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Infinite-Dimensional Lie Algebras

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Infinite-Dimensional Lie Algebras Infinite-dimensional Lie algebras Lecture course by Iain Gordon Edinburgh, 2008/9 Contents 1 Introduction 1 2 Central extensions 2 3 The Virasoro algebra 3 4 The Heisenberg algebra 4 5 Representations of the Virasoro algebra 5 5.1 Family 1: Chargeless representations .......................... 5 5.2 Family 2: Highest-weight representations ......................... 7 6 Enveloping algebras 8 7 The universal highest-weight representations of Vir 9 8 Irreducibilty and unitarity of Virasoro representations 10 9 A little infinite-dimensional surprise 12 9.1 Fermionic Fock ....................................... 12 9.2 Central extensions of gl ................................. 13 ∞ 10 Hands-on loop and affine algebras 14 11 Simple Lie algebras 16 12 Kac-Moody Lie algebras 17 12.1 Step I: Realisations ...................................... 17 12.2 Step II: A big Lie algebra .................................. 19 12.3 Step III: A smaller Lie algebra ............................... 20 13 Classification of generalised Cartan matrices 21 13.1 Types of generalised Cartan matrices ........................... 21 13.2 Symmetric generalised Cartan matrices ......................... 22 14 Dynkin diagrams 26 15 Forms, Weyl groups and roots 27 16 Root spaces 32 17 Affine Lie algebras and Kac-Moody Lie algebras 34 1 2 Iain Gordon 18 The Weyl-Kac formula 37 18.1 Category O ........................................... 37 18.2 Kac’s Casimir ......................................... 39 18.3 The Weyl-Kac formula ................................... 40 19 W − K + A = M 43 19.1 Affine Weyl groups ..................................... 43 19.2 Formulae ............................................ 44 20 KP hierarchy and Lie theory 45 21 The Kazhdan-Lusztig Conjecture 48 21.1 The Shapovalov form and Kac-Kazhdan formulas ................... 48 21.2 Bringing in the Weyl group ................................. 51 21.3 The Kazhdan-Lusztig Conjecture ............................. 52 A References 53 B Notational reference 53 C Generation of Lie algebras 54 1 Introduction Lie algebras may arise in the following ways in the wild: • Derivations of an associative algebra. • Vector fields on a smooth manifold. • Tangent spaces at the identity of Lie groups. We should often think of a Lie algebra of being one of those (intimately related) concepts. 1 Example (Witt algebra over C). Let A C[z,z− ] be the algebra of Laurent polynomials in one = variable and consider n d £ 1¤o n j 1 d o Der(A) f (z) : f C z,z− span L j : z + : j Z . = dz ∈ = = − dz ∈ We compute the structure of Der(A): £ ¤ h m 1 d n 1 d i Lm,Ln f z + , z + f = − dz − dz ¡ m n 1 m n 2 ¢ ¡ m n 1 m n 2 ¢ (n 1)z + + f 0 z + + f 00 (m 1)z + + f 0 z + + f 00 = + + − + + m n 1 df (n m)z + + (m n)Lm n f = − dz = − + This is called the Witt algebra over C, denoted Witt; its basis is ©L : j Zª and its structure is j ∈ £ ¤ Lm,Ln (m n)Lm n . = − + 2 Infinite-dimensional Lie algebras 3 Example (Vector fields on S1). Consider the complex vector fields on S1 ©eiθ : θ R± ª: = ∈ ∼ n d 1 o Vectfin : f : f : S C smooth with finite Fourier expansion , = dθ → d d i.e. we impose that each element is in the finite span of cos(nθ) dθ and sin(nθ) dθ , or equivalently in the finite span of ©einθ d : n Zª. Without the finiteness assumption, we would have to deal dθ ∈ with the Fourier analysis and convergence issues. Thus we have a basis n inθ d o Ln : ie : n Z . = dθ ∈ iθ n 1 d Set z : e , then L z + , and we recover the Witt algebra. = n = − dz Example (Diffeomorphisms of S1). Let G : Diff ¡S1¢ be the group of (orientation-preserving) 1 = + ¡ 1 ¢ ¡ 1 ¢ diffeomorphisms of S under composition. It acts on C ∞ S ,C via (g.f )(z) : f g − (z) for = g G. An element of the tangent space at 1 G has the form ∈ ∈ ¡ ¢ X∞ n 1 g(z) z 1 ²(z) z λn²z + . = + = + n =−∞ Then 1 ¡ ¢ X∞ n 1 g − (z) z 1 ²(z) z λn²z + . = − = − n =−∞ Thus ¡ ¢ ¡ ¢ ¡ X∞ n 1¢ g.f (z) f z(1 ²(z)) f z λn²z + = − = − n =−∞ ³ X∞ n 1 d ´ ¡ X∞ ¢ 1 λn²z + f (z) 1 λn²Ln f (z). = − n dz = + n =−∞ =−∞ The Ln for a topological basis for Lie(G). The three preceding examples all give the same Lie algebra structure. The Witt algebra has two further properties: 1. There is an anti-linear Lie algebra involution ω(Ln) : L n. = − 2. Thence we can build a real form of the Witt algebra as ©x Witt : ω(x) xª. ∈ = − 2 Central extensions Definition 2.1. A central extension of a Lie algebra g is a short exact sequence of Lie algebras 0 z g g 0 (2.1) −→ −→ b −→ −→ such that z Z ¡g¢ ©x g :[x,g] 0ª. ⊆ b = ∈ b b = As vector spaces the sequence (2.1) splits as g g0 z, say via a section σ: g g0. Let b = ⊕ → β: g g z ,(x, y) £σ(x),σ(y)¤ σ¡[x, y]¢ . × → 7→ − We check that β satisfies • β(x, y) β(y,x), and = − 3 4 Iain Gordon • β¡x,[y,z]¢ β¡y,[z,x]¢ β¡z,[x, y]¢ 0. + + = Conversely, given β C 2¡g;z¢ : ©β: g g z : satisfying the aboveª, we can construct ∈ = × → £ ¤ ¡ ¢ g : g z as vector spaces, (g,z),(g 0,z0) : [g,g 0],β(g,g 0) . bβ = ⊕ = Then bgβ is a central extension. However, there exist isomorphisms of such bgβ: z g g z g −−−−→ bβ = ⊕ −−−−→ ° ° ° θ ° ° y=∼ ° z g g z g −−−−→ bγ = ⊕ −−−−→ ¡ ¢ ¡ ¢ We find quickly that θ : bgβ bgγ is of the form θ (g,z) g,θ2(g) z , where θ2 : g z; and for θ → =¡ ¢ + → to be a Lie algebra homomorphism, we require that θ [g,g 0] γ(g,g 0) β(g,g 0). 2 = − So we find that central extensions are classified by C 2¡g;z¢ H 2¡g;z¢ , = dB 1¡g;z¢ 1 © ª ¡ ¢ where B θ : g z , and dθ(g,g 0) θ [g,g 0] . = → = Exercise 2.2. Show that if g is a finite-dimensional simple Lie algebra, then H 2¡g;C¢ 0. = Proposition 2.3. H 2¡Witt¢ : H 2¡Witt;C¢ C. In other words, there is a one-dimensional space = =∼ of central extensions of the Witt algebra. Proof. Let β: Witt Witt C,(L ,L ) β(m,n) satisfy the two conditions from above, namely × → m n 7→ • β(m,n) β(n,m), and = − • (m n)β(l,m n) (n l)β(m,l n) (l m)β(n,l m) 0. − + + − + + − + = Freedom of choice is provided by dθ for some θ : Witt C , L θ(m). → m 7→ Writing θ (m) : β(0,m)±m for m 0, replace β by β dθ : We see that w.l.o.g. β(0,m) 0 β = 6= + β = for all m Z. Now with l 0, we get nβ(m,n) mβ(n,m) 0, i.e. (m n)β(m,n) 0. Thus ∈ = − = + = β(m,n) δm, nβb(m), where βb(m) βb( m). = − = − − With l m n 0, we produce a relation which for n 1 gives + + = = (1 m)βb(m 1) (m 2)βb(m) (2m 1)βb(1) 0 . − + + + − + = This recurrence relation has a two-dimensional solution space: 3 βb(m) α1m α2m , with α1,α2 C . = + ∈ α1 3 Now replace β with β dθβ, where θβ(m) δm,0 , to get β(m,n) δm, nα2m . + = 2 = − 3 The Virasoro algebra Definition 3.1. The Virasoro algebra (denoted by Vir) is the central extension of the Witt algebra 1 ¡ ¢ with the choice βb(m) m3 m in the notation of Proposition 2.3. = 12 − The Virasoro Algebra is spanned by ©L : m Zª {c} such that c is central, i.e. [c,L ] 0 m ∈ ∪ m = for all m Z, and ∈ £ ¤ 1 ¡ 3 ¢ Lm,Ln (m n)Lm n δm, n m m c . = − + + − 12 − The number 1/12 is chosen so that c acts as the scalar 1 on a natural irreducible representation that we are about to see. 4 Infinite-dimensional Lie algebras 5 4 The Heisenberg algebra The Witt algebra was a simple point of departure for infinite dimensional Lie algebras. Another simple case is given by constructing loops on a (1-dimensional) abelian Lie algebra, and again making a central extension. This leads to © ª Definition 4.1. The Heisenberg algebra H is spanned by an : n Z {h}, where h is central £ ¤ ∈ ∪ and am,an mδm, nh. (We could even rescale to get rid of m.) = − This algebra has a representation as an algebra of operators on B(µ,h) : C[x1,x2,...], on ¡ ¢ = which the H -module structure is given by ρ : H gl B(µ,h) as follows: → ∂ n 0 , ∂xn > ρ : an ρ(an) hnx n n 0 , and ρ : h ρ(h) ( h) . (4.1) 7→ = − − < 7→ = − × µ n 0 , = Exercise 4.2. If h 0, then B(µ,h) is irreducible, while if h 0 it is not (e.g. the subspace of 6= = constants C B(µ,h) is invariant). < There exists an involution ω(an) a n, ω(h) h. = − = Link to Virasoro algebras. Define 1 X Lk : : a j a j k : for k Z . = 2 j Z − + ∈ ∈ The notation “: a j a j k :” means “normal ordering”, i.e. − + ( a j a j k if j j k, : a j a j k : : − + − ≤ + − + = a j k a j if j j k. + − − ≥ + Explicitly, we have X 1 a2 a a if k 2n, 2 n n j n j + j 0 − + = L > k = X an 1 j an j if k 2n 1.
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