Kac-Moody Algebras and Applications

Owen Barrett

December 24, 2014

Abstract

This article is an introduction to the theory of Kac-Moody algebras: their genesis, their con-struction, basic theorems concerning them, and some of their applications. We first recordsome of the classical theory, since the Kac-Moody construction generalizes the theory ofsimple finite-dimensional Lie algebras in a closely analogous way. We then introduce theconstruction and properties of Kac-Moody algebras with an eye to drawing natural connec-tions to the classical theory. Last, we discuss some physical applications of Kac-Moody alge-bras, including the Sugawara and Virosoro coset constructions, which are basic to conformalfield theory.

Contents

1 Introduction 2

2 Finite-dimensional Lie algebras 32.1 Nilpotency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Semisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4.2 The Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.3 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.4 The Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.5 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.6 Complex root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 The structure of semisimple Lie algebras . . . . . . . . . . . . . . . . . . . . . 72.5.1 Cartan subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.2 Decomposition of g . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.3 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6 Linear representations of complex semisimple Lie algebras . . . . . . . . . . . 92.6.1 Weights and primitive elements . . . . . . . . . . . . . . . . . . . . . 9

2.7 Irreducible modules with a highest weight . . . . . . . . . . . . . . . . . . . . 92.8 Finite-dimensional modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Kac-Moody algebras 103.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Construction of the auxiliary Lie algebra . . . . . . . . . . . . . . . . . 113.1.2 Construction of the Kac-Moody algebra . . . . . . . . . . . . . . . . . 123.1.3 Root space of the Kac-Moody algebra g(A) . . . . . . . . . . . . . . . 12

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3.2 Invariant bilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Casimir element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Integrable representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5.1 Weyl chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.6 Classification of generalized Cartan matrices . . . . . . . . . . . . . . . . . . . 16

3.6.1 Kac-Moody algebras of finite type . . . . . . . . . . . . . . . . . . . . 173.7 Root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.7.1 Real roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.7.2 Imaginary roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7.3 Hyperbolic type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.8 Kac-Moody algebras of affine type . . . . . . . . . . . . . . . . . . . . . . . . . 203.8.1 The underlying finite-dimensional Lie algebra . . . . . . . . . . . . . . 203.8.2 Relating g and g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Highest-weight modules over Kac-Moody algebras 224.1 Inductive Verma construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Primitive vectors and weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Complete reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Formal characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5 Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Physical applications of Kac-Moody algebras 285.1 Sugawara construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Virasoro algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Virasoro operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4 Coset Virasoro construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1 Introduction

Élie Cartan, in his 1894 thesis, completed the classification of the finite-dimensional simple Liealgebras over C. Cartan's thesis is a convenient marker for the start of over a half-century ofresearch on the structure and representation theory of finite-dimensional simple complex Liealgebras. This effort, marshalled into existence by Lie, Killing, Cartan, and Weyl, was contin-ued and extended by the work of Chevalley, Harish-Chandra, and Jacobson, and culminated inSerre's Theorem [Ser66], which gives a presentation for all finite-dimensional simple complexLie algebras.

In the mid-1960s, Robert Moody and Victor Kac independently introduced a collection of Liealgebras that did not belong to Cartan's classification. Moody, at the University of Toronto, con-structed Lie algebras from a generalization of the Cartan matrix, emphasizing the algebras thatwould come to be known as affine Kac-Moody Lie algebras, while Kac, at Moscow State Univer-sity, began by studying simple graded Lie algebras of finite growth. In this way, Moody and Kacindependently introduced a new class of generally infinite-dimensional, almost always simpleLie algebras that would come to be known as Kac-Moody Lie algebras.

The first major application of the infinite-dimensional Lie algebras of Kac and Moody camein 1972 with Macdonald's 1972 discovery [Mac71] of an affine analogue to Weyl's denominatorformula. Specializing Macdonald's formula to particular affine root systems yielded previously-known identities for modular forms such as the Dedekind η-function and the Ramanujan τ-function. The arithmetic connection Macdonald established to affine Kac-Moody algebras was

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subsequently clarified and explicated independently by both Kac and Moody, but only markedthe beginning of the applications of these infinite-dimensional Lie algebras. It didn't take longfor the importance of Kac-Moody algebras to be recognized, or for many applications of them toboth pure mathematics and physics to be discovered.

We will structure this article as follows. We will first recall some basic results in the theoryof finite-dimensional Lie algebras so as to ease the introduction of their infinite-dimensionalrelatives. Next, we will define what is a Kac-Moody algebra, and present some fundamentaltheorems concerning these algebras. With the basic theory established, we will turn to someapplications of Kac-Moody algebras to physics.

We include proofs for none of the results quoted in this review, but we do take care to cite thereferences where appropriate proofs may be found.

2 Finite-dimensional Lie algebras

We collect in this section some basic results pertaining to the structure and representation the-ory of finite-dimensional Lie algebras. This is the classical theory due to Lie, Killing, Cartan,Chevalley, Harish-Chandra, and Jacobson. The justification for reproducing the results in thefinite-dimensional case is that both Moody and Kac worked initially to generalize the methodsof Harish-Chandra and Chevalley as presented by Jacobson in [Jac62, Chapter 7]. For this reason,the general outline of the approach to studying the structure of Kac-Moody algebras resemblesthat of the approach to studying the finite-dimensional case.

We follow the presentation of Serre [Ser00] in this section. We suppose throughout this sec-tion that g is a finite-dimensional Lie algebra over the field of complex numbers, though we willomit both ‘finite-dimensional’ and ‘complex’ in the statements that follow. The Lie bracket ofx and y is denoted by [x, y] and the map y 7→ [x, y] is denoted ad x. For basic definitionssuch as that of an ideal of a Lie algebra (and for proofs of some statements given without proofin [Ser00]), c.f. Humphreys [Hum72].

2.1 Nilpotency

The lower central series of g is the descending series (Cng)n>1 of ideals of g defined by the for-mulæ

C1g = g (2.1)

Cng = [g, Cn−1g] ifn > 2. (2.2)

Definition. A Lie algebra g is said to be nilpotent if there exists an integern such thatCng = 0. g isnilpotent of class r ifCr+1g = 0.

Proposition 2.1. The following conditions are equivalent.

(i) g is nilpotent of class6 r.

(ii) There is a descending series of ideals

g = a0 ⊃ a1 ⊃ · · · ⊃ ar = 0 (2.3)

such that [g, ai] ⊂ ai+1 for 0 6 i 6 r− 1.

Recall that the center of a Lie algebra is the set of x ∈ g such that [x, y] = 0 for all y ∈ g. It is anabelian ideal of g. We denote the center of g byZ(g).

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Proposition 2.2. Let g be a Lie algebra and let a ⊂ Z(g) be an ideal. Then

g is nilpotent ⇔ g/a is nilpotent. (2.4)

We now present the basic results pertaining to nilpotent Lie algebras.

Theorem 2.3 (Engel). For a Lie algebra g to be nilpotent, it is necessary and sufficient for ad x to benilpotent∀x ∈ g.

Theorem 2.4. Letφ : g → End(V) be a linear representation of a Lie algebra g on a nonzero finite-dimensional vector spaceV . Suppose thatφ(x) is nilpotent∀x ∈ g. Then∃0 6= v ∈ Vwhich is invariantunder g.

2.2 Solvability

The derived series of g is the descending series (Dng)n>1 of ideals of g defined by the formulæ

D1g = g (2.5)

Dng = [Dn−1g, Dn−1g] ifn > 2. (2.6)

Definition. A Lie algebrag is said to be solvable if∃n ∈ N s.t.Dng = 0. g is solvable of derived length6 r ifDr+1g = 0.

n.b. Every nilpotent algebra is solvable.

Proposition 2.5. The following conditions are equivalent.

(i) g is solvable of derived length6 r.

(ii) There is a descending central series of ideals of g:

g = a0 ⊃ a1 ⊃ · · · ⊃ ar = 0 (2.7)

such that [ai, ai] ⊂ ai+1 for 0 6 i 6 r− 1 (⇒ successive quotientsai/ai+1 are abelian).

In analogy with Engel's theorem, we now record Lie's theorem.

Theorem 2.6 (Lie). Letφ : g→ End(V) be a finite-dimensional linear representation of g. Ifg is solv-able, there is a flagDofV such thatφ(g) ⊂ b(D), whereb(D) is the Borel algebra ofD (the subalgebraof EndV that stabilizes the flag).

Theorem 2.7. If g is solvable, all simple g-modules are one-dimensional.

Theorem 2.7 leads us to conclude that the representation theory of solvable Lie algebras is veryboring. The following section will make this statement more transparent.

2.3 Semisimplicity

Recall that the radical rad g is largest solvable ideal in g. rad g exists because if a and b are solv-able ideals of g, the ideal a + b is also solvable since it is an extension of b/(a ∩ b) = (a + b)/a

by a.

Theorem 2.8. Let g be a Lie algebra.

(a) rad(g/ rad g) = 0.

(b) There is a Lie subalgebra s ⊂ g that is a complement for rad g; i.e. such that the projection s →g/ rad g is an isomorphism.

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In this way, we may easily strip away the radical of a Lie algebra g and are left with a Lie algebrawith no nontrivial solvable ideal.

Definition. A Lie algebra g is semisimple if rad g = 0.

Thus Theorem 2.8 says that any Lie algebra g is a semidirect product of a semisimple algebraand a solvable ideal (the so-called ‘Levi decomposition’ of g). The above justifies the restrictionof study to semisimple Lie algebras. The following definition and theorem imply that we mayrestrict our concern even further.

Definition. A Lie algebra s is said to be simple if

(a) it is not abelian,

(b) its only ideals are 0 and s.

Theorem 2.9. A Lie algebrag is semisimple iff it is isomorphic to a product of simple algebras. Indeed, letgbe a semisimple Lie algebra and (ai)be its minimal nonzero ideals. The idealsai are simple Lie algebras,and g can be identified with their product.

Theorem 2.9, together with the above discussion, shows that the Cartan classification of simplecomplex finite-dimensional Lie algebras allows us to classify all finite-dimensional Lie algebras.Hermann Weyl proved, via his ‘unitarian trick,’ the following theorem about representations ofsemisimple algebras.

Theorem 2.10. Every finite-dimensional linear representation of a semisimple algebra is completely re-ducible.

2.4 Root systems

Before turning to the structure theory of complex semisimple Lie algebras, we need to estab-lish some basic definitions and results pertaining to symmetries of vector spaces. This theorycontains the concrete ideas (most notably the Cartan matrix) that are generalized in the formu-lation of Kac-Moody algebras, since, as we will see in section 2.5, the theory of symmetries ofvector spaces contains the information about the decomposition of complex semisimple Lie al-gebras.

2.4.1 Symmetries

We begin by introducing the notion of a real root system, which, as we will see, is all we needto study the complex root systems that arise in the structure theory of semisimple complex Liealgebras. Let V be a (real) vector space and 0 6= α ∈ V . We begin by defining a symmetry withvectorα to be any s ∈ AutV that satisfies the conditions

(i) s(α) = −α.

(ii) The setHs ⊂ V of s-invariant elements is a hyperplane ofV .

The setHs is actually a complement for the line Rα, and s is an automorphism of order 2. Thesymmetry is totally determined by the choice ofRα and ofHs. If we denote the dual space ofVbyV∗ and letα∗ be the unique element ofV∗ which vanishes onHs and takes the value 2 onα.Then

s(x) = x− 〈α∗, x〉α ∀x ∈ V. (2.8)

Since EndV ' V∗ ⊗ V , we may instead write

s = 1− α∗ ⊗ α. (2.9)

We make the following

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Definition. LetV be a finite-dimensional vector space. A subsetR ⊂ V is called a root system if

(1) R is finite, spansV , and does not contain 0.

(2) ∀α ∈ R there is a unique symmetry sα = 1− α∗ ⊗ αwith vectorαwhich leavesR invariant.

(3) Ifα,β ∈ R, sα(β) − β is an integer multiple ofα.

We say the rank of R is dimV and the elements of R are called the roots ofV (relative to R). WesayR is reduced if ∀α ∈ R,α and −α are the only roots proportional toα. The elementα∗ ∈ V∗as in (2.9) is called the inverse root ofα.

2.4.2 The Weyl group

Definition. LetRbe a root system in a vector spaceV . The Weyl group ofR is the subgroupW ⊂ GL(V)generated by the symmetries sα, α ∈ R.

Given any positive definite symmetric bilinear form on V , since W is finite we may average itoverW 's action onV to obtain aW-invariant form.

Proposition 2.11. LetR be a root system inV . There is a positive definite symmetric bilinear form ( , )

onV invariant under the Weyl groupW ofR.

The choice of ( , )givesV a Euclidean structure on whichW acts by orthogonal transformations.We may re-write sα as

sα(x) = x− 2(x, α)

(α,α)α ∀x ∈ V (2.10)

and introduce the notion of relative position of two rootsα,β ∈ R by the formula

n(β,α) = 2(α,β)

(α,α). (2.11)

2.4.3 Bases

Definition. A subsetS ⊂ R is called a base forR if

(i) S is a basis for the vector spaceV .

(ii) Eachβ ∈ R can be written as a linear combination

β =∑α∈S

mαα, (2.12)

where the coefficientsmα are integers with the same sign (i.e. all non-negative or all non-positive).

The elements of S are also sometimes called ‘simple roots.’ Every root system admits a base.LettingR+ denote the set of roots which are linear combinations with non-negative integer co-efficients of elements of S, we say an element ofR+ is a positive root (with respect to S).

Proposition 2.12. Every positive rootβ can be written as

β = α1 + · · ·+ αk withαi ∈ S, (2.13)

in such a way that the partial sums

α1 + · · ·+ αh, 1 6 h 6 k, (2.14)

are all roots.

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Theorem 2.13. SupposeR is reduced and letW be the Weyl group ofR.

(a) ∀t ∈ V∗∃w ∈W : 〈w(t), α〉 > 0∀α ∈ S.

(b) IfS ′ is a base forR,∃w ∈W such thatw(S ′) = S.

(c) For eachβ ∈ R,∃w ∈W such thatw(β) ∈ S.

(d) The groupW is generated by the symmetries sα,α ∈ S.

2.4.4 The Cartan matrix

We are now prepared to make the following definition, which is the jumping-off point for thestudy of Kac-Moody algebras.

Definition. The Cartan matrix ofR (with respect to a fixed baseS) is the matrix (n(α,β))α,β∈S.

The following proposition shows that the Cartan matrix encodes all the information containedin a reduced root system.

Proposition 2.14. A reduced root system is determined up to isomorphism by its Cartan matrix. Thatis, let R ′ be a reduced root system in a vector space V ′, let S ′ be a base for R ′, and letφ : S → S ′ bea bijection such that n(φ(α), φ(β)) = n(α,β) for all α,β ∈ S. If R is reduced, then there is anisomorphism f : V → V ′which is an extension ofφ and mapsR ontoR ′.

That is, any bijection of two bases that preserves the Cartan matrix can be extended linearly toa vector space isomorphism.

2.4.5 Irreducibility

Proposition 2.15. Suppose thatV = V1 ⊕ V2 and thatR ⊂ V1 ∪ V2. PutRi := R ∩ Vi. Then

(a) V1 andV2 are orthogonal.

(b) Ri is a root system inVi.

One says thatR is the sum of the subsystemsRi. If this can happen only trivially (that is, withV1orV2 equal to 0), and ifV 6= 0, thenR is said to be irreducible.

Proposition 2.16. Every root system is a sum of irreducible systems.

2.4.6 Complex root systems

The definition of complex root system is identical to the definition in the real case after replacingthe real vector space V with a complex one. Actually, it is a theorem that every complex rootsystem may be obtained by extension of scalars of some real root system. Namely, if R is a rootsystem in a complex vector space V , then we may form the R-subspace V0 ⊂ V . Then R is a(real) root system in V0, the canonical mapping V0 ⊗ C → V is an isomorphism, and ∀α ∈ Rthe symmetry sα of V is a linear extension of the corresponding symmetry in V0. This showsthat the theory of complex root systems reduces totally to the real theory.

2.5 The structure of semisimple Lie algebras

We now turn to the structure theory of semisimple complex Lie algebras. As throughout thissection, g is a finite-dimensional Lie algebra over C.

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2.5.1 Cartan subalgebras

Let g be a Lie algebra (not necessarily semisimple) and a ⊂ g a subalgebra. Recall that the nor-malizer of a in g is defined to be the set n(a) of all x ∈ g such that ad(x)(a) ⊂ a; it is the largestsubalgebra of g that contains a and in which a is an ideal.

Definition. A subalgebra h ⊂ g is called a Cartan subalgebra of g if it satisfies the following twoconditions.

(a) h is nilpotent.

(b) h = n(h).

If h is a Cartan subalgebra of a semisimple Lie algebra g, then h is abelian and the centralizer of his h. It follows that h is a maximal abelian subalgebra of g.

2.5.2 Decomposition of g

For the rest of this section, g is assumed to be semisimple and h is assumed to be a Cartan sub-algebra of g.

Ifα ∈ h∗, we let gα be the eigensubspace of g; i.e.

gα := x ∈ g : [H, x] = α(H)x∀H ∈ h . (2.15)

An element of gα is said to have weightα. In particular, g0 is the set of elements in g that com-mute with h; since g is semisimple, h is maximal abelian subalgebra of g and so actually g0 = h.

Theorem 2.17. One has g = h⊕α∈R g

α.

Any element 0 6= α ∈ h∗ s.t. gα 6= 0 is called a root of g (relative to h); the set of roots will bedenotedR. This suggestive terminology is not a coincidence.

Theorem 2.18. (a) R is a complex reduced root system inh∗.

(b) Letα ∈ R. Then dim gα = 1 = dim hα, where hα := [gα, g−α]. ∃!Hα ∈ hα s.t. α(Hα) =2; it is the inverse root ofα.

(c) The subspaces gα and gβ are orthogonal ifα+ β 6= 0. The subspaces gα and g−α are dual withrespect to ( , ). The restriction of ( , ) toh is nondegenerate.

2.5.3 Existence and uniqueness

We conclude our remarks on the structure of complex semisimple Lie algebras with the follow-ing theorems, which will make clear the significance of the root system (encoded in the Cartanmatrix) associated to the Cartan decomposition of g.

Theorem 2.19 (Existence). Let R be a reduced root system. There exists a semisimple Lie algebra g

whose root system is isomorphic toR.

Theorem 2.20 (Uniqueness). Two semisimple Lie algebras corresponding to isomorphic root systemsare isomorphic. More precisely, let g, g ′ be semismple Lie algebras, h, h ′ respective Cartan subalgebrasof g, g ′, S, S ′ bases for the corresponding root systems, and r : S → S ′ a bijection sending the Cartanmatrix ofS to that ofS ′. For each i ∈ S, j ∈ S ′, letXi, X ′i be nonzero elements ofgi andg ′j, respectively.Then there is a unique isomorphism of Lie algebras f : g → g ′ sendingHi toH ′r(i) andXi toX ′r(i) forall i ∈ S.

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2.6 Linear representations of complex semisimple Lie algebras

After having described the structure of a complex semisimple Lie algebra g, we now pass to itslinear representations. Let h ⊂ g be a Cartan subalgebra of g and R the corresponding rootsystem. We fix a base S = α1, . . . , αn of R and denote by R+ the set of positive roots withrespect to S. We recall the definition ofHα, the inverse root ofα.

Theorem 2.21. ∀α ∈ R∃Xα ∈ gα such that

[Xα, X−α] = Hα ∀α ∈ R. (2.16)

Fix suchXα ∈ gα, Yα ∈ g−α so that [Xα, Yα] = Hα. Whenα is one of the simple rootsαi, wewriteXi, Yi, Hi instead ofXαi , etc. We put n :=

∑α>0 g

α, n− :=∑α<0 g

α, and b := h⊕ n.

2.6.1 Weights and primitive elements

LetV be a g-module of possibly infinite dimension and letω ∈ h∗ be a linear form on h. LetVωdenote the vector subspace of all v ∈ V such thatHv = ω(H)v∀H ∈ h. An element of Vω issaid to have weightω. The dimension of Vω is called the multiplicity ofω in V . If Vω 6= 0,ωcalled a weight ofV .

Proposition 2.22. (a) One has gαVω ⊂ Vω+α ifω ∈ h∗,α ∈ R.

(b) The sumV ′ :=∑ω V

ω is direct and a g-submodule ofV .

One says that some v ∈ V is a primitive element of weightω of it satisfies the conditions

(i) v 6= 0 has weightω.

(ii) Xαv = 0∀α ∈ R+ (equivalently, ∀α ∈ S).

Given a primitive element of weightω, it generates an indecomposableg-module in whichω isa weight of multiplicity 1.

2.7 Irreduciblemoduleswith ahighestweight

The notion of a highest weight representation of g is basic and is generalized to the representa-tion theory of Kac-Moody algebras. We present two basic results here.

Theorem 2.23. LetV be an irreducible g-module containing a primitive element v of weightω. Then

(a) v is the only primitive element ofV up to scaling; its weightω is called the ‘highest weight’ ofV .

(b) The weightsπ ofV have the form

π = ω−∑

miαi withmi ∈ N. (2.17)

They have finite multiplicity; in particular,ω has multiplicity 1. One hasV =∑π V

π.

(c) For two irreducible g-modulesV1 andV2 with highest weightsω1 andω2 to be isomorphic, it isnecessary and sufficient forω1 = ω2.

Remark. One can give examples of irreducible g-modules with no highest weight. These modules arenecessarily infinite-dimensional.

Theorem 2.24. For eachω ∈ h∗, there exists an irreducibleg-module with highest weight equal toω.

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Remark. The above theorems establish a bijection between the elements of h∗ and the classes of irre-ducible g-modules with highest weight.

2.8 Finite-dimensionalmodules

Proposition 2.25. LetV be a finite-dimensional g-module. Then

(a) V =∑π V

π.

(b) Ifπ is a weight ofV ,π(Hα) ∈ Z∀α ∈ R.

(c) IfV 6= 0,V contains a primitive element.

(d) IfV is generated by a primitive element, thenV is irreducible.

Theorem 2.26. Letω ∈ h∗ and letEω be an irreducibleg-module havingωas highest weight. ForEωto be finite-dimensional, it is necessary and sufficient that

∀α ∈ R+,ω(Hα) ∈ Z>0. (2.18)

3 Kac-Moody algebras

Having summarized the basics of the structure theory of semisimple Lie algebras overCand thebasics of their highest-weight linear representations, we may now proceed naturally to the for-mulation of their generalization. The key idea is to generalize the Cartan matrix of section 2.4.4;as the existence and uniqueness theorems show, every Cartan matrix encoding the data corre-sponding to a reduced root system determines a unique finite-dimensional semisimple com-plex Lie algebra. Therefore, generalizing the Cartan matrix provides a neat way to generalizethe Lie algebra.

Note that the formulation of the generalized Cartan matrix is not the only way to approachKac-Moody algebras. Early on, Kac considered arbitrary simpleZ-graded Lie algebrasg =

⊕j gj

of ‘finite growth,’ meaning dim gj grew only polynomially in j. Nevertheless, we will present theapproach of the generalized Cartan matrix here, following Kac [Kac94].

3.1 Basic definitions

We start with a complex n × nmatrixA = (aij)ni,j=1 of rank ` and we will associate with it a

complex Lie algebra g(A).

Definition. We callA a generalized Cartan matrix if it satisfies the following conditions:

aii = 2 for i = 1, . . . , n (C1)aij are nonpositive integers for i 6= j (C2)aij = 0 impliesaji = 0. (C3)

However, we may start with an even more general object than a generalized Cartan matrix; letnow A be any arbitrary matrix. A realization of A is a triple

(h, II, II∨

), where h is a complex

vector space, II = α1, . . . , αn ⊂ h∗ and II∨ =α∨1 , . . . , α

∨n

⊂ h are indexed subsets of h∗

and h, respectively, satisfying the conditions

both sets II and II∨ are linearly independent; (D1)⟨α∨i , αj

⟩= aij for i, j = 1, . . . , n; (D2)

n− ` = dim h − n. (D3)

Two realizations(h, II, II∨

)and

(h ′, II ′, II ′∨

)are called isomorphic if there exists a vector space

isomorphismφ : h→ h ′ such thatφ(II∨)= II ′∨ andφ∗ (II ′) = II.

10

Proposition 3.1. Fix ann × nmatrixA. Then there exists a realization ofA, and this realization isunique up to isomorphism.

Given two matricesA1 andA2 and their realizations(h1, II1, II

∨1

)and

(h2, II2, II

∨2

), we ob-

tain a realization of the direct sumA1 ⊕A2 of the two matrices as(h1 ⊕ h2, II1 × 0 ∪ 0× II2, II∨1 × 0, ∪ 0× II∨2

), (3.1)

which establishes the notion of direct sum of realizations. Therefore, we can talk about decom-posing realizations; a matrix A is called decomposable if, after reordering its indices (permutingthe rows and columns by the same permutation), A decomposes into a nontrivial direct sum.Hence every realization decomposes into a direct sum of indecomposable realizations.

We now introduce some terminology that strengthens the analogy with the finite-dimensionalcase. We call II the root base, II∨ the coroot base, and elements from II (resp. II∨) simple roots(resp. simple coroots). We also put

Q :=

n∑i=1

Zαi, Q+ :=

n∑i=1

Z+αi, (3.2)

and call the latticeQ the root lattice. Given anyα =∑i kiαi ∈ Q, the number htα :=

∑i ki

is called the height ofα. We put a partial order on h∗ by settingα > µ if λ− µ ∈ Q+.

3.1.1 Construction of the auxiliary Lie algebra

Given ann×n complex matrixA and a realization(h, II, II∨

)ofA, we first introduce an auxil-

iary Lie algebra g(A) with the generators ei, fi (i = 1, . . . , n) and h and the following definingrelations.

[ei, fj] = δijα∨i i, j = 1, . . . , n,

[h, h ′] = 0 h, h ′ ∈ h,

[h, ei] = 〈αi, h〉 ei[h, fi] = − 〈αi, h〉 fi i = 1, . . . , n;h ∈ h.

(DR)

The uniqueness of the realization guarantees that g(A)depends only onA. Denote by n+ (respn−) the subalgebra of g(A) generated by e1, . . . , en (resp. f1, . . . , fn). We now record the fol-lowing properties of the auxiliary Lie algebra g(A).

Theorem 3.2. (a) g(A) = n− ⊕ h⊕ n+ as vector spaces.

(b) n+ (resp. n−) is freely generated by e1, . . . , en (resp. f1, . . . , fn).

(c) The map ei 7→ −fi, fi 7→ −ei (i = 1, . . . , n),h 7→ −h (h ∈ h) can be uniquely extended toan involution of the Lie algebra g(A).

(d) With respect toh one has the root space decomposition

g(A) =( ⊕α∈Q+α6=0

g−α

)⊕ h⊕

(⊕α∈Qα 6=0

gα

), (3.3)

where gα = x ∈ g(A) : [h, x] = α(h)x. Additionally, dim gα < ∞, and gα ⊂ n± for±α ∈ Q+, α 6= 0.

(e) Among the ideals of g(A) intersecting h trivially, there exists a unique maximal ideal r. Further-more, there is a direct sum decomposition of ideals

r = (r ∩ n−)⊕ (r ∩ n+) . (3.4)

The analogy to the root space of a finite-dimensional semisimple Lie algebra is clear.

11

3.1.2 Construction of the Kac-Moody algebra

We now may proceed to construct the Lie algebra g(A) that is the main object of study. With(h, II, II∨

)as before and g(A) the auxiliary Lie algebra associated to this realization on gener-

ators ei, fi (i = 1, . . . , n), and h, and the definining relations (DR). By Theorem 3.2, the naturalmaph→ g(A) is an embedding. Let rbe the maximal ideal in g(A) intersectingh trivially, as inTheorem 3.2(e). Then, put

g(A) := g(A)/r. (3.5)

The matrixA is called the Cartan matrix of the Lie algebra g(A), andn is called the rank of g(A).We say

(g(A), h, II, II∨

)is the quadruple associated toA, and say two quadruples

(g(A), h, II, II∨

)and

(g(A ′), h ′, II ′, II ′∨

)are isomorphic if there exists a Lie algebra isomorphismφ : g(A) →

g(A ′) such thatφ(h) = h ′,φ(II∨)= II ′∨, andφ∗ (II ′) = II. Recall the conditions (C1–C3)

on a generalized Cartan matrix. We are in a position to make the following

Definition. The Lie algebra g(A) whose Cartan matrix is a generalized Cartan matrix is called a Kac-Moody algebra.

We re-use the notationei, fi, h for the images of these generators ing(A)after the quotient by r.In analogy with the classical finite-dimensional case, we call the subalgebrah ⊂ g(A) the Cartansubalgebra and the elements ei, fi the Chevalley generators of g(A). The Chevalley generatorsgenerate the derived subalgebra g ′(A) := D2g(A) = [g(A), g(A)], and

g(A) = g ′(A) + h, (3.6)

where this decomposition is not necessarily direct.

Remark. g(A) = g ′(A) if and only if detA 6= 0.

Quadruples decompose like realizations.

Proposition 3.3. The quadruple associated to a direct sum of matricesAi is isomorphic to a direct sumof the quadruples associated to theAi. The root system of g(A) is the union of the root systems of theg (Ai).

We now characterize the center of the Lie algebra g(A).

Proposition 3.4. The center of the Lie algebra g(A) is equal to

c := h ∈ h : 〈αi, h〉 = 0∀i = 1, . . . , n . (3.7)

Furtherore, dim c = n− `, where ` = rkA.

We record the necessary and sufficient conditions on the matrixA for the algebra g(A) to besimple.

Proposition 3.5. g(A) is simple if and only if detA 6= 0 and for each pair of indices i and j, there existindices i1, i2, . . . , is such that the productaii1ai1i2 · · ·aisj 6= 0.

3.1.3 Root space of the Kac-Moody algebra g(A)

Continuing in analogy, put h ′ :=∑ni=1Cα

∨i . Then g ′(A) ∩ h = h ′, and g ′(A) ∩ gα = gα if

α 6= 0. One may deduce from Theorem 3.2(d) the following root space decomposition with respectto h.

g(A) =⊕α∈Q

gα. (3.8)

12

In further analogy with section 2.5.2, we have g0 = h, and we define the multiplicity multα :=

dim gα. Note that multα 6 n| htα|.Call a nonzero elementα ∈ Q a root if multα 6= 0. A rootα > 0 (resp.< 0) is called positive

(resp. negative), and (3.3) assures us thatαmust be either positive or negative. Denoting the setof roots (resp. positive, negative roots) by∆ (resp.∆+,∆−),∆ decomposes disjointly as

∆ = ∆+ t ∆−. (3.9)

Letn+ (respn−) denote the subalgebra ofg(A)generated bye1, . . . , en (resp. f1, . . . , fn). The-orem 3.2(a) establishes the triangular decomposition of vector spaces

g(A) = n− ⊕ h⊕ n+. (3.10)

It follows from Theorem 3.2(e) that r ⊂ g(A) is ω-invariant. Filtering through the quotient, ωinduces an involutionω of g(A) called the Chevalley involution of g(A). Asω (gα) = g−α, wehave that multα = mult(−α) and∆− = −∆+.

3.2 Invariant bilinear form

Though we omitted it in our discussion, every finite-dimensional Lie algebra carries a distin-guished symmetric bilinear form called the Killing form. We now present some results pertain-ing to the existence of invariant bilinear forms on Kac-Moody algebras.

Such a form only exists on symmetrizable Kac-Moody algebras. We first explain what this ter-minology means. On the level of matrices, ann×nmatrixA is called symmetrizable if there ex-ists an invertible diagonal matrixDwithdii = εi, and a symmetric matrixB such thatA = DB.The matrixB is called a symmetrization ofA and g(A) is called a symmetrizable Lie algebra, sincerescaling the Chevalley generators ei 7→ ei, fi 7→ εifi for εi 6= 0 rescalesα∨

i 7→ εiα∨i , which

extends to an isomorphism h 7→ h (nonunique, if detA = 0), which in turn extends to an iso-morphism of Lie algebras g(A)→ g(DA).

Let A be a symmetrizable matrix with a fixed decomposition A = DB as above, and let(h, II, II∨

)be a realization ofA. Fix a complementary subspace h ′′ to h ′ =

∑Cα∨

i in h, anddefine a symmetric bilinear C-valued form ( , ) on h by the two equations(

α∨i , h

)= 〈αi, h〉εi forh ∈ h, i = 1, . . . , n (3.11)

(h ′, h ′′) = 0 forh ′, h ′′ ∈ h ′′. (3.12)

By the decompositionA = DB and since α∨1 , . . . , α

∨n are linearly independent, we conclude

from (3.11) that(α∨i , α

∨j

)= bijεiεj for i = 1, . . . , n, (3.13)

so there is no ambiguity in the definition of ( , ).

Proposition 3.6. The bilinear form ( , ) is nondegenerate onh.

Since the bilinear form ( , )|h is nondegenerate, we have an isomorphismν : h→ h∗ given by

〈ν(h), h1〉 = (h, h1), h, h1 ∈ h, (3.14)

and the induced bilinear form ( , ) on h∗. Then (3.11) implies

ν(α∨i ) = εiαi, i = 1, . . . , n. (3.15)

Combining with (3.13) yields

(αi, αj) = bij = aij/εi, i, j = 1, . . . , n. (3.16)

13

Theorem 3.7. Lt g(A) be a symmetrizable Lie algebra; fix a decomposition ofA. Then there exists anondegenerate symmetric bilinear complex-valued form ( , ) on g(A) such that

(a) ( , ) is invariant; i.e. ([x, y], z) = (x, [y, z])∀x, y ∈ g(A).

(b) ( , )|h is defined by (3.11) and (3.12) and is nondegenerate.

(c) (gα, gβ) = 0 ifα+ β 6= 0.

(d) ( , )|gα+g−αis nondegenerate forα 6= 0, and hence gα and g−α are nondegenerately paired by

( , ).

(e) [x, y] = (x, y)ν−1(α) forx ∈ gα, y ∈ g−α, α ∈ ∆.

If, moreover, (αi, αi) > 0 for i = 1, . . . , n, ( , ) is called a standard invariant form.

3.3 Casimir element

Letg(A)be a Lie algebra associated to a matrixA,h the Cartan subalgebra, andg =⊕α gα the

root space decomposition with respect to h.

Definition. A g(A)-moduleV is called restricted if for every v ∈ V , we have gα(v) = 0 for all but afinite number of positive rootsα.

Remark. Every submodule or quotient of a restricted module is restricted. Direct sums and tensor prod-ucts of a finite number of restricted modules are restricted.

Assume now thatA is symmetrizable and ( , ) an invariant bilinear form (guaranteed by The-orem 3.7). Given a restricted g(A)-moduleV , we construct a linear operatorΩ onV , called the(generalized) Casimir operator, as follows.

Define implicity a linear function ρ ∈ h∗ by

〈ρ, α∨i 〉 =

1

2aii i = 1, . . . , n. (3.17)

If detA = 0, this does not specify ρ uniquely, and hence we must choose one. Then (3.15)and (3.16) imply

(ρ, αi) =1

2(αi, αi) i = 1, . . . , n. (3.18)

Next, for eachα ∈ ∆+, choose a basise(i)α

of the space gα, and form the dual basis

e(i)−α

of g−α. We then define an operatorΩ0 onV by

Ω0 := 2∑α∈∆+

∑i

e(i)−αe

(i)α . (3.19)

Ω0 is independent of the choice of basis. Since ∀v ∈ V , only a finite number of summandsare nonzero, Ω0 is well-defined on V . Let u1, u2, . . . and u1, u2, . . . be dual bases of h. Thegeneralized Casimir operator is defined by

Ω = 2ν−1(ρ) +∑i

uiui +Ω0. (3.20)

We now record the following theorem that justifies this construction.

Theorem 3.8. Let g(A) be a symmetrizable Lie algebra and V a restricted g(A)-module. Then theCasimir operatorΩ commutes with the action of g(A) onV .

14

We now give a result about eigenvalues ofΩ.

Corollary 3.9. Under the same hypotheses as Theorem 3.8, if there exists some v ∈ V such that theChevalley generators ei vanish on v for i = 1, . . . , n, and h(v) = 〈Λ,h〉v for someΛ ∈ h∗ and allh ∈ h, then

Ω(v) = (Λ+ 2ρ,Λ)v. (3.21)

3.4 Integrable representations

We first introduce some definitions in close analogy with the classical case. A g(A)-moduleV iscalled h-diagonalizable ifV =

⊕λ∈h∗ Vλ, where

Vλ = v ∈ V : h(v) = 〈λ, h〉v∀h ∈ h . (3.22)

Vλ is called a weight space, λ ∈ h∗ is called a weight ifVλ 6= 0, and dimVλ is called the multiplic-ity of λ and is denoted multV λ. A h-diagonalizable module over a Kac-Moody algebra g(A) iscalled integrable if all ei and fi are locally nilpotent onV . (An operator T is locally nilpotent on avector spaceV if T is nilpotent on every finite dimensional subspace ofV .)

Remark. g(A) is an integrable g(A)-module under the adjoint representation; i.e. ad ei and ad fi arelocally nilpotent on g(A).

The following two propositions together justify the terminology ‘integrable.’

Proposition 3.10. Let g(A) be a Kac-Moody algebra and let ei, fi (i = 1, . . . , n) be its Chevalley gen-erators. Put g(i) = Cei +Cα∨

i +Cfi. Then

g(i) ' sl2(C) (3.23)

asC-algebras with standard basisei, α

∨i , fi

.

Proposition 3.11. LetV be an integrable g(A)-module. As a g(i)-module,V decomposes into a directsum of finite-dimensional irreducible h-invariant modules. (Hence the action of g(i) can be ‘integrated’to the action of the group SL2(C).)

3.5 Weyl group

Definition. For each i = 1, . . . , n, we define the fundamental reflection ri of the spaceh∗ by

ri(λ) = λ−⟨λ, α∨

i

⟩αi, α ∈ h∗. (3.24)

One can verify ri is a reflection since its fixed point set is Ti =λ ∈ h∗ :

⟨λ, α∨

i

⟩= 0

, andri(αi) = −αi.

Definition. The subgroupW ⊂ GL (h∗) generated by all fundamental reflections is called the WeylgroupW of g(A).

The following proposition justifies this definition.

Proposition 3.12. LetV be an integrable module over a Kac-Moody algebra g(A). Then

(a) multV w(λ)∀λ ∈ h∗, w ∈W. In particular, the set of weights ofV isW-invariant.

(b) The root system∆ of g(A) isW-invariant, and multα = multw(α)∀α ∈ ∆,w ∈W.

We now consider the special case of a symmetrizable Kac-Moody algebra.

15

Proposition 3.13. LetA be a symmetrizable generalized Cartan matrix and let ( , ) be a standard in-variant bilinear form on g(A). The restriction of the bilinear form ( , ) toh∗ isW-invariant.

3.5.1 Weyl chambers

It is basic to understand the geometry of the action of the Weyl group. First, letA be ann × nmatrix overR, and let

(hR, II, II

∨)

be a realization of the matrixAoverR; i.e. hR is a real vectorspace of dimension 2n − `, so that, extending hR by scalars to h := hR ⊗ C,

(h, II, II∨

)is a

realization ofA over C.

Remark. hR is stable underW sinceQ∨ ⊂ hR, whereQ is the coroot lattice (c.f. (3.2) for the definitionof the root lattice).

Definition. The set

C := h ∈ hR : 〈αi, h〉 > 0 for i = 1, . . . , n (3.25)

is called the fundamental chamber.

Definition. The setsw(C), w ∈W are called chambers, and their union

X =⋃w∈W

w(C) (3.26)

is called the Tits cone.

Remark. We have the corresponding dual notions ofC∨ andX∨ inh∗R.

We now record a proposition that characterizes the action of the Weyl group on chambers.

Proposition 3.14. (a) For h ∈ C, the groupWh = w ∈ W : w(h) = h is generated by thefundaental reflections that it contains.

(b) The fundamental chamberC is a fundamental domain for the action ofW onX; i.e. any orbitW ·hofh ∈ X intersectsC in exactly one point. In particular,W acts simply transitively on chambers.

(c) X = x ∈ hR : 〈α, h〉 < 0 only for a finite number ofα ∈ ∆+. In particular,X is a convex cone.

(d) C =h ∈ hR : ∀w ∈W,h−w(h) =

∑i ciα

∨i , where ci > 0

.

(e) The following conditions are equivalent:

(i) |W| <∞;

(ii) X = hR;

(iii) |∆| <∞;

(iv)∣∣∆∨

∣∣ <∞.

(f) Ifh ∈ X, then |Wh| <∞ if and only ifh lies in the interior ofX.

3.6 Classification of generalized Cartanmatrices

It is a convenient fact about Kac-Moody algebras that basic properties of the generalized CartanmatrixA translate into essential facts about the algebra g(A). SupposeA is a realn×nmatrixsatisfying the following three properties (we may reduce to the first and a generalized Cartanmatrix satisfies the rest).

(m1) A is indecomposable.

16

(m2) aij 6 0 for i 6= j.

(m3) aij = 0 impliesaji = 0.

The main result is then as follows.

Theorem 3.15. LetA be a realn× nmatrix satisfying (m1), (m2), and (m3). Then one and only one ofthe following three possibilities holds.

(Fin) detA 6= 0;∃u > 0 : Au > 0;Av > 0 implies v > 0 or v = 0.

(Aff) corankA = 1;∃u > 0 : Au = 0;Av > 0 impliesAv = 0;

(Ind) ∃u > 0 : Au < 0;Av > 0, v > 0 imply v = 0.

Definition. We say that a matrix satisfying (Fin), (Aff), and (Ind) is of finite, affine, or indefinite type,respectively.

Corollary 3.16. LetAbe a matrix satisfying (m1), (m2), and (m3). ThenA is of finite, affine, or indefinitetype if and only if there exists anα > 0 such thatAα > 0,= 0, or< 0, respectively.

3.6.1 Kac-Moody algebras of finite type

One success of the generalization of Kac-Moody is that one recovers the full collection of classicalfinite-dimensional complex semisimple Lie algebras in the set of Kac-Moody algebras of finitetype.

Proposition 3.17. LetA be an indecomposable generalized Cartan matrix. Then the following condi-tions are equivalent.

(i) A is a generalized Cartan matrix of finite type;

(ii) A is symmetrizable and the bilinear form ( , )hRis positive-definite;

(iii) |W| <∞;

(iv) |∆| <∞;

(v) g(A) is a simple finite-dimensional Lie algebra;

(vi) ∃α ∈ ∆+ : α+ αi 6∈ ∆∀i = 1, . . . , n.

3.7 Root system

The main innovation in the theory of roots of a Kac-Moody algebra g(A) vis-à-vis the classicaltheory, is the notion of an imaginary root, which is an entirely new notion. The classical theoryof roots is recovered in the complementary notion of a real root.

3.7.1 Real roots

Definition. A rootα ∈ ∆ is called real if there exists aw ∈W such thatw(α) is a simple root.

We denote the set of all real roots by∆re and the set of all positive real roots by∆re+. Given any

α ∈ ∆re, then α = w(αi) for some αi ∈ II,w ∈ W. Define the dual (real) root α∨ ∈ ∆∨re

by α∨ = w(α∨i

). This is independent of the choice of presentation α = w (αi). We define a

reflection rα with respect toα ∈ ∆re by

rα(λ) = λ−⟨λ, α∨

⟩α, λ ∈ h∗. (3.27)

The following proposition strongly evokes the classical setting.

17

Proposition 3.18. Letα be a real root of a Kac-Moody algebra g(A). Then

(a) multα = 1.

(b) kα is a root if and only ifk = ±1.

(c) Ifβ ∈ ∆, then∃p, q ∈ Z>0 related by the equation

p− 1 =⟨β,α∨

⟩, (3.28)

such thatβ+ kα ∈ ∆ ∪ 0 if and only if−p 6 k 6 q, k ∈ Z.

(d) Suppose thatA is symmetrizable and let ( , )be a standard invariant bilinear form ong(A). Then

(i) (α,α) > 0,

(ii) α∨ = 2ν−1(α)/(α,α), and

(iii) ifα =∑i kiαi, thenki(αi, αi) ∈ (α,α)Z.

(e) Provided that±α 6∈ II,∃i such that

|ht ri(α)| < |htα| . (3.29)

Definition. LetAbe a symmetrizable generalized Cartan matrix and ( , )a standard invariant bilinearform. Then, given a real rootα, we have (α,α) = (αi, αi) for some simple rootαi. We callα a short(resp. long) real root if (α,α) = mini(αi, αi) (resp. (α,α) = maxi(αi, αi)).

Remark. The definitions of short and long roots are independent of the choice of a standard form. IfA issymmetric, then all simple roots and hence all real roots have the same square length.

3.7.2 Imaginary roots

Definition. A rootα ∈ ∆ is called imaginary if it is not real.

We denote the set of all imaginary roots by∆im and the set of all positive imaginary roots by∆im+ .

By definition,∆ decomposes as a disjoint union like

∆ = ∆re t ∆im. (3.30)

Imaginary roots enjoy the following properties, some very different from those of real roots.

Proposition 3.19. (a) The set∆im+ isW-invariant.

(b) Forα ∈ ∆im+ ,∃!β ∈ −C∨ (i.e.

⟨β,α∨

i

⟩6 0∀i) which isW-equivalent toα.

(c) IfA is symmetrizable and ( , ) is a standard invariant bilinear form, thenα ∈ ∆ is imaginary ifand only if (α,α) 6 0.

(d) Ifα ∈ ∆im+ and r is a nonzero rational number such that rα ∈ Q, then rα ∈ ∆im. In particular,

nα ∈ ∆im ifn ∈ Z− 0.

We now state when imaginary roots exist.

Theorem 3.20. LetA be an indecomposable generalized Cartan matrix.

(a) IfA is of finite type,∆im = ∅.

18

(b) IfA is of affine type, then

∆im+ = nδ, n ∈ Z , (3.31)

where δ =∑`i=0 aiαi is a linear combination of rootsαi in terms of constants ai, which are

effectively computable from the matrixA,1 and

(c) IfA is of indefinite type, then ∃α ∈ ∆im+ , α =

∑i kiαi, such that ki > 0 and

⟨α,α∨

i

⟩<

0∀i = 1, . . . , n.

The Tits coneXmay be described naturally in terms of imaginary roots.

Proposition 3.21. (a) IfA is of finite type, thenX = hR.

(b) IfA is of affine type, then

X = h ∈ hR : 〈δ, h〉 > 0 ∪ Rν−1(δ). (3.32)

(c) IfA is of indefinite type, then

X =h ∈ hR : 〈α, h〉 > 0 for allα ∈ ∆im

+

, (3.33)

whereX denotes the closure ofX in the metric topology ofhR.

Remark. Proposition 3.21 has a geometric interpretation in terms of the imaginary coneZ, the convexhull inh∗R of the set∆im

+ ∪ 0. Namely, th conesZ andX are dual to each other;

X =h ∈ hR : 〈α, h〉 > 0∀α ∈ Z

. (3.34)

In particular,Z is a convex cone.

We introduce the notion of a Kac-Moody root base and consider the action of the Weyl group onit.

Definition. A linearly independent set of roots II ′ = α ′1, α′2, . . . is called a root base of∆ if each

rootα can be written in the formα = ±∑i kiα

′i, whereki ∈ Z+.

Proposition 3.22. LetA be an indecoposable generalized Cartan matrix. Then any root base II ′ of∆ isW-conjugate to II or−II.

Remark. II isW-conjugate to−II if and only ifA is of finite type.

3.7.3 Hyperbolic type

Definition. A generalized Cartan matrixA is called a matrix of hyperbolic type if it is indecompos-able symmetrizable of indefinite type, and if every proper connected subdiagram of the Dynkin diagramassociated toA is of finite or affine type.

Proposition 3.23. LetA be a generalized Cartan matrix of finite, affine, or hyperbolic type. Then

(a) The set of all short real roots isα ∈ Q : |α|2 = a = min

i|αi|

2

. (3.35)

1The coefficients ai are the labels of the Dynkin diagram associated toA, which we omit in our discussion, butwhich is treated in [Kac94, Ch. 4].

19

(b) The set of all real roots isα =∑j

kjαj ∈ Q : |α|2 > 0 andkj|αj|2 ∈ Z∀j

. (3.36)

(c) The set of all imaginary roots isα ∈ Q− 0 : |α|2 6 0

. (3.37)

(d) IfA is affine, then there exist roots of intermediate squared lengthm if and only ifA = A(2)2` ,

` > 2.2 The set of such roots coincides withα ∈ Q : |α|2 = m

.

3.8 Kac-Moody algebras of affine type

A convenient feature of Kac-Moody algebras of affine type is that all the basic information aboutthem can be expressed in terms of corresponding objects for an ‘underlying’ simple finite-dimensionalLie algebra. We make this notion precise in what follows. LetA be a generalized Cartan matrixof affine type of order `+1 (and rank `). Letg = g(A)be the affine algebra associated to a matrixA of affine type; let h be its Cartan subalgebra, II = α0, . . . , α` ⊂ h∗ the set of simple roots,II∨ =

α∨0 , . . . , α

∨`

⊂ h the set of simple coroots,∆ the root system, andQ andQ∨ the root

and coroot lattices. Proposition 3.4 implies that the center of g is one-dimensional and spannedby a linear combination of coroots3 called the canonical central element and denoted byK. Moreprecisely, with ai as in Theorem 3.20(b), let a∨i be the corresponding labels corresponding tothe dual algebra. Then

K =∑i=0

a∨i α∨i . (3.38)

Note that [g, g] has corank 1 in g. Fixing an element d ∈ h satisfying 〈αi, d〉 = 0∀i = 1, . . . , nand 〈α0, d〉 = 1, which we call the scaling element, it is clear that the elements α∨

0 , . . . , α∨` , d

form a basis of h, and

g = [g, g] +Cd. (3.39)

There is a distinguished standard form on g satisfying the properties of Theorem 3.7, called thenormalized invariant form.

3.8.1 The underlying finite-dimensional Lie algebra

Denote by h (resp. hR) the linear span over C (resp. R) of the coroots α∨1 , . . . , α

∨` . The dual

notions h∗ and h∗R are defined similarly. Then we have an orthogonal direct sum of subspaces

h = h⊕ (CK+Cd) h∗ = h∗ ⊕ (Cδ+CΛ0) , (3.40)

where δ is as in Theorem 3.20(b), andΛ0 ∈ h∗ is defined implicitly by the equations⟨Λ0, α

∨i

⟩= δ0i for i = 0, . . . , `; 〈Λ0, d〉 = 0. (3.41)

Denote now by g the subalgebra of g generated by the ei and fi with i = 1, . . . , `. g is a Kac-Moody algebra associated to the matrix Aobtained fromAby deleting the 0th row and column.

2This ‘type’ belongs to the classification of generalized Cartan matrices of affine type, and is given in terms of itsDynkin diagram in [Kac94, p.55].

3The coefficients in the linear combination again depend on the combinatorics of the Dynkin diagram.

20

The elements ei, fi (i = 1, . . . , `) are the Chevalley generators of g, and h = g ∩ h is its Cartansubalgebra. II = α1, . . . , α` is the root basis and II∨ =

α∨1 , . . . , α

∨`

the coroot basis for g.

It follows from Proposition 3.17 that g = g(A) is a simple finite-dimensional Lie algebra. The set∆ = ∆∩ h∗ is the root system of g; it is finite and consists of real roots (again by Proposition 3.17),the set ∆+ = ∆ ∩∆+ being the set of positive roots. Denote by ∆s and ∆` the sets of short andlong roots, respectively, in ∆. Put Q = Z∆. Let W be the Weyl group of ∆.

3.8.2 Relating g and g

Recall the classification of imaginary and positive imaginary roots ofg, following Theorem 3.20(b).We now characterize the set of real roots∆re and positive real roots∆re

+ in terms of ∆ and δ.

Proposition 3.24. (a) ∆re =α+ nδ : α ∈ ∆, n ∈ Z

if r = 1.

(b) ∆re = α+ nδ : α ∈ α∆s, n ∈ Z ∪ α+ nrδ : α ∈ α∆`, n ∈ Z if r = 2 or 4, butA isnot of typeA(2)2`.4

(c) IfA is of typeA(2)2` , then

∆re =

1

2(α+ (2n− 1)δ : α ∈ ∆`, n ∈ Z

∪

α+ nδ : α ∈ ∆s, n ∈ Z∪

α+ 2nδ : α ∈ ∆`, n ∈ Z.

(3.42)

(d) ∆re + rδ = ∆re.

(e) ∆re+ = α ∈ ∆re withn > 0 ∪ ∆+.

Remark. There is an orthogonal decomposition of the coroot lattice as

Q∨ = Q∨ ⊕ ZK, (3.43)

and an isomorphism of lattices equipped with bilinear forms

Q∨(A) ' Q(tA). (3.44)

Thusfar, it has been possible to describe the structure of the affine Kac-Moody algebrag in termsof a finite-dimensional Kac-Moody algebra g. We extend this progress to a characterization ofthe Weyl group of the affine algebra g. Recall thatW is generated by fundamental reflectionsr0, r1, . . . , r` which act on h∗ by

ri(λ) = λ−⟨λ, α∨

i

⟩, λ ∈ h∗. (3.45)

As⟨δ, α∨

i

⟩= 0 for i = 0, . . . , `, we have that

w(δ) = δ∀w ∈W. (3.46)

Recall that the invariant standard form is alsoW-invariant.Denote by W the subgroup ofW generated by r1, . . . , r`. As ri(Λ0) = Λ0 for i = 1, . . . , `,

W acts trivially onCΛ0 +Cδ; h∗ is also W-invariant. We conclude that W acts faithfully on h∗,and we can identify W with the Weyl group of the Lie algebra g, operating on h∗. By Proposi-tion 3.14(e),

∣∣∣W∣∣∣ <∞.

4c.f. the note attached to Proposition 3.23(d).

21

Recall that for a real rootα, we have the reflection rα ∈W given by

rα(λ) = −λ−⟨λ, α∨

⟩, λ ∈ h∗. (3.47)

Form the distinguished element

θ := δ− a0α0 =∑i=1

aiαi ∈ Q, (3.48)

where theai are as in Theorem 3.20(b). Given anyα ∈ h∗, we can form an operator on tα on h∗

given by, for any λ ∈ h∗,

tα(λ) = λ+ 〈λ, K〉α−

((λ, α) +

1

2|α|2〈α,K〉

)δ. (3.49)

Now form the distinguished latticeM ⊂ h∗R. Let Z(W · θ∨) denote the lattice in hR spannedoverZby the (finite) setW ·θ∨, whereθ∨ denotes theθelement from the dual algebrag ( tA ),and setM = ν(Z(W · θ∨)).M has the following properties.

M = Q = Q ifA is symmetric or r > a0;

M = ν(Q∨) = ν(Q∨) otherwise.(3.50)

The latticeM, considered as an abelian group, acts faithfully on h∗ by (3.49). We call the corre-sponding subgroup of GL(h∗) the group of translations and denote it by T . We may finally statethe decomposition of the Weyl groupW of g in terms of the Weyl group W of g.

Proposition 3.25. W = W n T .

Hence, we can obtain a rich description of the root and coroot lattices, the root system, and theWeyl group of an affine algebra g in terms of the corresponding objects for the ‘underlying’ sim-ple finite-dimensional Lie algebra g.

4 Highest-weight modules over Kac-Moody algebras

So far, we have emphasized the construction and structure of Kac-Moody algebras, and paid lessattention to their representation theory. The representation theory of Kac-Moody algebras is arich subject with many surprising connections to other areas of mathematics. In this section,we will introduce the notions of highest-weight modules over Kac-Moody algebras, and the cat-egory O and its Verma modules. We follow [Kac94, Ch. 9].

We start with an arbitrary complexn × nmatrixA and consider the associated Lie algebrag(A). We have the usual triangular decomposition

g(A) = n− ⊕ h⊕ n+, (4.1)

and the corresponding decomposition of the universal enveloping algebra

U(g(A)) = U(n−)⊗ U(h)⊗ U(n+). (4.2)

Recall that a g(A)-module V is called h-diagonalizable if it admits a weight space decomposi-tionV =

⊕λ∈h∗ Vλ by weight spacesVλ (c.f. §3.4).

Definition. A nonzero vector fromVλ is called a weight vector of weightλ.

22

Let P(V) = λ ∈ h∗ : Vλ 6= 0 denote the set of weights of V . Finally, for λ ∈ h∗, setD(λ) =

µ ∈ h∗ : µ 6 λ.

Definition. The categoryO is defined as follows. Objects inOareg(A)-modulesVwhich areh-diagonalizablewith finite-dimensional weight spaces and such that there exists a finite number of elementsλ1, . . . , λs ∈h∗ such that

P(V) ⊂s⋃i=1

D (λi) . (4.3)

Morphisms inO are homomorphisms of g(A)-modules.

It follows from the following general fact that any submodule or quotient module of a modulein O is in O.

Proposition 4.1. Leth be a commutative Lie algebra,V a diagonalizableh-module, i.e.

V =⊕λ∈h∗

Vλ, whereVλ = v ∈ V : h(v) = λ(h)v∀h ∈ h . (4.4)

Then any submoduleU ofV is graded with respect to the gradation in (4.4).

It is clear that a sum or tensor product of a finite number of modules from O is again in O. It islikewise the case that every module from O is restricted.

Definition. A g(A)-module V is called a highest-weight module with highest weightΛ ∈ h∗ ifthere exists a nonzero vector vΛ ∈ V such that

n+ (vΛ) = 0, h (vΛ) = Λ(h)vΛ forh ∈ h, andU(g(A)) (vΛ) = V. (4.5)

The vector vΛ is called a highest weight vector.

By the tensor product decomposition of the universal enveloping, we may restate the last con-dition as

U (n−) (vΛ) = V. (4.6)

It follows that

V =⊕λ6Λ

Vλ; VΛ = CvΛ; dimVλ <∞. (4.7)

Therefore, a highest-weight module lies in O, and every two highest-weight vectors are propor-tional.

Definition. A g(A)-moduleM(Λ) with highest weightΛ is called a Verma module if every g(A)-module with highestΛ is a quotient ofM(Λ).

Proposition 4.2. (a) For everyΛ ∈ h∗, there exists a unique (up to isomorphism) Verma moduleM(Λ).

(b) Viewed as aU (n−)-module,M(Λ) is a free module of rank1 generated by a highest-weight vec-tor.

(c) M(Λ) contains a unique proper maximal submoduleM ′(Λ).

23

4.1 Inductive Verma construction

Recall the definition of an induced module. That is, let V be a left module over a Lie algebra a,and suppose there is a Lie algebra homomorphism a→ b. Then, by identifying awith its imagein b, we have that the induced b-module is given by

U(b)⊗U(a) V, (4.8)

where the homomorphism from a to b is used to translate between the left action onV and onU(b) in the bilinear tensor product. Here, the action ofb is induced by left multiplication inU(b).Then, we may construct a Verma module as an induced (n+h)-module as follows. Define the left(n+h)-module Cλ with underlying space C by n+(1) = 0,h(1) = 〈λ, h〉1 forh ∈ h. Then

M(λ) = U(g(A))⊗U(n++h) Cλ. (4.9)

4.2 Primitive vectors andweights

It follows from Proposition 4.2(c) that among the modules with highest weight Λ there is aunique irreducible one, namely

L(Λ) :=M(Λ)/M ′(Λ). (4.10)

It is evident thatL(Λ) is a quotient of any module with highest weightΛ. To show that the mod-ulesL(Λ) exhaust all irreducible modules from the categoryO, as well as for other purposes, wemake the following definitions. LetV be a g(A)-module.

Definition. A vector v ∈ Vλ is called primitive if there exists a submoduleU ⊂ V such that

v 6∈ U; n+(v) ⊂ U. (4.11)

Thenλ is called a primitive weight.

One defines primitive vectors and weights for a g ′(A)-module similarly. A weight vector v suchthat n+(v) = 0 is obviously primitive.

Proposition 4.3. LetV be a nonzero module from the categoryO. Then

(a) V contains a nonzero weight vector v such thatn+(v) = 0.

(b) The following conditions are equivalent.

(i) V is irreducible;(ii) V is a highest weight module and any primitive vector ofV is a highest-weight vector;

(iii) V ' L(Λ) for someΛ ∈ h∗.

(c) V is generated by its primitive vectors as a g(A)-module.

Thus, we have a bijection between h∗ and irreducible modules from the category O given byΛ 7→ L(Λ). Note that L(Λ) can also be defined as an irreducible g(A)-module which admits anonzero vector vΛ such that

n+ (vΛ) = 0 andh (vΛ) = Λ(h)vΛ forh ∈ h. (4.12)

Henceforth, let U0(g) denote the augmentation ideal gU(g) of U(g). We have the following‘Schur-style’ lemma.

Lemma 4.4. Endg(A) L(Λ) = CIL(Λ).

24

LetL(Λ)∗ now e theg(A)-module dual toL(Λ). ThenL(Λ)∗ =∏λ (L(Λ)λ)

∗. The subspace

L∗(Λ) :=⊕

(L(Λ)λ)∗ (4.13)

is a submodule of the g(A)-module L(Λ)∗. It is clear that the module L∗(Λ) is irreducible, andthat for v ∈ (L(Λ)Λ)

∗ one has

n−(v) = 0; h(v) = −〈Λ,h〉v forh ∈ h. (4.14)

Such a module is called an irreducible module with lowest weight −Λ. As before, we have a bijec-tion between h∗ and irreducible lowest-weight modules given byΛ 7→ L∗(−Λ).

Denote by πΛ the action of g(A) on L(Λ), and introduce the new action π∗Λ on the spaceL(Λ) by

π∗Λ(g)v = πΛ(ω(g))v, (4.15)

whereω is the Chevalley involution of g(A). It is clear that (L(Λ), π∗Λ) is an irreducible g(A)-module with lowest weight−Λ. By the uniqueness theorem, this module can be identified withL∗(Λ), and the pairing between L(Λ) and L∗(Λ) gives us a nondegenerate bilinear form B onL(Λ) such that

B(g(x), y) = −B(x,ω(g)(y))∀g ∈ g(A) and x, y ∈ L(Λ). (4.16)

A bilinear form on L(Λ) which satisfies (4.16) is called a contravariant bilinear form.

Proposition 4.5. Everyg(A)-moduleL(Λ) carries a unique-up-to-constant-factor nondegenerate con-travariant bilinear formB. The form is symmetric and L(Λ) decomposes into an orthogonal direct sumof weight spaces with respect to this form.

4.3 Complete reducibility

We now may attack the question of reducibility.

Lemma 4.6. LetV be a g(A)-module from the categoryO. If for any two primitive weights λ andµ ofV the inequalityλ > µ, then the moduleV is completely reducible.

It is not the case that each module V ∈ O admits a composition series (a sequence of sub-modulesV ⊃ V1 ⊃ V2 ⊃ · · · such that eachVi/Vi+1 is irreducible. However, there is a kind ofsubstitute for this property that does hold.

Lemma 4.7. LetV ∈ O and λ ∈ h∗. Then there exists a filtration by a sequence of submodulesV =

Vt ⊃ Vt−1 ⊃ · · · ⊃ V1 ⊃ V0 = 0 and a subset J ⊂ 1, . . . , t such that

(i) if j ∈ J, thenVj/Vj−1 ' L(λj) for someλj > λ.

(ii) if j 6∈ J, then (Vj/Vj−1)µ = 0 for everyµ > λ.

Let V ∈ O and µ ∈ h∗. Fix a λ ∈ h∗ such that µ > λ, and construct a filtration as perLemma 4.7.

Definition. Let the multiplicity ofL(µ) inV , denoted by [V : L(µ)], be the number of timesµappearsamong λj : j ∈ J, which is independent of the filtration and the choice ofλ.

Remark. L(µ) has a nonzero multiplicity inV if and only ifµ is a primitive weight ofV .

25

4.4 Formal characters

To study the formal characters of modules fromO, we define an algebraEoverC in the followingway. Elements of E are sums∑

λ∈h∗cλe(λ), (4.17)

where cλ ∈ C and cλ = 0 for λ outside the union of a finite number of sets of the formD(λ).E becomes a commutative associative algebra if we set e(λ)e(µ) = e(λ + µ) and extend bylinearity in the usual way. e(0) is our identity.

Definition. The elements e(λ) are called formal exponents.

Formal exponents are linearly independent and in one-to-one correspondence with the elementsλ ∈ h∗.

Now fixV a module from the categoryO and letV =⊕λ∈h∗ Vλ be its weight space decom-

position.

Definition. Define the formal character ofV by

chV =∑λ∈h∗

(dimVλ) e(λ) ∈ E. (4.18)

Proposition 4.8. LetV be a g(A)-module fromO. Then

chV =∑λ∈h∗

[V : L(λ)] chL(λ). (4.19)

We record now a formula for the formal character of a Verma moduleM(Λ).

Lemma 4.9.

chM(Λ) = e(Λ)∏α∈∆+

(1− e(−α))−multα. (4.20)

Suppose now thatA is a symmetrizable matrix and let ( , )be a bilinear form on g(A). Thenthe generalized Casimir operatorΩ acts on each module from O.

Lemma 4.10. (a) IfV is a g(A)-module with highest weightΛ, then

Ω =(|Λ+ ρ|2 − |ρ|2

)IV . (4.21)

(b) IfV is a module from the categoryO and v is a primitive vector with weightλ, then there exists asubmoduleU ⊂ V such that v 6∈ U and

Ω(v) =(|λ+ ρ|2 − |ρ|2

)v mod U. (4.22)

Proposition 4.11. LetV be a g(A)-module with highest weightΛ. Then

chV =∑λ6Λ

|λ+ρ|2=|Λ+ρ|2

cλ chM(λ), where cλ ∈ Z, cλ = 1. (4.23)

4.5 Reducibility

Staying in the symmetrizable case, we now use the Casimir operator to study questions of irre-ducibility and complete reducibility in O.

26

Proposition 4.12. LetA be a symmetrizable matrix.

(a) If 2(Λ+ ρ, β) 6= (β,β)∀0 6= β ∈ Q+, then the g(A)-moduleM(Λ) is irreducible.

(b) IfV is a g(A)-module from the categoryO such that for any two primitive weightsλ andµ ofV ,such thatλ− µ = β > 0, one has 2(λ+ ρ, β) 6= (β,β), thenV is completely reducible.

Now we consider the derived subalgebra g ′(A) = [g(A), g(A)] of g(A) instead of g(A), fora general A (not necessarily symmetrizable). Recall that g(A) = g ′(A) + h, and that h ′ =∑iCα

∨i = g ′(A) ∩ h. Recall the free abelian groupQ. We have the following decomposition

of the universal enveloping algebra U(g(A)) with respect to h:

U(g(A)) =⊕β∈Q

Uβ, where (4.24)

Uβ := x ∈ U(g(A)) : [h, x] = 〈β, h〉x∀h ∈ h . (4.25)

PutU ′β = U(g ′(A)) ∩Uβ, so that U(g ′(A)) =⊕βU

′β.

Definition. A g ′(A)-moduleV is called a highest-weight module with highest weightΛ ∈ (h ′)∗

ifV admits aQ+-gradationV =⊕β∈Q+

VΛ−β such thatU ′β (VΛ−α) ⊂ VΛ−α+β, dimVΛ = 1,h(v) = Λ(h)v forh ∈ (h ′)∗, v ∈ VΛ, andV = U(g ′(A)) (VΛ).

In other words, this is a restriction of a highest-weight module over g(A) to g ′(A). We definethe Verma moduleM(Λ)overg ′(A) in the same way and show that it contains a unique propermaximal garded submoduleM ′(Λ). We put L(Λ) = M(Λ)/M ′(Λ), which is a restriction ofan irreducible g(A)-module to g ′(A).

Lemma 4.13. The g ′(A)-moduleL(Λ) is irreducible.

Definition. The labels ofΛ ∈ h∗ are the values 〈Λ,α∨i 〉 (i = 1, . . . , n).

IfΛ,M ∈ h∗ have the same labels, they may differ only off h ′. Then Lemma 4.13 implies thatthe modulesL(Λ)andL(M), when restricted tog ′(A), are irreducible and isomorphic asg ′(A)-modules, and that the actions of elements of g(A) on them differ only by scalar operations. Fi-nally, note that

dimL(Λ) = 1⇔ Λ|h′ = 0. (4.26)

Now we return to the case whereA is symmetrizable. We may fix a symmetric invariant bilinearform ong ′(A)which is defined also onh ′, and whose kernel isc. Put(β, γ) = (ν−1(β), ν−1(γ))

forβ, γ ∈ Q, and define ρ ∈ (h ′)∗ by 〈ρ, λ∨i 〉 = 12aii for i = 1, . . . , n.

Proposition 4.14. LetA be a (possibly infinite) symmetrizable matrix.

(a) If 2〈Λ+ ρ, ν−1(β)〉 6= (β,β)∀β ∈ Q+ − 0, then the g ′(A)-moduleM(Λ) is irreducible.

(b) LetV be a g ′(A)-module such that the following conditions are satisfied:

(i) ei(v) = 0∀v ∈ V and all but a finite number of the ei;(ii) ∀v ∈ V∃k > 0 : ei1 · · · eis(v) = 0whenever s > k;

(iii) V =⊕λ∈(h′)∗ Vλ, whereVλ = v ∈ V : h(v) = 〈λ, h〉v∀h ∈ h ′;

(iv) ifλandµ ∈ (h ′)∗ are primitive weights such thatλ−µ = β|h′ for someβ ∈ Q+− 0,then 2〈λ+ ρ, ν−1(β)〉 6= (β,β).

Then V is completely reducible; i.e. is isomorphic to a direct sum of g ′(A)-modules of the formL(Λ),Λ ∈ (h ′)∗.

27

Corollary 4.15. LetA be a symmetrizable matrix with nonpositive real entries and let V be a g ′(A)-module satisfying conditions (i)–(iii) of Proposition 4.14(b). Suppose that for every weightλ ofV one has〈λ, α∨

i 〉 > 0∀i. Then V is a direct sum of irreducible g ′(A)-modules, which are free of rank 1whenviewed ofU(n−)-modules.

We end our discussion of highest-weight modules over Kac-Moody algebras to mention that thetheory of these modules has ready application to finding the defining generators and relationsof symmetrizable Kac-Moody Lie algebras (c.f. [Kac94, §9.11–12]).

5 Physical applications of Kac-Moody algebras

We will conclude our exposition by discussing some applications to physics, namely the Sug-awara construction, the Virasoro algebra, and the coset construction, which are fundamentalto string theory and the basic constructions of conformal field theory. In this section, we fol-low [Kac94, Ch. 12].

5.1 Sugawara construction

The origin of the Sugawara construction in physics is Hirotaka Sugarawara's 1968 paper [Sug68].At this time, quantum chromodynamics (QCD) was not yet developed, and Sugawara was work-ing to find a theory of strong interactions that quantized the interaction in terms of ‘currents.’ Histheory was superseded by QCD, but as it was one of the first applications of Kac-Moody algebras,we describe it here.

As part of the classification of affine Kac-Moody algebras, there is a split of twisted and non-twisted affine algebras. These two subcollections are classified by their Dynkin diagrams, andare treated in [Kac94, Ch. 8]. We suppose our affine algebra g ′ is untwisted in what follows. Wethen have an isomorphism

g ′ ' C[t, t−1]⊗ g +CK (5.1)

with commutation relations[x(m), y(n)

]= [x, y](m+n) +mδm,−n(x, y)K. (5.2)

Here g is a simple finite-dimensional Lie algebra, (x, y) is the normalized invariant bilinear formon g, and x(n) stands for tn ⊗ x (n ∈ Z, x ∈ g).

Let ui andui

be dual bases of g; i.e. (ui, uj) = δij. Then the Casimir operator of g,

Ω =∑i

uiui, (5.3)

is independent of the choice of dual bases. Additionally, the Casimir operator

Ω = 2(K+ h∨)d+ Ω+ 2

∞∑n=1

∑i

u(−n)i ui(n) (5.4)

is the Casimir operator of the affine algebra

g = g ′ +Cd, where[d, x(n)

]= nx(n). (5.5)

Recall the definition of a restricted g ′-module in section 3.3; i.e. any module V such that ∀v ∈V , x(j)(v) = 0∀x ∈ g, j 0. Consider all series

∑∞j=1 uj with uj ∈ U(g ′), the universal

enveloping algebra of g ′, such that for any restricted g ′-module V and any v ∈ V , uj(v) = 0

for all but finitely mnay uj. We identify two such series if they represent the same operator inevery restricted g ′-module, thereby obtaining an algebraUc(g ′)which containsU(g ′) and actson every restricted g ′-module.

28

Definition. Define the restricted completion of the universal enveloping algebraU(g ′) to be thisalgebraUc(g ′).

We are now in a position to introduce our main object of study. Let Tn (n ∈ Z) be an operatorgiven by

T0 =∑i

uiui + 2

∞∑n=1

∑i

u(−n)i ui(n),

Tn =∑m∈Z

∑i

u(−m)i ui(m+n) ifn 6= 0.

(5.6)

Definition. The operators Tn are the Sugawara operators.

In fact, these operators are contained inUc(g′), and are independent of the choice of dual bases

of g. We have the following basic information about how the Sugawara operators act.

Lemma 5.1. (a) Forx ∈ g andn,m ∈ Z, one has[x(m), Tn

]= 2(K+ h∨)mx(m+n). (5.7)

(b) LetV be a restricted g-module and let v ∈ V be such that n+(v) = 0 andh(v) = 〈λ, h〉v forsomeλ ∈ h∗. Then

T0(v) = (λ, λ+ 2ρ)v. (5.8)

We now write down the commutator [Tm, Tn] of two Sugawara operators.

Proposition 5.2.

[Tm, Tn] = 2(K+ h∨)

((m− n)Tm+n + δm,−n

m3 −m

6(dim g)K

). (5.9)

Lemma 5.1 and Proposition 5.2 immediately yield the following

Corollary 5.3. LetV be a restricted g ′-module such thatK is a scalar operatorkI,k 6= −h∨. Let

Ln =1

2(k+ h∨)Tn, n ∈ Z (5.10)

c(k) =k(dim g)

k+ h∨, (5.11)

hΛ =(Λ+ 2ρ,Λ)

2(k+ h∨)ifV = L(λ), (5.12)

whereΛ ∈ h∗ is a highest weight, and L(Λ) is the unique irreducible module with highest weightΛ(c.f. [Kac94, §9.2]).

(a) Letting

dn 7→ Ln, c 7→ c(k) (5.13)

extendsV to a module over g ′ + Vir, in particular,V extends to a module over g (= g ′ +Cd).

(b) IfV is the g-moduleL(Λ), thenL0 = hλI− d.

Definition. The numberc(k) is called the conformal anomoly of theg ′-moduleV , and the number iscalled the vacuum anomoly ofL(Λ).

The appearance of Vir in the above corollary requires some explanation.

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5.2 Virasoro algebra

Let gbe a simple finite-dimensional Lie algebra, putL = C[t, t−1], and consider the loop algebra

L(g) := L⊗C g. (5.14)

This is an infinite-dimensional Lie algebra with bracket [ , ]0 given by

[P ⊗ x,Q⊗ y]0 = PQ⊗ [x, y] (P,Q ∈ L; x, y ∈ g). (5.15)

Denote by L(g) the extension of the Lie algebraL(g)by a 1-dimensional center; explicitly L(g) :=L(g)⊕CK, with bracket

[a+ λK, b+ µK] = [a, b]0 +ψ(a, b)K (a, b ∈ L(g); λ, µ ∈ C), (5.16)

whereψ is the C-valued 2-cocycle on the Lie algebra L(g)

ψ(a, b) := res(

dadt, b

)t

, (5.17)

where res is the usual residue, a linear functional on L, and ( , )t is the extension of a nonde-generate invariant symmetric bilinear complex-valued form ( , ) on g to an L-valued bilinearform on L(g) by

(P ⊗ x,Q⊗ y)t = PQ(x, y). (5.18)

Also, we extend every derivationD of the algebra L to a derivation of the Lie algebra L(g) by

D(P ⊗ x) = D(P)⊗ x. (5.19)

Then, letds denote the endomorphism of the space L(g) defined by

ds|L(g) = −ts+1d

dt, ds(K) = 0. (5.20)

Proposition 5.4. ds is a derivation of L(g).

Note that

d :=⊕j∈Z

Cdj (5.21)

is a Z-graded subalgebra in der L(g) with the commutation relations

[di, dj] = (i− j)di+j. (5.22)

This is the Lie algebra of regular vector fields on C× (i.e. derivations of L). The Lie algebra d hasa nontrivial central extension by a 1-dimensional center, say Cc, that is unique up to isomor-phism.

Definition. The unique-up-to-isomorphism nontrivial central extensiond⊕Cc ofd is called the Vira-soro algebra Vir, and is defined by the commutation relations

[di, dj] = (i− j)di+j +1

12

(i3 − i

)di,−jc, i, j ∈ Z. (5.23)

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5.3 Virasoro operators

We now may extend the Sugawara construction to the case of a reductive finite-dimensional Liealgebra g (we use this notation in place of g for this section). We have the decomposition of ginto a direct sum of ideals as

g = g(0) ⊕ g(1) ⊕ g(2) ⊕ · · · , (5.24)

where g(0) is the center of g and g(i) with i > 1 are simple. We fix on g a non-degenerate invari-ant bilinear form ( , ) so that (5.24) is an orthogonal decomposition, and we assume that therestriction of ( , ) to each g(i) for i > 1 is the normalized invariant form. We shall call such aform a normalized invariant form on g. Put

L(g) :=⊕i>0

L(g(i)), where L

(g(i))= L

(g(i))+CKi (5.25)

and

L(g) := L(g) +Cd, where d|L(g(i)) = −t

ddt, d (Ki) = 0. (5.26)

Definition. The Lie algebras L(g) and L(g) are called affine algebras associated to the reductive Liealgebrag. The subalgebras L

(g(i))

(resp. L(g(i))+Cd) are called components of L(g) (resp. L(g)).

Remark. c := g(0) +∑i>1CKi is the center of L(g) and L(g).

We identify g with the subalgebra 1 ⊗ g, let h be a Cartan subalgebra of g, and let g = n− ⊕h⊕ n+ be a triangular decomposition of g. The subalgebra h = h+ c +Cd is called the Cartansubalgebra of L(g). We also have a triangular decomposition

L(g) = n− ⊕ h⊕ n+, where (5.27)

n− =(t−1C

[t−1]⊗(n+ + h

))+C

[t−1]⊗ n−, and (5.28)

n+ =(tC[t]⊗

(n− + h

))+C[t]⊗ n+. (5.29)

For λ ∈ h∗, we denote its restriction to h by λ. We define δ ∈ h∗ by

δ|h+c = 0, 〈δ, d〉 = 1. (5.30)

GivenΛ ∈ h∗, there is a unique irreducible L(g)-module that admits a non-zero vector vΛ suchthat n+ (vλ) = 0 andh (vΛ) = 〈Λ,h〉vΛ forh ∈ h. By the uniqueness of L(Λ), we have

L(Λ) =⊗i>0

L(Λ(i)

), (5.31)

whereΛ(i) denotes the restriction ofΛ to h(i) := h ∩ L(g(i))

and L(Λ(i)

)is the L

(g(i))

-module with highest weightΛ(i).

We letki, the eigenvalue ofKi onL(Λ), be the ith level ofΛ, and letk = (k0, k1, . . .). Define

c(k) :=∑i

c (ki) , (5.32)

hΛ :=∑i

hΛ(i)(5.33)

mΛ =∑i

mΛ(i). (5.34)

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LetV be a restricted L(g)-module such thatKi acts askiIandki is distinct from the dual Coxeternumber of L

(g(i))

, which we will denote byh∨i , and which is generically defined as

h :=∑i=0

ai, (5.35)

where ` is the rank of the generic affine algebra and ai are the labels of the Dynkin diagram asbefore, anda∨i those of the diagram associated to the dual algebra or transpose matrix. LetT (i)nbe the Sugawara operators for L

(g(i))

, and let, ∀n ∈ Z,

L(i)n :=1

2(ki + h∨i

)T (i)n , Lgn :=∑i

L(i)n . (5.36)

Definition. The operatorsLgn are called the Virasoro operators for the g-moduleV .

Lettingdn 7→ Lgn, c 7→ c(k) extendsV to a module over L(g) + Vir. We also have the formula

Lg0 =∑i

Ωi

2(ki + h∨i

) − d, (5.37)

whereΩi is the Casimir operator for L(g(i))

.

5.4 Coset Virasoro construction

Let g be a reductive finite-dimensional Lie algebra with a normalized invariant form ( , ), andlet g be a reductive subalgebra of g such that ( , )|g is non-degenerate. Let

g =⊕i>0

g(i) and g =⊕i>0

g(i) (5.38)

be the decompositions of g and g. Let ( , ) be a normalized invariant form on g that coincideswith ( , ) on g(0). Due to the uniqueness of the invariant bilinear form on a simple Lie algebrawe have for x, y ∈ g(s), s > 1 that(

x(r), y(r))= jsr(x, y), (5.39)

where x(r) denotes the projection of x on g(r) and jsr is a (positive) number independent of xandy; we let j0r = 1.

Definition. The numbers jsr (s, r > 0) are called Dynkin indices.

Let V be a restricted L(g)-module such that Ki acts as kiI, ki 6= −h∨i . This is a L(g) modulewith Ki acting as kiI, where

ks =∑i

jsikj. (5.40)

Assume that ki 6= −h∨i . Put

Lg,gn := Lgn − Lgn. (5.41)

Proposition 5.5. (a) The operatorsLg,gn commute with L(g).

(b) The mapdn 7→ Lg,gn , c→ c(k) − c(k) defines a representation of Vir onV .

Definition. The Vir-module defined in Proposition 5.5 is called the coset Vir-module.

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The benefit of the coset construction is that it is a method of constructing unitary highest-weightrepresentations of the Virasoro algebra Vir, and further allows the classification of unitary high-est weight representations of the Virasoro algebra.

The Virasoro algebra is widely used in conformal field theory and string theory. It is the quan-tum version of the conformal algebra in two dimensions. The representation theory of stringson the worldsheet, the two-dimensional manifold which describes the embedding of a string inspacetime, reduces to the representation theory of the Virasoro algebra. The trace of the stress-energy tensor, the tensor that describes the density and flux of energy and momentum in space-time, and which generalizes the stress tensor of Newtonian physics, vanishes in the constraintsof the Virasoro algebra. In this way, the Virasoro algebra encodes two-dimensional conformalsymmetry and is therefore a fundamental object in physics.

References

[BP02] Stephen Berman and Karen Hunger Parshall, Victor Kac and Robert Moody: Their Pathsto Kac-Moody Lie Algebras, The Mathematical Intelligencer 24 (2002), no. 1, 50–60.

[Hum72] James Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.

[Jac62] Nathan Jacobson, Lie Algebras, John Wiley & Sons, 1962.

[Kac94] Victor G. Kac, Infinite-Dimensional Lie Algebras, Cambridge University Press, 1994.

[KP84] Victor G. Kac and Dale H. Peterson, Infinite-dimensional Lie algebras, theta functions andmodular forms, Advances in Mathematics 53 (1984), no. 2, 125 – 264.

[Mac71] I.G. Macdonald, Affine root systems and Dedekind'sη-function, Inventiones mathemati-cae 15 (1971), no. 2, 91–143.

[Ser66] J-P. Serre, Algèbres de Lie Semi-Simples Complexes, W. A. Benjamin, New York, 1966.

[Ser00] J-P. Serre, Complex Semisimple Lie Algebras, Springer Monographs in Mathematics,Springer-Verlag, New York, 2000, Reprint of the 1987 Edition.

[Sug68] Hirotaka Sugawara, A field theory of currents, Phys. Rev. 170 (1968), 1659–1662.

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