B|B B|(B ⊗B)1|m

��B|B

A⊗B

B|(B ⊗B)

A⊗B

��

(�1,2⊗1)∗

��B ⊗B B|Bτ ��B ⊗B

B ⊗B ⊗B

m

��

��

��

��

��

��

��

��

��

B|B B|(B ⊗B)1|m

��B|B

B ⊗B ⊗B

B|(B ⊗B)

B ⊗B ⊗B

��

(τ1,2⊗1+1⊗τ1,3+ρL)∗

��

��

��

��

��

��

��

��

��

where ρL has yet to be determined. If we take an element x|(y ⊗ z) ∈ V0|(V0 ⊗ V0) and

apply the dual to the maps above we get the following

y ⊗ d[x, z]− τ(x, yz) + d[x, y]⊗ z + �(τ(x, y)⊗ z + y ⊗ τ(x, z) + ρL(x, y, z))

= y ⊗ xz − y ⊗ zx + x⊗ yz − yz ⊗ x + xy ⊗ z − yx⊗ z + xy ⊗ z + x⊗ yz

− yz ⊗ x + y ⊗ zx + �(ρL(x, y, z))

= 0

Note that the component (τ1,2 ⊗ 1 + 1 ⊗ τ1,3)∗ was determined by extending τ to a

derivation on B ⊗ B ⊗ B. We also have the component ρL which is determined by the

previously determined relations and the fact that δ2 = 0. Solving for ρL we see that it

must be dual to the map

x|(y ⊗ z) �−→ y ⊗ x⊗ z.


We perform a similar calculation and get a map ρR : B⊗B⊗B �� (B⊗B)|B which

is dual to the map

(x⊗ y)|z �−→ x⊗ z ⊗ y

Theorem 3.3.8. The complex F�(W ∗[1]) along with the differential δ described above

gives the structure of a hemistrict commutative 2-algebra.

Proof. By lemmas 3.3.3-3.3.7 this complex gives us the relations of a hemistrict commu-

tative 2-algebra.

Now we take the complex F(W ∗[1]) and add one component p : A �� B ⊗ B ⊗ B

to the differential. We will interpret p as being dual to a triple bracket [−,−,−] :

V0⊗V0⊗V0�� V−1. We call the resulting differential ∂. We see in what follows that the

consequences of adding this extra component to the differential gives us a commutative

2-algebra.


Proof. Note that since the we have the associative relation on B⊗B⊗B we can describe

the map B ⊗B ⊗B �� B ⊗B ⊗B ⊗B immediately as dual to

x⊗ y ⊗ z ⊗ w �→ xy ⊗ z ⊗ w + x⊗ yz ⊗ w + x⊗ y ⊗ zw.

The relation then follows from the dual to the diagram

A A⊗Bm �� A⊗B B ⊗B ⊗B ⊗B

1⊗p ��

B ⊗ A

A

��

m

��B ⊗ A

A⊗B

B ⊗ A

B ⊗B ⊗B ⊗B

p⊗1

��

A A⊗Bm ��A

B ⊗B ⊗B

p

�� A⊗B B ⊗B ⊗B ⊗B1⊗p ��A⊗B

B ⊗B ⊗B

B ⊗B ⊗B ⊗B

B ⊗B ⊗B

��

m⊗1⊗1+1⊗m⊗1+1⊗1⊗m��

��



Proof. Considering the calculations following lemma 3.3.6 these two relations follow di-

rectly from the following diagrams

A B|Bn �� B|B B|(B ⊗B)1|m

��

B ⊗ A

A

��

m

��B ⊗ A

B|B

B ⊗ A

B|(B ⊗B)

(1⊗�1,3)∗

��

A B|Bn ��A

A⊗B

m

�� B|B B|(B ⊗B)1|m

��B|B

A⊗B

B|(B ⊗B)

A⊗B

��

(�1,2⊗1)∗


B ⊗B ⊗B

p

��

��

��

��

��

��

��

��

��

� B|B B|(B ⊗B)1|m

��B|B

B ⊗B ⊗B

B|(B ⊗B)

B ⊗B ⊗B

��

(τ1,2⊗1+1⊗τ1,3+ρL)∗

��

��

��

��

��

��

��

��

��

A B|Bn �� B|B (B ⊗B)|Bm|1��

B ⊗ A

A

��

m

��B ⊗ A

B|B

B ⊗ A

(B ⊗B)|B

(1⊗�2,3)∗

��

A B|Bn ��A

A⊗B

m

�� B|B (B ⊗B)|Bm|1��B|B

A⊗B

(B ⊗B)|B

A⊗B

��

(�1,3⊗1)∗


B ⊗B ⊗B

p

��

��

��

��

��

��

��

��

��

� B|B (B ⊗B)|Bm|1��B|B

B ⊗B ⊗B

(B ⊗B)|B

B ⊗B ⊗B

��

(1⊗τ2,3+τ1,3⊗1+ρR)∗

��

��

��

��

��

��

��

��

��

Theorem 3.3.11. The complex given by FQ!(W ∗[1]) and the differential ∂ gives the

structure of a commutative 2-algebra.


Proof. This follows from theorem 3.3.8, lemma 3.3.9 and lemma 3.3.10.

Applying the same reasoning as in chapter 2 we see that by theorem 1.4.2 the above

theorem tells us that a commutative 2-algebra is a homotopy algebra over a quadratic

operad. This concludes the results of this section.

Chapter 4

Distributive Laws and the

Koszulness of R

In this chapter we take care of some of the technical details on which our results relied

upon. To apply our main tool, which was the theorem from [4] describing homotopy

algebras, we need our operads to be Koszul. The goal is to show that the dg-operads

from chapters 2 and 3 are Koszul. The idea behind the proof is the following. Each

operad is in some sense built from two seperate quadratic operads P and Q (with no

differential), which are both Koszul. We show that these operads are built from P and Q

using what is called a distributive law. Then a theorem from [7] tells us that the operads

we constructed in this way are Koszul. Finally we add a differential d to our operad to

obtain a dg-operad and using spectral sequences we prove that (R, d) is also Koszul.

4.1 Distributive Laws

In this section we describe how two Koszul operads can be combined in a suitable way so

that the resulting operad is also Koszul. We use the results of this section to prove the

Koszulness of the operads controlling the Lie-2 and Comm-2 algebras described earlier.

The information on distributive laws is taken from [7]

52

Chapter 4. Distributive Laws and the Koszulness of R 53

Suppose we have two operads P and Q that are Koszul. We consider a new operad

R that consists of the operations belonging to P and Q along with all possible mixed

operations (composing one type of operation with the other). In what case does the

resulting operad turn out to be Koszul? This, as one would expect, depends on what

relations are imposed on these mixed operations. The following section makes this notion

precise with whats called a distributive law.

In what follows we need the following grading on a free operad over a graded operad.

Definition 4.1.1. Let E = ⊕Ni=1Ei be a graded k-collection, meaning that E(m) =

⊕Ni=1Ei(m) and the decomposition is Σm invariant. Then F(E) has a natural multigrad-

ing, F(E) = ⊕i1,...,iNFi1,...,iN (E) described by the following two properties

1. F0,...,0(E) = F(E)(1) = k and Ei = F0,...,0,1,0,...,0(E) where 1 is in the i-th place,

1 ≤ i ≤ N .

2. Let m, n ≥ 1, 1 ≤ l ≤ m and let a ∈ Fi1,...,iN (E)(m), b ∈ Fj1,...,jN (E)(n). Then

a ◦l b ∈ Fk1,...,kN (E)(m + n− 1) with ki = ii + ji for 1 ≤ i ≤ N . (Here if µ ∈ E(m)

and ν ∈ E(n) then for 1 ≤ l ≤ m, µ ◦l ν := γ(µ; 1, . . . , 1, ν, 1, . . . , 1) with ν at the

i-th place).

We give an example of this grading for a particular k-collection (in fact an operad)

in a later section.

We now introduce some notation necessary to describe distributive laws. Suppose

that U and V are two k-collections and let E := U ⊕ V . Then

• F(U, V ) := F(U ⊕ V )

• U ⊙V ⊂ F(E) is the collection generated by elements of the form γ(u; v1, . . . , vm),

u ∈ U(m) and vi ∈ V (ni), 1 ≤ i ≤ m.

• Let P = �U, S� and Q = �V, T � be two quadratic operads. Then V • U is the

subcollection of F(U, V ) generated by elements of the form γ(v; u, 1) or γ(v; 1, u).


Note that (V • U)(m) = 0 for m �= 3.

Now suppose that we have a map d : V • U �� U • V and let D := {z − d(z) :

z ∈ V • U} ⊂ F(U, V )(3) and R = P ◦D Q := �U ⊕ V, S ⊕ D ⊕ T �. The inclusion

F(U) ⊙ F(V ) ⊂ F(U, V ) induces a map ξ : P ⊙Q �� R of collections. If we consider

U⊕V as a graded collection with U in degree 0 and V in degree 1 then F(U, V ) is naturally

graded according to definition 4.1.1 and the relations S, T and D preserve this grading,

hence R is also graded. There is also a bigrading on P⊙Q where (P⊙Q)i,j is generated

by elements of the form γ(u; v1, . . . , vi+1), a ∈ Pi = P(i + 1) and bk ∈ Qjk= Q(jk + 1)

for 1 ≤ k ≤ i + 1 and j1 + . . . + ji+ = j. The map ξ preserves this bigrading (so

ξ((P ⊙Q)i,j) ⊂ Ri,j). We define ξi,j := ξ|(P⊙Q)i,j .

The following are the main definition and theorems for this section.

Definition 4.1.2. The map d : V • U �� U • V defines a distributive law if

ξi,j : (P ⊙Q)i,j�� Ri,j is an isomorphism for (i, j) ∈ {(1, 2), (2, 1)}.

Theorem 4.1.3. Suppose d is a distributive law. Then the map ξi,j : (P ⊙Q)i,j��Ri,j

is an isomorphism for all (i, j).

Theorem 4.1.4. Let R = �U ⊕ V, S ⊕D ⊕ T � be an operad with a distributive law and

let P = �U, S� and Q = �V, T �. If the operads P and Q are Koszul, then R is Koszul as

well.

The proofs of these theorems and more information about distributive laws can be

found in [7]

We wish to apply theorem 4.1.4 to the opeards involved in describing the operads con-

trolling the Lie 2 and commutative 2-algebras. We first check the condition of definition

4.1.2 for the case of the operad describing hemistrict Lie 2-algebras.

Recall in the Lie 2-case we have two operations [−,−], which satisfies the Leibniz

identity, and �−,−�, which is skew symmetric and all iterated operations are zero (so as


a quadratic operad its defining ternary relations would be R = (�−, �−,−��, ��−,−�,−�)).

We call the first operad P and the second operad Q (as in chapter 2) for the remainder

of the section. When describing a semistrict Lie-2 algebra we mixed the above operations

and supposed that the new operations we obtained satisfied the relations

[a, �b, c�] = �[a, b], c� − �b, [a, c]� (4.1)

[�a, b�, c] = 0 (4.2)

In terms of the notation above letting µ = [−,−] and ν = �−,−� we define a map

d :P(2) • Q(2) �� Q(2) • P(2)

γ(µ; 1, ν) �−→ γ(ν; µ, 1) + (γ(ν; 1, µ))1,2

γ(µ; ν, 1) �−→ 0

The goal next is to show that d defines a distributive law, i.e. that the maps ξi,j

for (i, j) ∈ {(1, 2), (2, 1)} in definition 4.1.2 are isomorphisms. Let P(2) = U and let

Q(2) = V .

First we recall the bigrading on F(U, V ), which in turn describes the bigrading on R.

The grading is defined recursively with F1,0 = U and F0,1 = V . Then elements of F1,1 are

of the form [a, �b, c�], [�a, b�, c], �[a, b], c�, . . . etc. The elements of F2,0 are generated by

[[a, b], c], [a, [b, c]], . . . (permutations) . . . and the elements of F0,2 are generated similarly

with �−,−�. Finally the elements of F1,2 and F2,1 are the quaternary operations in which

both brackets occur, for example [[a, b], �c, d�], ��a, [b, c]�, d�, [a, �[b, c], d�], . . . etc.

Next note that it is immediate that the maps ξ1,2 and ξ2,1 are surjective however

the fact that they are injective needs to be checked. This amounts to showing that an

operation a ∈ Fi(U)⊙Fj(V ) ⊂ Fi,j(U, V ) for (i, j) ∈ {(1, 2), (2, 1)} is zero mod (S,D, T )


if and only if it is zero mod (S, T ), where S and T are the ternary relations defining

P and Q. To ensure this condition we only need to check the following. Suppose we

have an element η ∈ Ri,j, (i, j) ∈ {(1, 2), (2, 1)} and that through some rearranging using

relations from S, T and D we get two expressions η1 and η2. Of course η1 = η2 when

thought of as elements of Ri,j but for the map ξi,j to be an isomorphism we need to check

that indeed η1 = η2 when thought of as elements of (P ⊙Q)i,j.

First we consider the operation [[a, b], �c, d�]. We expand this expression first using

the Leibniz rule twice and then equation (4.1) to obtain

[[a, b], �c, d�] = [a, [b, �c, d�]] + [b, [a, �c, d�]]

= [a, �[b, c], d�] + [a, �c, [b, d]�] + [b, �[a, c], d�] + [b, �c, [a, d]�]

= �[a, [b, c]], d� − �[b, c], [a, d]�+ �[a, c], [b, d]� − �c, [a, [b, d]]�

+ �[b, [a, c]], d� − �[a, c], [b, d]�+ �[b, c], [a, d]� − �c, [b, [a, d]]�

= �[a, [b, c]], d�+ �c, [a, [b, d]]� − �[b, [a, c]], d� − �c, [b, [a, d]]�

We could also expand [[a, b], �c, d�] first using equation (4.1) and then Leibniz which

would give

[[a, b], �c, d�] = �[[a, b], c], d�+ �c, [[a, b], d]�

= �[a, [b, c]], d� − �[b, [a, c]], d�+ �c, [a, [b, d]]� − �c, [b, [a, d]]�

Hence both ways of rearranging our original expression give the same result. Now consider

the operation [[a, �b, c�], d]. Again we first expand this expression using the Leibniz rule


and then use equation (4.2)

[[a, �b, c�], d] = [a, [�b, c�, d]] + [�b, c�, [a, d]]

= [a, 0]− 0

= 0

If we use equation (4.1) first and then equation (4.2) we get

[[a, �b, c�], d] = [�[a, b], c�, d] + [�b, [a, c]�, d]

= 0

in a similar manner both ways to expand the expression [[�a, b�, c], d] lead to zero.

This shows that equations (4.1) and (4.2) are compatible with the Leibniz relations

given by S. Since �a, �b, c�� = ��a, b�, c� = 0 equations (4.1) and (4.2) are automatically

compatible with T . The last thing we have to check is that both ways of expanding the

operation [�a, b�, �c, d�] are the same, so that no new relations are introduced in this way

by D. Expanding using equation (4.1) and then applying equation (4.2) we get

[�a, b�, �c, d�] = �[�a, b�, c], d� − �c, [�a, b�, d]�

= �0, d� − �c, 0�

= 0

Alternatively we could have applied equation (4.2) right away to get [�a, b�, �c, d�] = 0.

The preceding discussion shows that d does define a distributive law. Note that by

theorem (4.1.3) we can be sure that we have not introduced any new relations in R for

n ≥ 2, that is, loosely speaking, any way of rewriting a bracketing leads to the same

expression. Finally we obtain the main result of this section


Theorem 4.1.5. The operad given by the leibniz bracket µ = [−,−] and the skew-

symmetric bracket ν = �−,−� satisfying γ(ν; 1, ν) = γ(ν; ν, 1) = 0 and subject to the

relations

γ(µ; 1, ν) = γ(ν; µ, 1) + (γ(ν; 1, µ))1,2

γ(µ; ν, 1) = 0

is Koszul.

Proof. The operad described is the one given by R in the preceding discussion. The fact

that d is a distributive law and that P and Q are Koszul along with theorem 4.1.4 give

the result.

In a similar manner we can show that the operad controlling commutative 2-algebras

is Koszul. What needs to be checked is the fact that the Poisson relation

[a, bc] = [a, b]c + b[a, c]

defines a distributive law. This can be readily checked and is also in [7].

4.2 The Spectral Sequence of a Double Complex

The following section is for the most part a list of facts about spectral sequences. The

results are technical in nature and as a result the section can be skipped without loosing

much of this papers content.

The technical detail to verify is the following. We have an operad R = (R, d) con-

sisting of some operations (coming from R(n), n ≥ 2) and a differential d. We know

that R (the operad with no differential) is Koszul and we want to conclude R = (R, d) is

Koszul. Since D(R) is obtained from the total complex of a double complex and that R

is Koszul tells us about the cohomology of the rows we naturally use spectral sequences.


Definition 4.2.1. A double complex M is a bigraded object M =�

p,q∈Z Mp,q with

differentials d : Mp,q �� Mp+1,q and δ : M

p,q �� Mp,q+1 such that d2 = δ

2 = dδ + δd = 0.

To each double complex M we have a complex Tot(M) called the total complex defined

by Totn(M) =�

p+q=n Mp,q with differential D = d + δ

Definition 4.2.2. For a fixed integer r0 a spectral sequence consists of the following

• For each r ≥ r0 a bigraded object Er =�

p,q∈Z Ep,qr

• Differentials dr : Er�� Er such that dr(Ep,q

r ) ⊂ Ep+r,q−r+1 and H(Er) = Er+1

Definition 4.2.3. Given a double complex (M, d, δ) we have two canonical filtrations

Fp1 (Totn(M)) =

�

r+s=nr≥p

Mr,s and F

q1 (Totn(M)) =

�

r+s=ns≥q

Mr,s

Definition 4.2.4. Suppose Ep,qr is a spectral sequence and that for every pair (p0, q0)

we have that Ep0,q0r stabilizes as r �� ∞ and denote this value by E

p,q∞ . Suppose that

{Hn}n∈Z is a collection of objects with finite filtrations 0 ⊂ F1nH

n ⊂ . . . ⊂ Ftnn H

n = Hn.

Then Ep,qr converges to H

·, we write Ep,qr ⇒ H

p+q, if

Ep,q∞ = F

pH

p+q/F

p+1H

p+q = GrpH

p+q

The following two lemmas are standard results about spectral sequences.

Lemma 4.2.5. Suppose K· is a filtered complex. Then there exists a spectral sequence

Ep,qr such that

• Ep,q0 = Grp(Kp+q)

• Ep,q1 = H

p+q(GrpK

·)

• Ep,qr = Grp(Hp+q

K·) for sufficiently large r

where Gr(K)n = Kn/K

n−1 is the graded complex associated to the filtered complex K.


Let 1E and 2

E be the spectral sequences obtained from the above lemma and the

filtered complexes F1 and F2 respectively. Then we see immediately that 1E

p,q0 = M

p,q.

The less immediate fact that can be seen in the proof of the above lemma is that the

map 1E

p,q0

�� 1Ep,q+10 arising from the construction of the spectral sequence 1

E is the

differential δ. Hence we get 1E

p,q1 = H

qδ (M

p,·). The other differential 1E

p,q1

�� 1Ep+1,q1 is

induced by d and so 1E

p,q2 = H

pd(Hq

δ (M)). In summary we have the following lemma.

Lemma 4.2.6. Let M be a double complex with corresponding spectral sequences 1E and

2E as in the preceding lemma. We have the following equalities

1E

p,q0 = M

p,q 1E

p,q1 = H

qδ (M

p,·) 1E

p,q2 = H

pd(Hq

δ (M))

2E

p,q0 = M

p,q 2E

p,q1 = H

qd(M

·,p) 2E

p,q2 = H

pδ (Hq

d(M))

If M is 1st quadrant (Mp,q = 0 unless p, q > 0) or third quadrant (Mp,q = 0 unless

p, q < 0) then 1E

p,qr ⇒ H

p+q(Tot(M)) and 2E

p,qr ⇒ H

p+q(Tot(M)).

4.3 The Koszulness of R

Notation 4.3.1. We will let k ·µ denote the k vector space spanned by the operation µ.

We begin by describing the dg-operad R = (R, d) explicitly.

For each n, R(n) is a dg-vector space. These complexes for small values of n are given

below

degree -1 degree 0

R(2):

k · �a, b� k · [a, b]⊕ k · [b, a]��


R(3):

k · �a, [b, c]� ⊕ k · �a, [c, b]��

σ∈Σ3k · [aσ(1), [aσ(2), aσ(3)]]��

R(4):�

σ,τ∈Σ2k · �[aσ(1), aσ(2)], [bτ(1), bτ(2)]�

�σ∈Σ4

k · [aσ(1), [aσ(2), [aσ(3), aσ(4)]]]��

�σ∈Σ3

k · �[aσ(1), [aσ(2), aσ(3)]], b�

�σ∈Σ4

k · [aσ(1), [aσ(2), [aσ(3), aσ(4)]]]��

where all arrows are induced by the differential d. One can see that in general the complex

given by R(n) is of the form

�

p≤q:p+q=n,σ∈Σp,τ∈Σq

k · �µσ, µτ � ��

ω∈Σn

k · µω

where for σ ∈ Σn we have

µσ(a1, . . . , an) = [aσ(1), [aσ(2), [. . . , [aσ(n−1), aσ(n)]] . . .]]

To get a feeling for what follows we will now describe part of the complex ΣR.

Consider ΣR(3) and the following tree T

v1��

��

�

v1��

��

v1

v2

e

��

��

v2��

��

��

��

�

v2

��

Then R(T ) = ⊗vi∈V (T )R(In(vi)) = R(2)⊗R(2) which is the complex formed by the

right column of the following figure, which is the dual picture


(k · �[a, b], c�)⊕ (k · [�a, b�, c]) (k · �a, b� ⊗ k · [a, b])⊕ (k · [a, b]⊗ �a, b�)�� δ∗

k · ��a, b�, c�

(k · �[a, b], c�)⊕ (k · [�a, b�, c])

d

��

k · ��a, b�, c� k · �a, b� ⊗ �a, b�� δ∗k · �a, b� ⊗ �a, b�

(k · �a, b� ⊗ k · [a, b])⊕ (k · [a, b]⊗ �a, b�)

d

��

k · [[a, b], c] k · [a, b]⊗ k · [a, b]�� δ∗

(k · �[a, b], c�)⊕ (k · [�a, b�, c])

k · [[a, b], c]

d

��

(k · �[a, b], c�)⊕ (k · [�a, b�, c]) (k · �a, b� ⊗ k · [a, b])⊕ (k · [a, b]⊗ �a, b�)δ∗ (k · �a, b� ⊗ k · [a, b])⊕ (k · [a, b]⊗ �a, b�)

k · [a, b]⊗ k · [a, b]

d

��

The left hand column forms part of the complex corresponding to the tree T� = T/e.

As the above picture shows it is clear that the map δ has no nonzero components

that lower the degree in the columns. Because of this the fact that Hn(ΣR)(m) = 0 for

n > 0 implies that Hp,q0δ (ΣR) = 0 for q0 > 0.

Next we use some results from section 4.2 on spectral sequences to show that R is

Koszul. We will need to make reference to the double complex (used to define the cobar

construction) preceding theorem 1.3.4. For a given operad P and a value n ≥ 0 call this

double complex ΣP (n) 1. Let R be the operad R forgetting the differential d. In this

case the complex ΣR is no longer a double complex and the results from section 4.1 tell

us that Hqδ (ΣR) ∼= R

!, so in particular H

qδ (ΣR) = 0 for q > 0. Now if we add in the

differential d to obtain the operad R we have that

Hpd(R!) ∼= H

pd(Hq

δ (ΣR)) ∼= Hp+q(Tot(ΣR)) = H

p(D(R))

The above discussion proves the main theorem of this chapter.

Theorem 4.3.2. The operad R is Koszul.

Proof. See the discussion preceding the statement of the theorem.

1This notation is often used to denote whats called the suspension of P, which is related to thiscomplex but is not identical to it.

Conclusion

From this point there are two problems to be investigated. The first is to use the fact

that a Courant algebroid is a homotopy algebra over an operad to apply the notion of

a module over an algebra over an operad to Courant algebroids. Next we would like to

extend the fact that Poiss = Lie ◦ Com to arrive at the definition of a Poisson 2-algebra

by gluing together a Lie 2-algebra and a Commutative 2-algebra together in a way that

is not yet clear. The Lie and commutative 2-algebras described in this paper both have

categorical descriptions and we wish to have a categorical description of Poisson algebras.

In this paper we introduce the new notion of a commutative 2-algebra however there

is more to say about these structures. A full description of the 2-category of commutative

2-algebras should be defined including morphisms between commutative 2-algebras, 2-

morphisms between those, isomorphisms, etc. One could also seek a classification of

skeletal commutative 2-algebras similar to that given for skeletal Lie 2-algebras in [9].

There are other connections between Lie algebras and commutative algebras for which

we we would like to find counterparts in the 2-algebra setting. For instance the derivations

of a commutative algebra form a Lie algebra so perhaps there is a notion of a 2-derivation

such that the 2-derivations of a commutative 2-algebra form a Lie 2-algebra. We also

know that Lie algebras and commutative algebras are quadratic dual to each other so

this begs the question of whether there is a notion of duality for 2-algebras for which Lie

2-algebras and commutative 2-algebras are dual.

63

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