Kac-Moody Algebras and Applications

Home , Category O, Robert Moody, Unitarian trick, Victor Kac

Owen Barrett December 24, 2014

Abstract This article is an introduction to the theory of Kac-Moody algebras: their genesis, their construction, basic theorems concerning them, and some of their applications. We first record some of the classical theory, since the Kac-Moody construction generalizes the theory of simple finite-dimensional Lie algebras in a closely analogous way. We then introduce the construction and properties of Kac-Moody algebras with an eye to drawing natural connections to the classical theory. Last, we discuss some physical applications of Kac-Moody algebras,includingtheSugawaraandVirosorocosetconstructions,whicharebasictoconformal field theory.

Contents

1 Introduction2

2 Finite-dimensional Lie algebras3 2.1 Nilpotency...... 3 2.2 Solvability...... 4 2.3 Semisimplicity...... 4 2.4 Root systems...... 5 2.4.1 Symmetries...... 5 2.4.2 The Weyl group...... 6 2.4.3 Bases...... 6 2.4.4 The Cartan matrix...... 7 2.4.5 Irreducibility...... 7 2.4.6 Complex root systems...... 7 2.5 The structure of semisimple Lie algebras...... 7 2.5.1 Cartan subalgebras...... 8 2.5.2 Decomposition of g ...... 8 2.5.3 Existence and uniqueness...... 8 2.6 Linear representations of complex semisimple Lie algebras...... 9 2.6.1 Weights and primitive elements...... 9 2.7 Irreducible modules with a highest weight...... 9 2.8 Finite-dimensional modules...... 10

3 Kac-Moody algebras 10 3.1 Basic definitions...... 10 3.1.1 Construction of the auxiliary Lie algebra...... 11 3.1.2 Construction of the Kac-Moody algebra...... 12 3.1.3 Root space of the Kac-Moody algebra g(A) ...... 12

1 3.2 Invariant bilinear form...... 13 3.3 Casimir element...... 14 3.4 Integrable representations...... 15 3.5 Weyl group...... 15 3.5.1 Weyl chambers...... 16 3.6 Classification of generalized Cartan matrices...... 16 3.6.1 Kac-Moody algebras of finite type...... 17 3.7 Root system...... 17 3.7.1 Real roots...... 17 3.7.2 Imaginary roots...... 18 3.7.3 Hyperbolic type...... 19 3.8 Kac-Moody algebras of affine type...... 20 3.8.1 The underlying finite-dimensional Lie algebra...... 20 3.8.2 Relating g and g˚ ...... 21

4 Highest-weight modules over Kac-Moody algebras 22 4.1 Inductive Verma construction...... 24 4.2 Primitive vectors and weights...... 24 4.3 Complete reducibility...... 25 4.4 Formal characters...... 26 4.5 Reducibility...... 26

5 Physical applications of Kac-Moody algebras 28 5.1 Sugawara construction...... 28 5.2 Virasoro algebra...... 30 5.3 Virasoro operators...... 31 5.4 Coset Virasoro construction...... 32

1 Introduction

Élie Cartan, in his 1894 thesis, completed the classification of the finite-dimensional simple Lie algebras over C. Cartan's thesis is a convenient marker for the start of over a half-century of research on the structure and representation theory of finite-dimensional simple complex Lie algebras. 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 in Serre's Theorem [Ser66], which gives a presentation for all finite-dimensional simple complex Lie algebras. In the mid-1960s, Robert Moody and Victor Kac independently introduced a collection of Lie algebras 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 that would 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 Kac independently introduced a new class of generally infinite-dimensional, almost always simple Lie 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 came in 1972 with Macdonald's 1972 discovery [Mac71] of an affine analogue to Weyl's denominator formula. 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

2 subsequently clarified and explicated independently by both Kac and Moody, but only marked the beginning of the applications of these infinite-dimensional Lie algebras. It didn't take long for the importance of Kac-Moody algebras to be recognized, or for many applications of them to both pure mathematics and physics to be discovered. We will structure this article as follows. We will first recall some basic results in the theory of finite-dimensional Lie algebras so as to ease the introduction of their infinite-dimensional relatives. Next, we will define what is a Kac-Moody algebra, and present some fundamental theorems concerning these algebras. With the basic theory established, we will turn to some applications 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 the references 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 theory 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 the finite-dimensional case is that both Moody and Kac worked initially to generalize the methods of 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 resembles that of the approach to studying the finite-dimensional case. We follow the presentation of Serre [Ser00] in this section. We suppose throughout this section that g is a finite-dimensional Lie algebra over the field of complex numbers, though we will omit both ‘finite-dimensional’ and ‘complex’ in the statements that follow. The Lie bracket of x and y is denoted by [x, y] and the map y 7→ [x, y] is denoted ad x. For basic definitions such as that of an ideal of a Lie algebra (and for proofs of some statements given without proof in [Ser00]), c.f. Humphreys [Hum72].

2.1 Nilpotency

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

C1g = g (2.1) n n−1 C g = [g,C g] if n > 2. (2.2)

Definition. A Lie algebra g is said to be nilpotent if there exists an integer n such that Cng = 0. g is nilpotent of class r if Cr+1g = 0.

Proposition 2.1. The following conditions are equivalent.

(i) g is nilpotent of class 6 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 an abelian ideal of g. We denote the center of g by Z(g).

3 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 be nilpotent ∀x ∈ g.

Theorem 2.4. Let φ : g → End(V) be a linear representation of a Lie algebra g on a nonzero finite- dimensionalvectorspaceV. Supposethatφ(x)isnilpotent∀x ∈ g. Then∃0 6= v ∈ V whichisinvariant under g.

2.2 Solvability

n The derived series of g is the descending series (D g)n>1 of ideals of g defined by the formulæ D1g = g (2.5) n n−1 n−1 D g = [D g,D g] if n > 2. (2.6) Definition. A Lie algebra g is said to be solvable if ∃n ∈ N s.t. Dng = 0. g is solvable of derived length r+1 6 r if D g = 0. n.b. Every nilpotent algebra is solvable. Proposition 2.5. The following conditions are equivalent.

(i) g is solvable of derived length 6 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 quotients ai/ai+1 are abelian). In analogy with Engel's theorem, we now record Lie's theorem.

Theorem 2.6 (Lie). Let φ : g → End(V) be a finite-dimensional linear representation of g. If g is solvable,thereisaflagDofV suchthatφ(g) ⊂ b(D),whereb(D)istheBorelalgebraofD(thesubalgebra of End V 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 very boring. 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 solvable 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.

4 In this way, we may easily strip away the radical of a Lie algebra g and are left with a Lie algebra with 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 algebra and a solvable ideal (the so-called ‘Levi decomposition’ of g). The above justifies the restriction of study to semisimple Lie algebras. The following definition and theorem imply that we may restrict 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. Theorem2.9. ALiealgebragissemisimpleiffitisisomorphictoaproductofsimplealgebras. Indeed,let gbeasemisimpleLiealgebraand(ai) beitsminimalnonzeroideals. Theidealsai aresimpleLiealgebras, and g can be identified with their product. Theorem 2.9, together with the above discussion, shows that the Cartan classification of simple complex 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 of semisimple algebras. Theorem 2.10. Every finite-dimensional linear representation of a semisimple algebra is completely reducible. 2.4 Root systems Before turning to the structure theory of complex semisimple Lie algebras, we need to establish some basic definitions and results pertaining to symmetries of vector spaces. This theory contains the concrete ideas (most notably the Cartan matrix) that are generalized in the formulation of Kac-Moody algebras, since, as we will see in section 2.5, the theory of symmetries of vector spaces contains the information about the decomposition of complex semisimple Lie algebras.

2.4.1 Symmetries We begin by introducing the notion of a real root system, which, as we will see, is all we need to study the complex root systems that arise in the structure theory of semisimple complex Lie algebras. Let V be a (real) vector space and 0 6= α ∈ V. We begin by defining a symmetry with vector α to be any s ∈ Aut V that satisfies the conditions (i) s(α) = −α. (ii) The set Hs ⊂ V of s-invariant elements is a hyperplane of V. The set Hs is actually a complement for the line Rα, and s is an automorphism of order 2. The symmetry is totally determined by the choice of Rα and of Hs. If we denote the dual space of V by V∗ and let α∗ be the unique element of V∗ which vanishes on Hs and takes the value 2 on α. Then s(x) = x − hα∗, xiα ∀x ∈ V. (2.8) Since End V ' V∗ ⊗ V, we may instead write s = 1 − α∗ ⊗ α. (2.9) We make the following

5 Definition. Let V be a finite-dimensional vector space. A subset R ⊂ V is called a root system if (1) R is finite, spans V, and does not contain 0.

∗ (2) ∀α ∈ R there is a unique symmetry sα = 1 − α ⊗ α with vector α which leaves R invariant.

(3) If α, β ∈ R, sα(β) − β is an integer multiple of α. We say the rank of R is dim V and the elements of R are called the roots of V (relative to R). We say R is reduced if ∀α ∈ R, α and −α are the only roots proportional to α. The element α∗ ∈ V∗ as in (2.9) is called the inverse root of α.

2.4.2 The Weyl group Definition. LetRbearootsysteminavectorspaceV. TheWeylgroupofRisthesubgroupW ⊂ 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 it over W's action on V to obtain a W-invariant form. Proposition 2.11. Let R be a root system in V. There is a positive definite symmetric bilinear form ( , ) on V invariant under the Weyl group W of R. Thechoiceof( , )givesV aEuclideanstructureonwhichW actsbyorthogonaltransformations. We may re-write sα as (x, α) s (x) = x − 2 α ∀x ∈ V (2.10) α (α, α) and introduce the notion of relative position of two roots α, β ∈ R by the formula (α, β) n(β, α) = 2 . (2.11) (α, α)

2.4.3 Bases Definition. A subset S ⊂ R is called a base for R if (i) S is a basis for the vector space V.

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

β = mαα, (2.12) α∈S X wherethecoefficientsmα areintegerswiththesamesign(i.e. allnon-negativeorallnon-positive). The elements of S are also sometimes called ‘simple roots.’ Every root system admits a base. Letting R+ denote the set of roots which are linear combinations with non-negative integer coefficients of elements of S, we say an element of R+ is a positive root (with respect to S). Proposition 2.12. Every positive root β can be written as

β = α1 + ··· + αk with αi ∈ S, (2.13) in such a way that the partial sums

α1 + ··· + αh, 1 6 h 6 k, (2.14) are all roots.

6 Theorem 2.13. Suppose R is reduced and let W be the Weyl group of R.

∗ (a) ∀t ∈ V ∃w ∈ W : hw(t), αi > 0∀α ∈ S. (b) If S0 is a base for R, ∃w ∈ W such that w(S0) = S.

(d) The group W 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 the study of Kac-Moody algebras.

Definition. The Cartan matrix of R (with respect to a fixed base S) is the matrix (n(α, β))α,β∈S. The following proposition shows that the Cartan matrix encodes all the information contained in a reduced root system.

Proposition 2.14. A reduced root system is determined up to isomorphism by its Cartan matrix. That is, let R0 be a reduced root system in a vector space V 0, let S0 be a base for R0, and let φ : S → S0 be a bijection such that n(φ(α), φ(β)) = n(α, β) for all α, β ∈ S. If R is reduced, then there is an isomorphism f : V → V 0 which is an extension of φ and maps R onto R0. That is, any bijection of two bases that preserves the Cartan matrix can be extended linearly to a vector space isomorphism.

2.4.5 Irreducibility

Proposition 2.15. Suppose that V = V1 ⊕ V2 and that R ⊂ V1 ∪ V2. Put Ri := R ∩ Vi. Then

(a) V1 and V2 are orthogonal.

(b) Ri is a root system in Vi.

One says that R is the sum of the subsystems Ri. If this can happen only trivially (that is, with V1 or V2 equal to 0), and if V 6= 0, then R is said to be irreducible. Proposition 2.16. Every root system is a sum of irreducible systems. 2.4.6 Complex root systems Thedefinitionofcomplexrootsystemisidenticaltothedefinitionintherealcaseafterreplacing the real vector space V with a complex one. Actually, it is a theorem that every complex root system may be obtained by extension of scalars of some real root system. Namely, if R is a root system 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 ∈ R the symmetry sα of V is a linear extension of the corresponding symmetry in V0. This shows that 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 this section, g is a finite-dimensional Lie algebra over C.

7 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 largest subalgebra 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 two conditions.

(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 h is 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 subalgebra of g. If α ∈ 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 weight α. In particular, g0 is the set of elements in g that commute with h; since g is semisimple, h is maximal abelian subalgebra of g and so actually g0 = h.

L α Theorem 2.17. One has g = h α∈R g . 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 be denoted R. This suggestive terminology is not a coincidence.

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

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

(c) The subspaces gα and gβ are orthogonal if α + β 6= 0. The subspaces gα and g−α are dual with respect to ( , ). The restriction of ( , ) to h is nondegenerate. 2.5.3 Existence and uniqueness We conclude our remarks on the structure of complex semisimple Lie algebras with the following theorems, which will make clear the significance of the root system (encoded in the Cartan matrix) 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 to R.

Theorem 2.20 (Uniqueness). Two semisimple Lie algebras corresponding to isomorphic root systems are isomorphic. More precisely, let g, g0 be semismple Lie algebras, h, h0 respective Cartan subalgebras of g, g0, S, S0 bases for the corresponding root systems, and r : S → S0 a bijection sending the Cartan 0 0 0 i 0j matrixofStothatofS . Foreachi ∈ S, j ∈ S ,letXi,Xi benonzeroelementsofg andg ,respectively. 0 0 0 Then there is a unique isomorphism of Lie algebras f : g → g sending Hi to Hr(i) and Xi to Xr(i) for all i ∈ S.

8 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 its linear representations. Let h ⊂ g be a Cartan subalgebra of g and R the corresponding root + system. We fix a base S = {α1, . . . , αn} of R and denote by R the set of positive roots with respect to S. We recall the definition of Hα, the inverse root of α.

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

[Xα,X−α] = Hα ∀α ∈ R. (2.16)

α −α Fix such Xα ∈ g , Yα ∈ g so that [Xα,Yα] = Hα. When α is one of the simple roots αi, we α − α write Xi,Yi,Hi instead of Xαi , etc. We put n := α>0 g , n := α<0 g , and b := h ⊕ n. P P 2.6.1 Weights and primitive elements Let V be a g-module of possibly infinite dimension and let ω ∈ h∗ be a linear form on h. Let Vω denote the vector subspace of all v ∈ V such that Hv = ω(H)v∀H ∈ h. An element of Vω is said to have weight ω. The dimension of Vω is called the multiplicity of ω in V. If Vω 6= 0, ω called a weight of V.

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

0 ω (b) The sum V := ω V is direct and a g-submodule of V. One says that some vP∈ V is a primitive element of weight ω of it satisfies the conditions

(i) v 6= 0 has weight ω.

+ (ii) Xαv = 0∀α ∈ R (equivalently, ∀α ∈ S). Given a primitive element of weight ω, it generates an indecomposable g-module in which ω is a weight of multiplicity 1.

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

Theorem 2.23. Let V be an irreducible g-module containing a primitive element v of weight ω. Then

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

(b) The weights π of V have the form

π = ω − miαi with mi ∈ N. (2.17)

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

(c) For two irreducible g-modules V1 and V2 with highest weights ω1 and ω2 toP be isomorphic, it is necessary and sufficient for ω1 = ω2. Remark. One can give examples of irreducible g-modules with no highest weight. These modules are necessarily infinite-dimensional.

Theorem 2.24. For each ω ∈ h∗, there exists an irreducible g-module with highest weight equal to ω.

9 Remark. The above theorems establish a bijection between the elements of h∗ and the classes of irreducible g-modules with highest weight. 2.8 Finite-dimensional modules Proposition 2.25. Let V be a finite-dimensional g-module. Then

π (a) V = π V .

(b) If π isP a weight of V, π(Hα) ∈ Z∀α ∈ R. (c) If V 6= 0, V contains a primitive element. (d) If V is generated by a primitive element, then V is irreducible.

∗ Theorem2.26. Let ω ∈ h andletEω beanirreducible g-modulehavingω ashighestweight. For Eω to be finite-dimensional, it is necessary and sufficient that

+ ∀α ∈ R , ω(Hα) ∈ Z>0. (2.18) 3 Kac-Moody algebras

HavingsummarizedthebasicsofthestructuretheoryofsemisimpleLiealgebrasover C andthe basics of their highest-weight linear representations, we may now proceed naturally to the formulation 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 corresponding to a reduced root system determines a unique finite-dimensional semisimple complex Lie algebra. Therefore, generalizing the Cartan matrix provides a neat way to generalize the Lie algebra. Note that the formulation of the generalized Cartan matrix is not the only way to approach L Kac-Moodyalgebras. Earlyon,KacconsideredarbitrarysimpleZ-gradedLiealgebrasg = j gj of ‘finite growth,’ meaning dim gj grew only polynomially in j. Nevertheless, we will present the approach of the generalized Cartan matrix here, following Kac [Kac94].

3.1 Basic definitions n We start with a complex n × n matrix A = (aij)i,j=1 of rank ` and we will associate with it a complex Lie algebra g(A). Definition. We call A 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 implies aji = 0. (C3) However, we may start with an even more general object than a generalized Cartan matrix; let now 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 = α1 , . . . , αn ⊂ h are indexed subsets of h and h, respectively, satisfying the conditions both sets II and II∨ are linearly independent; (D1) ∨ αi , αj = aij for i, j = 1, . . . , n; (D2) n − ` = dim h − n. (D3)

Tworealizations h, II, II∨ and h0, II0, II0∨ arecalledisomorphicifthereexistsavectorspace isomorphism φ : h → h0 such that φ II∨ = II0∨ and φ∗ (II0) = II.

10 Proposition 3.1. Fix an n × n matrix A. Then there exists a realization of A, and this realization is unique up to isomorphism. ∨ ∨ Given two matrices A1 and A2 and their realizations h1, II1, II1 and h2, II2, II2 , we obtain a realization of the direct sum A1 ⊕ A2 of the two matrices as ∨ ∨ h1 ⊕ h2, II1 × {0} ∪ {0} × II2, II1 × {0, } ∪ {0} × II2 , (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 (permuting the 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. Wenowintroducesometerminologythatstrengthenstheanalogywiththefinite-dimensional case. We call II the root base, II∨ the coroot base, and elements from II (resp. II∨) simple roots (resp. simple coroots). We also put n n

Q := Zαi,Q+ := Z+αi, (3.2) i=1 i=1 X X and call the lattice Q the root lattice. Given any α = i kiαi ∈ Q, the number ht α := i ki ∗ is called the height of α. We put a partial order on h by setting α µ if λ − µ ∈ Q+. P > P 3.1.1 Construction of the auxiliary Lie algebra Given an n× n complex matrix A and a realization h, II, II∨ of A, we first introduce an auxiliary Lie algebra g(˜ A) with the generators ei, fi (i = 1, . . . , n) and h and the following defining relations. ∨ [ei, fj] = δijαi i, j = 1, . . . , n, [h, h0] = 0 h, h0 ∈ h, (DR) [h, ei] = hαi, hi ei

[h, fi] = − hαi, hi fi i = 1, . . . , n; h ∈ h.

The uniqueness of the realization guarantees that g(˜ A) depends only on A. Denote by n˜+ (resp n˜−) the subalgebra of g(˜ A) generated by e1, . . . , en (resp. f1, . . . , fn). We now record the following 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 to an involution of the Lie algebra g(˜ A). (d) With respect to h one has the root space decomposition M M g(˜ A) = g˜−α ⊕ h ⊕ g˜α , (3.3)

α∈Q+ α∈Q α6=0 α6=0

where g˜α = {x ∈ g(˜ A):[h, x] = α(h)x}. Additionally, dimg ˜α < , and g˜α ⊂ 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 generators ei, fi (i = 1, . . . , n), and h, and the definining relations (DR). By Theorem 3.2, the natural map h → g(˜ A) is an embedding. Let r be the maximal ideal in g(˜ A) intersecting h trivially, as in Theorem 3.2(e). Then, put

g(A) := g(˜ A)/r. (3.5)

The matrix A is called the Cartan matrix of the Lie algebra g(A), and n is called the rank of g(A). Wesayg(A), h, II, II∨isthequadrupleassociatedtoA,andsaytwoquadruplesg(A), h, II, II∨ and g(A0), h0, II0, II0∨ are isomorphic if there exists a Lie algebra isomorphism φ : g(A) → g(A0) such that φ(h) = h0, φ II∨ = II0∨, and φ∗ (II0) = 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.

Were-usethenotationei, fi, hfortheimagesofthesegeneratorsing(A)afterthequotientbyr. Inanalogywiththeclassicalfinite-dimensionalcase,wecallthesubalgebrah ⊂ g(A)theCartan subalgebra and the elements ei, fi the Chevalley generators of g(A). The Chevalley generators generate the derived subalgebra g0(A) := D2g(A) = [g(A), g(A)], and

g(A) = g0(A) + h, (3.6) where this decomposition is not necessarily direct.

Remark. g(A) = g0(A) if and only if det A 6= 0. Quadruples decompose like realizations.

Proposition 3.3. The quadruple associated to a direct sum of matrices Ai is isomorphic to a direct sum of the quadruples associated to the Ai. The root system of g(A) is the union of the root systems of the g (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 : hαi, hi = 0∀i = 1, . . . , n} . (3.7)

Furtherore, dim c = n − `, where ` = rk A.

We record the necessary and sufficient conditions on the matrix A for the algebra g(A) to be simple.

Proposition 3.5. g(A) is simple if and only if det A 6= 0 and for each pair of indices i and j, there exist indices i1, i2, . . . , is such that the product aii1 ai1i2 ··· aisj 6= 0. 3.1.3 Root space of the Kac-Moody algebra g(A)

0 n ∨ 0 0 0 Continuing in analogy, put h := i=1 Cαi . 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 spacedecomposition with respect P to h. M g(A) = gα. (3.8) α∈Q

12 In further analogy with section 2.5.2, we have g0 = h, and we define the multiplicity mult α := | ht α| dim gα. Note that mult α 6 n . Call a nonzero element α ∈ Q a root if mult α 6= 0. A root α > 0 (resp. < 0) is called positive (resp. negative), and (3.3) assures us that α must be either positive or negative. Denoting the set of roots (resp. positive, negative roots) by ∆ (resp. ∆+, ∆−), ∆ decomposes disjointly as

∆ = ∆+ t ∆−. (3.9)

Let n+ (resp n−) denote the subalgebra of g(A) generated by e1, . . . , 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, ω˜ induces an involution ω of g(A) called the Chevalley involution of g(A). As ω (gα) = g−α, we have that mult α = mult(−α) and ∆− = −∆+.

3.2 Invariant bilinear form Though we omitted it in our discussion, every finite-dimensional Lie algebra carries a distinguished symmetric bilinear form called the Killing form. We now present some results pertaining 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 terminology means. On the level of matrices, an n × n matrix A is called symmetrizable if there ex- istsaninvertiblediagonalmatrixDwithdii = i,andasymmetricmatrixBsuchthatA = DB. The matrix B is called a symmetrization of A and g(A) is called a symmetrizable Lie algebra, since ∨ ∨ rescaling the Chevalley generators ei 7→ ei, fi 7→ ifi for i 6= 0 rescales αi 7→ iαi , which extends to an isomorphism h 7→ h (nonunique, if det A = 0), which in turn extends to an isomorphism of Lie algebras g(A) → g(DA). Let A be a symmetrizable matrix with a fixed decomposition A = DB as above, and let ∨ 00 0 ∨ h, II, II be a realization of A. Fix a complementary subspace h to h = Cαi in h, and define a symmetric bilinear C-valued form ( , ) on h by the two equations P ∨ αi , h = hαi, hii for h ∈ h, i = 1, . . . , n (3.11) (h0, h00) = 0 for h0, h00 ∈ h00. (3.12)

∨ ∨ By the decomposition A = DB and since α1 , . . . , αn are linearly independent, we conclude from (3.11) that

∨ ∨ αi , αj = bijij for i = 1, . . . , n, (3.13) so there is no ambiguity in the definition of ( , ). Proposition 3.6. The bilinear form ( , ) is nondegenerate on h.

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

hν(h), h1i = (h, h1), h, h1 ∈ h, (3.14) and the induced bilinear form ( , ) on h∗. Then (3.11) implies

∨ ν(αi ) = iαi, i = 1, . . . , n. (3.15) Combining with (3.13) yields

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

13 Theorem 3.7. Lt g(A) be a symmetrizable Lie algebra; fix a decomposition of A. Then there exists a nondegenerate 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 if α + β 6= 0. (d) ( , )| is nondegenerate for α 6= 0, and hence g and g are nondegenerately paired by gα+g−α α −α ( , ).

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

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

3.3 Casimir element L Let g(A) be a Lie algebra associated to a matrix A, h the Cartan subalgebra, and g = α gα the root space decomposition with respect to h.

Definition. A g(A)-module V is called restricted if for every v ∈ V, we have gα(v) = 0 for all but a finite number of positive roots α. 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 that A is symmetrizable and ( , ) an invariant bilinear form (guaranteed by The- orem 3.7). Given a restricted g(A)-module V, we construct a linear operator Ω on V, called the (generalized) Casimir operator, as follows. Define implicity a linear function ρ ∈ h∗ by 1 hρ, α∨i = a i = 1, . . . , n. (3.17) i 2 ii If det A = 0, this does not specify ρ uniquely, and hence we must choose one. Then (3.15) and (3.16) imply 1 (ρ, α ) = (α , α ) i = 1, . . . , n. (3.18) i 2 i i

(i) (i) Next, for each α ∈ ∆+, choose a basis eα of the space gα, and form the dual basis e−α of g−α. We then define an operator Ω0 on V by

(i) (i) Ω0 := 2 e−αeα . (3.19) α∈∆ i X+ X

Ω0 is independent of the choice of basis. Since ∀v ∈ V, only a finite number of summands 1 2 are nonzero, Ω0 is well-defined on V. Let u1, u2,... and u , u ,... be dual bases of h. The generalized Casimir operator is defined by

−1 i Ω = 2ν (ρ) + u ui + Ω0. (3.20) i X 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 the Casimir operator Ω commutes with the action of g(A) on V.

14 We now give a result about eigenvalues of Ω.

Corollary 3.9. Under the same hypotheses as Theorem 3.8, if there exists some v ∈ V such that the ∗ Chevalley generators ei vanish on v for i = 1, . . . , n, and h(v) = hΛ, hiv for some Λ ∈ h and all h ∈ 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)-module V is L called h-diagonalizable if V = λ∈h∗ Vλ, where

Vλ = {v ∈ V : h(v) = hλ, hiv∀h ∈ h} . (3.22)

∗ Vλ is called a weight space, λ ∈ h is called a weight if Vλ 6= 0, and dim Vλ is called the multiplicity of λ and is denoted multV λ.A h-diagonalizable module over a Kac-Moody algebra g(A) is called integrable if all ei and fi are locally nilpotent on V. (An operator T is locally nilpotent on a vector space V if T is nilpotent on every finite dimensional subspace of V.)

Remark. g(A) is an integrable g(A)-module under the adjoint representation; i.e. ad ei and ad fi are locally 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 + Cαi + Cfi. Then

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

∨ as C-algebras with standard basis ei, αi , fi .

Proposition 3.11. Let V be an integrable g(A )-module. As a g(i)-module, V decomposes into a direct sum 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 space h by

∨ ∗ ri(λ) = λ − λ, αi αi, α ∈ h . (3.24)

∗ ∨ One can verify ri is a reflection since its fixed point set is Ti = λ ∈ h : λ, αi = 0 , and r (α ) = −α . i i i Definition. The subgroup W ⊂ GL (h∗) generated by all fundamental reflections is called the Weyl group W of g(A). The following proposition justifies this definition.

Proposition 3.12. Let V 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 of V is W-invariant.

(b) The root system ∆ of g(A) is W-invariant, and mult α = mult w(α)∀α ∈ ∆, w ∈ W. We now consider the special case of a symmetrizable Kac-Moody algebra.

15 Proposition 3.13. Let A be a symmetrizable generalized Cartan matrix and let ( , ) be a standard invariant bilinear form on g(A). The restriction of the bilinear form ( , ) to h∗ is W-invariant. 3.5.1 Weyl chambers It is basic to understand the geometry of the action of the Weyl group. First, let A be an n × n ∨ matrix over R, and let hR, II, II be a realization of the matrix A over R; i.e. hR is a real vector ∨ space of dimension 2n − `, so that, extending hR by scalars to h := hR ⊗ C, h, II, II is a realization of A over C.

∨ Remark. hR isstableunderW sinceQ ⊂ hR,whereQisthecorootlattice(c.f.(3.2)forthedefinition of the root lattice).

Definition. The set

C := {h ∈ hR : hαi, hi > 0 for i = 1, . . . , n} (3.25) is called the fundamental chamber.

Definition. The sets w(C), w ∈ W are called chambers, and their union [ X = w(C) (3.26) w∈W is called the Tits cone.

∨ ∨ ∗ Remark. We have the corresponding dual notions of C and X in hR. We now record a proposition that characterizes the action of the Weyl group on chambers.

Proposition 3.14. (a) For h ∈ C, the group Wh = {w ∈ W : w(h) = h} is generated by the fundaental reflections that it contains.

(b) ThefundamentalchamberCisafundamentaldomainfortheactionofW onX;i.e. anyorbitW·h of h ∈ X intersects C in exactly one point. In particular, W acts simply transitively on chambers.

∨ (d) C = h ∈ hR : ∀w ∈ W, h − w(h) = i ciαi , where ci > 0 . (e) The following conditions are equivalent: P

(i) |W| < ;

(ii) X = hR; ∞ (iii) |∆| < ;

(iv) ∆∨ < . ∞ (f) If h ∈ X, then |W | < if and only if h lies in the interior of X. ∞h 3.6 Classification of generalized Cartan matrices ∞ It is a convenient fact about Kac-Moody algebras that basic properties of the generalized Cartan matrix A translate into essential facts about the algebra g(A). Suppose A is a real n × n matrix satisfying the following three properties (we may reduce to the first and a generalized Cartan matrix satisfies the rest).

(m1) A is indecomposable.

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

(m3) aij = 0 implies aji = 0. The main result is then as follows.

Theorem 3.15. Let A be a real n × n matrix satisfying (m1), (m2), and (m3). Then one and only one of the following three possibilities holds.

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

(Aff) corank A = 1; ∃u > 0 : Au = 0; Av > 0 implies Av = 0;

(Ind) ∃u > 0 : Au < 0; Av > 0, v > 0 imply v = 0. Definition. Wesaythatamatrixsatisfying(Fin),(Aff),and(Ind)isof finite,affine,or indefinitetype, respectively. Corollary3.16. LetAbeamatrixsatisfying(m1),(m2),and(m3). ThenAisoffinite,affine,orindefinite type if and only if there exists an α > 0 such that Aα > 0, = 0, or < 0, respectively. 3.6.1 Kac-Moody algebras of finite type OnesuccessofthegeneralizationofKac-Moodyisthatonerecoversthefullcollectionofclassical finite-dimensional complex semisimple Lie algebras in the set of Kac-Moody algebras of finite type. Proposition 3.17. Let A be an indecomposable generalized Cartan matrix. Then the following conditions are equivalent. (i) A is a generalized Cartan matrix of finite type;

(ii) A is symmetrizable and the bilinear form ( , )hR is 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 classical theory, is the notion of an imaginary root, which is an entirely new notion. The classical theory of roots is recovered in the complementary notion of a real root.

3.7.1 Real roots Definition. A root α ∈ ∆ is called real if there exists a w ∈ W such that w(α) is a simple root.

re re We denote the set of all real roots by ∆ and the set of all positive real roots by ∆+. Given any re ∨ ∨re α ∈ ∆ , then α = w(αi) for some αi ∈ II, w ∈ W. Define the dual (real) root α ∈ ∆ ∨ ∨ by α = w αi . This is independent of the choice of presentation α = w (αi). We define a re reflection rα with respect to α ∈ ∆ by

∨ ∗ rα(λ) = λ − λ, α α, λ ∈ h . (3.27)

The following proposition strongly evokes the classical setting.

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

(a) mult α = 1.

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

such that β + kα ∈ ∆ ∪ {0} if and only if −p 6 k 6 q, k ∈ Z. (d) SupposethatAissymmetrizableandlet( , )beastandardinvariantbilinearformong(A). Then

(i) (α, α) > 0, (ii) α∨ = 2ν−1(α)/(α, α), and

(iii) if α = i kiαi, then ki(αi, αi) ∈ (α, α)Z. (e) Provided thatP±α 6∈ II, ∃i such that

|ht ri(α)| < |ht α| . (3.29)

Definition. LetAbeasymmetrizablegeneralizedCartanmatrixand( , )astandardinvariantbilinear form. Then, given a real root α, we have (α, α) = (αi, αi) for some simple root αi. We call α 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. If A is symmetric, then all simple roots and hence all real roots have the same square length. 3.7.2 Imaginary roots Definition. A root α ∈ ∆ is called imaginary if it is not real.

im im Wedenotethesetofallimaginaryrootsby ∆ andthesetofallpositiveimaginaryrootsby ∆+ . 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.

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

im ∨ ∨ (b) For α ∈ ∆+ , ∃!β ∈ −C (i.e. β, αi 6 0∀i) which is W-equivalent to α. (c) If A is symmetrizable and ( , ) is a standard invariant bilinear form, then α ∈ ∆ is imaginary if and only if (α, α) 6 0. im im (d) If α ∈ ∆+ and r is a nonzero rational number such that rα ∈ Q, then rα ∈ ∆ . In particular, nα ∈ ∆im if n ∈ Z − {0}. We now state when imaginary roots exist.

Theorem 3.20. Let A be an indecomposable generalized Cartan matrix.

(a) If A is of finite type, ∆im = ∅.

18 (b) If A is of affine type, then

im ∆+ = {nδ, n ∈ Z} , (3.31)

` where δ = i=0 aiαi is a linear combination of roots αi in terms of constants ai, which are effectively computable from the matrix A,1 and P im ∨ (c) If A is of indefinite type, then ∃α ∈ ∆+ , α = i kiαi, such that ki > 0 and α, αi < 0∀i = 1, . . . , n. P The Tits cone X may be described naturally in terms of imaginary roots.

Proposition 3.21. (a) If A is of finite type, then X = hR. (b) If A is of affine type, then

−1 X = {h ∈ hR : hδ, hi > 0} ∪ Rν (δ). (3.32)

im X = h ∈ hR : hα, hi > 0 for all α ∈ ∆+ , (3.33) where X denotes the closure of X in the metric topology of hR. Remark. Proposition 3.21 has a geometric interpretation in terms of the imaginary cone Z, the convex ∗ im hull in hR of the set ∆+ ∪ {0}. Namely, th cones Z and X are dual to each other;

X = h ∈ hR : hα, hi > 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 on it.

0 0 0 Definition. A linearly independent set of roots II = {α1, α2,...} is called a root base of ∆ if each 0 root α can be written in the form α = ± i kiαi, where ki ∈ Z+. Proposition 3.22. Let A be an indecoposableP generalized Cartan matrix. Then any root base II0 of ∆ is W-conjugate to II or −II.

Remark. II is W-conjugate to −II if and only if A is of finite type. 3.7.3 Hyperbolic type Definition. A generalized Cartan matrix A is called a matrix of hyperbolic type if it is indecomposable symmetrizable of indefinite type, and if every proper connected subdiagram of the Dynkin diagram associated to A is of finite or affine type.

Proposition 3.23. Let A be a generalized Cartan matrix of finite, affine, or hyperbolic type. Then (a) The set of all short real roots is

2 2 α ∈ Q : |α| = a = min |αi| . (3.35) i

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

19 (b) The set of all real roots is

α = k α ∈ Q : |α|2 > 0 and k |α |2 ∈ Z∀j . (3.36)  j j j j  j  X  (c) The set of all imaginary roots is 

2 α ∈ Q − {0} : |α| 6 0 . (3.37) (2) (d) If A is affine, then there exist roots of intermediate squared length m if and only if A = A2` , 2 2 ` > 2. The set of such roots coincides with α ∈ Q : |α| = m . 3.8 Kac-Moody algebras of affine type Aconvenient featureof Kac-Moodyalgebrasof affinetype isthatall thebasic informationabout themcanbeexpressedintermsofcorrespondingobjectsforan‘underlying’simplefinite-dimensional Lie algebra. We make this notion precise in what follows. Let A be a generalized Cartan matrix ofaffinetypeoforder`+1 (andrank`). Letg = g(A)betheaffinealgebraassociatedtoamatrix ∗ A of affine type; let h be its Cartan subalgebra, II = {α0, . . . , α`} ⊂ h the set of simple roots, ∨ ∨ ∨ ∨ II = α0 , . . . , α` ⊂ h the set of simple coroots, ∆ the root system, and Q and Q the root and coroot lattices. Proposition 3.4 implies that the center of g is one-dimensional and spanned by a linear combination of coroots3 called the canonical central element and denoted by K. More ∨ precisely, with ai as in Theorem 3.20(b), let ai be the corresponding labels corresponding to the dual algebra. Then

` ∨ ∨ K = ai αi . (3.38) i=0 X

Note that [g, g] has corank 1 in g. Fixing an element d ∈ h satisfying hαi, di = 0∀i = 1, . . . , n ∨ ∨ and hα0, di = 1, which we call the scaling element, it is clear that the elements α0 , . . . , 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 the normalized 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 α1 , . . . , α` . The dual ˚∗ ˚∗ notions h and hR 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), and Λ0 ∈ h is defined implicitly by the equations

∨ Λ0, αi = δ0i for i = 0, . . . , `; hΛ0, di = 0. (3.41)

Denote now by g˚ the subalgebra of g generated by the ei and fi with i = 1, . . . , `. g˚ is a Kac- Moodyalgebraassociatedtothematrix A˚ obtainedfrom A bydeletingthe0throwandcolumn.

2This ‘type’ belongs to the classification of generalized Cartan matrices of affine type, and is given in terms of its Dynkin 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 Cartan ˚ ˚ ∨ ∨ ∨ subalgebra. II = {α1, . . . , α`} is the root basis and II = α1 , . . . , 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 theroot system of g˚; it is finiteand consists ofreal roots (againby Proposition3.17), the set ∆˚ + = ∆˚ ∩ ∆+ being the set of positive roots. Denote by ∆˚ s and ∆˚ ` the sets of short and long roots, respectively, in ∆˚. Put Q˚ = Z∆˚. Let W˚ be the Weyl group of ∆˚.

3.8.2 Relating g and g˚ Recalltheclassificationofimaginaryandpositiveimaginaryrootsofg,followingTheorem3.20(b). re re ˚ We now characterize the set of real roots ∆ and positive real roots ∆+ in terms of ∆ and δ.

Proposition 3.24. (a) ∆re = α + nδ : α ∈ ∆,˚ n ∈ Z if r = 1.

re (b) ∆ = {α + nδ : α ∈ α∆s, n ∈ Z} ∪ {α + nrδ : α ∈ α∆`, n ∈ Z} if r = 2 or 4, but A is ( 4 not of type A 2)2`.

(2) (c) If A is of type A2` , then 1 ∆re = (α + (2n − 1)δ : α ∈ ∆˚ , n ∈ Z ∪ 2 `

(3.42) α + nδ : α ∈ ∆˚ s, n ∈ Z ∪

α + 2nδ : α ∈ ∆˚ `, n ∈ Z .

(d) ∆re + rδ = ∆re. re re ˚ (e) ∆+ = {α ∈ ∆ with n > 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 algebra g in terms of a finite-dimensional Kac-Moody algebra g˚. We extend this progress to a characterization of the Weyl group of the affine algebra g. Recall that W is generated by fundamental reflections ∗ r0, r1, . . . , r` which act on h by

∨ ∗ ri(λ) = λ − λ, αi , λ ∈ h . (3.45)

∨ As δ, αi = 0 for i = 0, . . . , `, we have that w(δ) = δ∀w ∈ W. (3.46)

Recall that the invariant standard form is also W-invariant. Denote by W˚ the subgroup of W generated by r1, . . . , r`. As ri(Λ0) = Λ0 for i = 1, . . . , `, ∗ ∗ W˚ acts trivially on CΛ0 + 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 < .

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

21 Recall that for a real root α, we have the reflection rα ∈ W given by

∨ ∗ rα(λ) = −λ − λ, α , λ ∈ h . (3.47)

Form the distinguished element

θ := δ − a0α0 = aiαi ∈ Q,˚ (3.48) i=1 X ∗ ∗ where the ai are as in Theorem 3.20(b). Given any α ∈ h˚ , we can form an operator on tα on h given by, for any λ ∈ h∗,

1 t (λ) = λ + hλ, Kiα − (λ, α) + |α|2hα, Ki δ. (3.49) α 2

˚∗ ˚ ∨ ˚ Now form the distinguished lattice M ⊂ hR. Let Z(W · θ ) denote the lattice in hR spanned over Z by the (finite) set W˚ ·θ∨, where θ∨ denotes the θ element from the dual algebra g ( tA), and set M = ν(Z(W˚ · θ∨)). M has the following properties.

M = Q = Q˚ if A is symmetric or r > a0; (3.50) M = ν(Q∨) = ν(Q˚ ∨) otherwise.

The lattice M, considered as an abelian group, acts faithfully on h∗ by (3.49). We call the corresponding subgroup of GL(h∗) the group of translations and denote it by T. We may finally state the decomposition of the Weyl group W 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 the Weyl group of an affine algebra g in terms of the corresponding objects for the ‘underlying’ simple 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 less attention to their representation theory. The representation theory of Kac-Moody algebras is a rich 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 category O and its Verma modules. We follow [Kac94, Ch. 9]. We start with an arbitrary complex n × n matrix A and consider the associated Lie algebra g(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- L tion V = λ∈h∗ Vλ by weight spaces Vλ (c.f. §3.4).

Definition. A nonzero vector from Vλ is called a weight vector of weight λ.

22 ∗ ∗ Let P(V) = {λ ∈ h : Vλ 6= 0} denote the set of weights of V. Finally, for λ ∈ h , set D(λ) = ∗ {µ ∈ h : µ 6 λ}. Definition. ThecategoryOisdefinedasfollows. ObjectsinOareg(A)-modulesV whichareh-diagonalizable withfinite-dimensionalweightspacesandsuchthatthereexistsafinitenumberofelementsλ1, . . . , λs ∈ h∗ such that

s [ P(V) ⊂ D (λi) . (4.3) i=1

Morphisms in O are homomorphisms of g(A)-modules. It follows from the following general fact that any submodule or quotient module of a module in O is in O.

Proposition 4.1. Let h be a commutative Lie algebra, V a diagonalizable h-module, i.e. M V = Vλ, where Vλ = {v ∈ V : h(v) = λ(h)v∀h ∈ h} . (4.4) λ∈h∗

Then any submodule U of V 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 is likewise the case that every module from O is restricted.

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

n+ (vΛ) = 0, h (vΛ) = Λ(h)vΛ for h ∈ h, and U(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 M V = Vλ; VΛ = CvΛ; dim Vλ < . (4.7) λ6Λ ∞ Therefore, a highest-weight module lies in O, and every two highest-weight vectors are proportional.

Definition. A g(A)-module M(Λ) with highest weight Λ is called a Verma module if every g(A)- module with highest Λ is a quotient of M(Λ).

Proposition 4.2. (a) For every Λ ∈ h∗, there exists a unique (up to isomorphism) Verma module M(Λ).

(b) Viewed as a U (n−)-module, M(Λ) is a free module of rank 1 generated by a highest-weight vector.

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 a with its image in 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 on V and on U(b) inthebilineartensorproduct. Here, theactionof b isinducedbyleftmultiplicationin U(b). Then,wemayconstructaVermamoduleasaninduced(n+h)-moduleasfollows. Definetheleft (n+h)-module Cλ with underlying space C by n+(1) = 0, h(1) = hλ, hi1 for h ∈ h. Then

M(λ) = U(g(A)) ⊗U(n++h) Cλ. (4.9)

4.2 Primitive vectors and weights It follows from Proposition 4.2(c) that among the modules with highest weight Λ there is a unique irreducible one, namely

L(Λ) := M(Λ)/M0(Λ). (4.10)

It is evident that L(Λ) is a quotient of any module with highest weight Λ. Toshow that the modules L(Λ) exhaust all irreducible modules from the category O, as well as for other purposes, we make the following definitions. Let V be a g(A)-module.

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

v 6∈ U; n+(v) ⊂ U. (4.11)

Then λ is called a primitive weight.

One defines primitive vectors and weights for a g0(A)-module similarly. A weight vector v such that n+(v) = 0 is obviously primitive. Proposition 4.3. Let V be a nonzero module from the category O. Then

(a) V contains a nonzero weight vector v such that n+(v) = 0. (b) The following conditions are equivalent.

(i) V is irreducible; (ii) V is a highest weight module and any primitive vector of V is a highest-weight vector; (iii) V ' L(Λ) for some Λ ∈ h∗.

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

n+ (vΛ) = 0 and h (vΛ) = Λ(h)vΛ for h ∈ 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 ∗ ∗ ∗ Let L(Λ) nowethe g(A)-moduledualto L(Λ). Then L(Λ) = λ (L(Λ)λ) . Thesubspace Q ∗ M ∗ 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, and ∗ that for v ∈ (L(Λ)Λ) one has

n−(v) = 0; h(v) = −hΛ, hiv for h ∈ h. (4.14)

Such a module is called an irreducible module with lowest weight −Λ. As before, we have a bijection between h∗ and irreducible lowest-weight modules given by Λ 7→ L∗(−Λ). ∗ Denote by πΛ the action of g(A) on L(Λ), and introduce the new action πΛ on the space L(Λ) by

∗ πΛ(g)v = πΛ(ω(g))v, (4.15)

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

Proposition4.5. Everyg(A)-moduleL(Λ)carriesaunique-up-to-constant-factornondegeneratecon- travariant bilinear form B. The form is symmetric and L(Λ) decomposes into an orthogonal direct sum of weight spaces with respect to this form.

4.3 Complete reducibility We now may attack the question of reducibility.

Lemma 4.6. Let V be a g(A)-module from the category O. If for any two primitive weights λ and µ of V the inequality λ > µ, then the module V is completely reducible. It is not the case that each module V ∈ O admits a composition series (a sequence of submodules V ⊃ V1 ⊃ V2 ⊃ · · · such that each Vi/Vi+1 is irreducible. However, there is a kind of substitute for this property that does hold.

Lemma 4.7. Let V ∈ O and λ ∈ h∗. Then there exists a filtration by a sequence of submodules V = Vt ⊃ Vt−1 ⊃ · · · ⊃ V1 ⊃ V0 = 0 and a subset J ⊂ {1, . . . , t} such that

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

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

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

Definition. Letthemultiplicityof L(µ) in V, denotedby [V : L(µ)], bethenumberoftimes µ appears among {λj : j ∈ J}, which is independent of the filtration and the choice of λ. Remark. L(µ) has a nonzero multiplicity in V if and only if µ is a primitive weight of V.

25 4.4 Formal characters TostudytheformalcharactersofmodulesfromO,wedefineanalgebraEoverCinthefollowing way. Elements of E are sums

cλe(λ), (4.17) λ∈h∗ X where cλ ∈ C and cλ = 0 for λ outside the union of a finite number of sets of the form D(λ). E becomes a commutative associative algebra if we set e(λ)e(µ) = e(λ + µ) and extend by linearity in the usual way. e(0) is our identity.

Definition. The elements e(λ) are called formal exponents. Formalexponentsarelinearlyindependentandinone-to-onecorrespondencewiththeelements λ ∈ h∗. L Now fix V a module from the category O and let V = λ∈h∗ Vλ be its weight space decomposition.

Definition. Define the formal character of V by

ch V = (dim Vλ) e(λ) ∈ E. (4.18) λ∈h∗ X Proposition 4.8. Let V be a g(A)-module from O. Then

ch V = [V : L(λ)] ch L(λ). (4.19) λ∈h∗ X We record now a formula for the formal character of a Verma module M(Λ). Lemma 4.9.

ch M(Λ) = e(Λ) (1 − e(−α))− mult α. (4.20) α∈∆ Y+ Suppose now that A is a symmetrizable matrix and let ( , ) be a bilinear form on g(A). Then the generalized Casimir operator Ω acts on each module from O.

Lemma 4.10. (a) If V is a g(A)-module with highest weight Λ, then

2 2 Ω = |Λ + ρ| − |ρ| IV . (4.21)

(b) If V is a module from the category O and v is a primitive vector with weight λ, then there exists a submodule U ⊂ V such that v 6∈ U and

Ω(v) = |λ + ρ|2 − |ρ|2 v mod U. (4.22)

Proposition 4.11. Let V be a g(A)-module with highest weight Λ. Then

ch V = cλ ch M(λ), where cλ ∈ Z, cλ = 1. (4.23) λ6Λ |λ+ρ|2X=|Λ+ρ|2

4.5 Reducibility Staying in the symmetrizable case, we now use the Casimir operator to study questions of irreducibility and complete reducibility in O.

26 Proposition 4.12. Let A be a symmetrizable matrix.

(a) If 2(Λ + ρ, β) 6= (β, β)∀0 6= β ∈ Q+, then the g(A)-module M(Λ) is irreducible. (b) If V is a g(A)-module from the category O such that for any two primitive weights λ and µ of V, such that λ − µ = β > 0, one has 2(λ + ρ, β) 6= (β, β), then V is completely reducible.

Now we consider the derived subalgebra g0(A) = [g(A), g(A)] of g(A) instead of g(A), for a general A (not necessarily symmetrizable). Recall that g(A) = g0(A) + h, and that h0 = ∨ 0 i Cαi = g (A) ∩ h. Recall the free abelian group Q. We have the following decomposition of the universal enveloping algebra U(g(A)) with respect to h: P M U(g(A)) = Uβ, where (4.24) β∈Q

Uβ := {x ∈ U(g(A)) : [h, x] = hβ, hix∀h ∈ h} . (4.25)

0 0 0 L 0 Put Uβ = U(g (A)) ∩ Uβ, so that U(g (A)) = β Uβ. Definition. A g0(A)-module V is called a highest-weight module with highest weight Λ ∈ (h0)∗ if V admits a Q -gradation V = L V such that U0 (V ) ⊂ V , dim V = 1, + β∈Q+ Λ−β β Λ−α Λ−α+β Λ 0 ∗ 0 h(v) = Λ(h)v for h ∈ (h ) , v ∈ VΛ, and V = U(g (A)) (VΛ). In other words, this is a restriction of a highest-weight module over g(A) to g0(A). We define the Verma module M(Λ) over g0(A) in the same way and show that it contains a unique proper maximal garded submodule M0(Λ). We put L(Λ) = M(Λ)/M0(Λ), which is a restriction of an irreducible g(A)-module to g0(A).

Lemma 4.13. The g0(A)-module L(Λ) is irreducible.

∗ ∨ Definition. The labels of Λ ∈ h are the values hΛ, αi i (i = 1, . . . , n). If Λ, M ∈ h∗ have the same labels, they may differ only off h0. Then Lemma 4.13 implies that themodulesL(Λ)andL(M),whenrestrictedtog0(A),areirreducibleandisomorphicasg0(A)- modules, and that the actions of elements of g(A) on them differ only by scalar operations. Fi- nally, note that

dim L(Λ) = 1 ⇔ Λ|h0 = 0. (4.26) Now we return to the case where A is symmetrizable. We may fix a symmetric invariant bilinear formong0(A)whichisdefinedalsoonh0,andwhosekernelisc. Put(β, γ) = (ν−1(β), ν−1(γ)) 0 ∗ ∨ 1 for β, γ ∈ Q, and define ρ ∈ (h ) by hρ, λi i = 2 aii for i = 1, . . . , n. Proposition 4.14. Let A be a (possibly infinite) symmetrizable matrix.

−1 0 (a) If 2hΛ + ρ, ν (β)i= 6 (β, β)∀β ∈ Q+ − {0}, then the g (A)-module M(Λ) is irreducible. (b) Let V be a g0(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) = 0 whenever s > k; L 0 (iii) V = λ∈(h0)∗ Vλ, where Vλ = {v ∈ V : h(v) = hλ, hiv∀h ∈ h }; 0 ∗ (iv) if λ and µ ∈ (h ) are primitive weights such that λ − µ = β|h0 for some β ∈ Q+ − {0}, then 2hλ + ρ, ν−1(β)i= 6 (β, β).

Then V is completely reducible; i.e. is isomorphic to a direct sum of g0(A)-modules of the form L(Λ), Λ ∈ (h0)∗.

27 Corollary 4.15. Let A be a symmetrizable matrix with nonpositive real entries and let V be a g0(A)- module satisfying conditions (i)–(iii) of Proposition 4.14(b). Suppose that for every weight λ of V one has ∨ 0 hλ, αi i > 0∀i. Then V is a direct sum of irreducible g (A)-modules, which are free of rank 1 when viewed of U(n−)-modules. We end our discussion of highest-weight modules over Kac-Moody algebras to mention that the theory of these modules has ready application to finding the defining generators and relations of 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 fundamental to string theory and the basic constructions of conformal field theory. In this section, we follow [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- ingtofindatheoryofstronginteractionsthatquantizedtheinteractionintermsof‘currents.’ His theory wassuperseded by QCD,but as it wasone of thefirst applications ofKac-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, and are treated in [Kac94, Ch. 8]. We suppose our affine algebra g0 is untwisted in what follows. We then have an isomorphism g0 ' C[t, t−1] ⊗ g˚ + CK (5.1) with commutation relations

h (m) (n)i (m+n) x , y = [x, y] + mδm,−n(x, y)K. (5.2)

Hereg˚ isasimplefinite-dimensionalLiealgebra, (x, y)isthenormalizedinvariantbilinearform on g˚, and x(n) stands for tn ⊗ x (n ∈ Z, x ∈ g˚). i j Let {ui} and u be dual bases of g˚; i.e. (ui, u ) = δij. Then the Casimir operator of g˚,

i Ω˚ = uiu , (5.3) i X is independent of the choice of dual bases. Additionally, the Casimir operator

∨ ˚ (−n) i(n) Ω = 2(K + h )d + Ω + 2 ui u (5.4) ∞ n=1 i X X is the Casimir operator of the affine algebra h i g = g0 + Cd, where d, x(n) = nx(n). (5.5)

Recall the definition of a restricted g0-module in section 3.3; i.e. any module V such that ∀v ∈ (j) 0 V, x (v) = 0∀x ∈ g˚, j 0. Consider all series j=1 uj with uj ∈ U(g ), the universal 0 0 enveloping algebra of g , such that for any restricted g -module V and any v ∈ V, uj(v) = 0 P∞ for all but finitely mnay uj. We identify two such series if they represent the same operator in 0 0 0 every restricted g -module, thereby obtaining an algebra Uc(g ) which contains U(g ) and acts on every restricted g0-module.

28 Definition. Define the restricted completion of the universal enveloping algebra U(g0) to be this 0 algebra Uc(g ).

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

i (−n) i(n) T0 = uiu + 2 ui u , ∞ i n=1 i X X X (5.6) (−m) i(m+n) Tn = ui u if n 6= 0. m∈Z i X X Definition. The operators Tn are the Sugawara operators.

0 In fact, these operators are contained in Uc(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) For x ∈ g˚ and n, m ∈ Z, one has

h (m) i ∨ (m+n) x ,Tn = 2(K + h )mx . (5.7)

(b) Let V be a restricted g-module and let v ∈ V be such that n+(v) = 0 and h(v) = hλ, hiv for some λ ∈ 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.

m3 − m [T ,T ] = 2(K + h∨) (m − n)T + δ (dimg) ˚ K . (5.9) m n m+n m,−n 6 Lemma 5.1 and Proposition 5.2 immediately yield the following Corollary 5.3. Let V be a restricted g0-module such that K is a scalar operator kI, k 6= −h∨. Let 1 L = T , n ∈ Z (5.10) n 2(k + h∨) n k(dimg) ˚ c(k) = , (5.11) k + h∨ (Λ + 2ρ, Λ) h = if V = L(λ), (5.12) Λ 2(k + h∨) where Λ ∈ h∗ is a highest weight, and L(Λ) is the unique irreducible module with highest weight Λ (c.f. [Kac94, §9.2]). (a) Letting

dn 7→ Ln, c 7→ c(k) (5.13)

extends V to a module over g0 + Vir, in particular, V extends to a module over g (= g0 + Cd).

(b) If V is the g-module L(Λ), then L0 = hλI − d. Definition. The number c(k) is called the conformal anomoly of the g0-module V, and the number is called the vacuum anomoly of L(Λ). The appearance of Vir in the above corollary requires some explanation.

29 5.2 Virasoro algebra Letg˚ beasimplefinite-dimensionalLiealgebra, putL = C[t, t−1], andconsidertheloopalgebra

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)

DenotebyL˜ (g)˚ theextensionoftheLiealgebraL(g)˚ bya1-dimensionalcenter;explicitlyL˜ (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) where ψ is the C-valued 2-cocycle on the Lie algebra L(g)˚

da ψ(a, b) := res , b , (5.17) dt t where res is the usual residue, a linear functional on L, and ( , )t is the extension of a nondegenerate invariant symmetric bilinear complex-valued form ( , ) on g˚ to an L-valued bilinear form on L(g)˚ by

(P ⊗ x, Q ⊗ y)t = PQ(x, y). (5.18)

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

D(P ⊗ x) = D(P) ⊗ x. (5.19)

Then, let ds denote the endomorphism of the space L˜ (g)˚ defined by d d | = −ts+1 , d (K) = 0. (5.20) s L(g)˚ dt s

Proposition 5.4. ds is a derivation of L˜ (g)˚ . Note that M d := Cdj (5.21) j∈Z 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 has a nontrivial central extension by a 1-dimensional center, say Cc, that is unique up to isomorphism.

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

1 [d , d ] = (i − j)d + i3 − i d c, i, j ∈ Z. (5.23) i j i+j 12 i,−j

30 5.3 Virasoro operators We now may extend the Sugawara construction to the case of a reductive finite-dimensional Lie algebra g (we use this notation in place of g˚ for this section). We have the decomposition of g into 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 invariant bilinear form ( , ) so that (5.24) is an orthogonal decomposition, and we assume that the restriction of ( , ) to each g(i) for i > 1 is the normalized invariant form. We shall call such a form a normalized invariant form on g. Put M L˜ (g) := L˜ g(i) , where L˜ g(i) = L˜ g(i) + CKi (5.25) i>0 and d Lˆ (g) := L˜ (g) + Cd, where d| ˜ = −t , d (Ki) = 0. (5.26) L(g(i)) dt

Definition. The Lie algebras L˜ (g) and Lˆ (g) are called affine algebras associated to the reductive Lie algebrag. ThesubalgebrasL˜ g(i) (resp. L˜ g(i) +Cd)arecalledcomponentsofL˜ (g)(resp. Lˆ (g)).

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

We identify g with theP 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 Cartan subalgebra of Lˆ (g). We also have a triangular decomposition

Lˆ (g) = n− ⊕ h ⊕ n+, where (5.27) −1 −1 −1 n− = t C t ⊗ n+ + h + C t ⊗ 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, hδ, di = 1. (5.30)

∗ Given Λ ∈ h , there is a unique irreducible Lˆ (g)-module that admits a non-zero vector vΛ such that n+ (vλ) = 0 and h (vΛ) = hΛ, hivΛ for h ∈ h. By the uniqueness of L(Λ), we have O L(Λ) = L Λ(i) , (5.31) i>0 where Λ(i) denotes the restriction of Λ to h(i) := h ∩ Lˆ g(i) and L Λ(i) is the Lˆ g(i) - module with highest weight Λ(i). We let ki, the eigenvalue of Ki on L(Λ), be the ithlevel of Λ, and let k = (k0, k1,...). Define

c(k) := c (ki) , (5.32) i X

hΛ := hΛ(i) (5.33) i X

mΛ = mΛ(i) . (5.34) i X

31 LetV bearestrictedL˜ (g)-modulesuchthatKi actsaskiIandki isdistinctfromthedualCoxeter ˜ ∨ number of L g(i) , which we will denote by hi , and which is generically defined as

h := ai, (5.35) i=0 X where ` is the rank of the generic affine algebra and ai are the labels of the Dynkin diagram as ∨ (i) before, and a those of the diagram associated to the dual algebra or transpose matrix. Let Tn i be the Sugawara operators for L˜ g(i) , and let, ∀n ∈ Z, 1 L(i) := T (i),Lg := L(i). (5.36) n 2 k + h∨ n n n i i i X g Definition. The operators Ln are called the Virasoro operators for the g-module V.

g Letting dn 7→ Ln, c 7→ c(k) extends V to a module over L˜ (g) + Vir. We also have the formula Ω Lg = i − d, (5.37) 0 2 k + h∨ i i i X where Ωi is the Casimir operator for Lˆ g(i) .

5.4 Coset Virasoro construction Let g be a reductive finite-dimensional Lie algebra with a normalized invariant form ( , ), and let g˙ be a reductive subalgebra of g such that ( , )|g˙ is non-degenerate. Let M M g = g(i) and g˙ = g˙ (i) (5.38) i>0 i>0 be the decompositions of g and g˙. Let ( , ) be a normalized invariant form on g˙ that coincides with ( , ) on g˙ (0). Due to the uniqueness of the invariant bilinear form on a simple Lie algebra we 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 x and y; 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= −hi . This is a L(g)˙ module with K˙ i acting as k˙ iI, where

k˙ s = jsikj. (5.40) i X ˙ ˙ ∨ Assume that ki 6= −hi . Put

g,g˙ g g˙ Ln := Ln − Ln. (5.41)

g,g˙ Proposition 5.5. (a) The operators Ln commute with L˜ (g)˙ .

g,g˙ (b) The map dn 7→ Ln , c → c(k) − c(k˙ ) defines a representation of Vir on V. Definition. The Vir-module defined in Proposition 5.5 is called the coset Vir-module.

32 Thebenefitofthecosetconstructionisthatitisamethodofconstructingunitaryhighest-weight representations of the Virasoro algebra Vir, and further allows the classification of unitary highest weight representations of the Virasoro algebra. TheVirasoroalgebraiswidelyusedinconformalfieldtheoryandstringtheory. Itisthequan- tum version of the conformal algebra in two dimensions. The representation theory of strings on the worldsheet, the two-dimensional manifold which describes the embedding of a string in spacetime, reduces to the representation theory of the Virasoro algebra. The trace of the stress- energytensor,thetensorthatdescribes thedensityandflux ofenergyand momentuminspace- time, and which generalizes the stress tensor of Newtonian physics, vanishes in the constraints of the Virasoro algebra. In this way, the Virasoro algebra encodes two-dimensional conformal symmetry and is therefore a fundamental object in physics.

References

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