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Kac-Moody Algebras and Applications

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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 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 connec- tions to the classical theory. Last, we discuss some physical applications of Kac-Moody alge- bras,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 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 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 sec- tion 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 2 g such that [x; y] = 0 for all y 2 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 8x 2 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)isnilpotent8x 2 g. Then90 6= v 2 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 9n 2 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 solv- able,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 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.
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