5 1 Weyl modules associated to Kac-Moody Lie 0 2 n a algebras J 0 2 ] S. Eswara Rao, V. Futorny and Sachin S. Sharma T R . h t a Abstract m [ Weyl modules were originally defined for affine Lie algebras by 1 v Chari and Pressley in [4]. In this paper we extend the notion of Weyl 2 0 modules for a Lie algebra g ⊗ A, where g is any Kac-Moody alge- 8 4 0 bra and A is any finitely generated commutative associative algebra . 1 0 with unit over C, and prove a tensor product decomposition theorem 5 1 generalizing [4]. : v i X r a 1 Introduction Let g be a Kac-Moody Lie algebra and let h be a Cartan subalgebra of g. Set g′ = [g,g] and h′ = g ∩ h. Let h′′ be a vector subspace of h such that h′ ⊕h′′ = h. Let A be a finitely generated commutative associative algebra ∼ with unit over C. Denote g= g′ ⊗ A ⊕ h′′ and let g = N− ⊕ h ⊕ N+ be a 1 standard triangular decomposition into positive and negative root subspaces ∼− ∼+ ∼ andaCartansubalgebra. LetN = N−⊗A, N = N+⊗Aandh= h′⊗A⊕h′′. ∼ Consider a linear map ψ :h→ C. In [4] Chari and Pressley defined the Weyl modules for the loop alge- bras, which are nothing but the maximal integrable highest weight modules. Feigin and Loktev [8] generalized the notion of Weyl module by replacing Laurent polynomial ring by any commutative associative algebra with unit and generalized the tensor decomposition theorem of [4]. Chari and Thang [2] studied Weyl modules for double affine Lie algebra. In [1], a functorial ◦ approach used to study Weyl modules associated with the Lie algebra g⊗A, ◦ where g is finite dimensional simple Lie algebra and A is a commutative al- gebra with unit over C. Using this approach they [1] defined a Weyl functor from category commutative associative algebra modules to simple Lie alge- bra modules, and studied tensor product properties of this functor. Neher and Savage [13] using generalized evaluation representation discussed more general case by replacing finite dimensional simple algebra with an infinite dimensional Lie algebra. ◦ ◦ Let τ = g⊗A ⊕Ω /dA be a toroidal algebra, where g is finite dimen- n An n sionalsimpleLiealgebraandA = C[t±,··· ,t±]isaLaurentpolynomialring n 1 n in commutating n variables(see [5]). It is proved in [6] that any irreducible module with finite dimensional weight spaces of τ is in fact a module for 2 ◦ g⊗A where g is affinization of g. Thus it is important to study g⊗A - n−1 n−1 modules. Rao and Futorny [7] initiated the study of g ⊗ A -modules in n−1 their recent work. In our paper we consider the g ⊗ A-module, where g is any Kac-Moody Lie algebra and A is any finitely generated commutative associative algebra with unit over C. Our work is kind of generalization of the tensor product results in [1, 8, 4]. For a cofinite ideal I of A we define ∼ a module M(ψ,I), and a Weyl module W(ψ,I) of g (Section 4). The main result of the paper is the tensor product decomposition of W(ψ,I), where I is a finite intesection of maximal ideals. The paper is organised as follows. We begin with preliminaries by stating some basic facts about Kac-Moody algebras and Weyl modules. In Section ∼ 3 we define the modules M(ψ,I) over g and show that they have finite dimensional weight spaces andprove tensor decomposition theorem for them. Section4isdevotedtothetensordecompositiontheoremfortheWeylmodule ∼ W(ψ,I) over g. 2 PRELIMINARIES ◦ Let g be a finite dimensional simple Lie algebra of rank r with a Cartan ◦ ◦ ◦ ◦ ◦ subalgebra h. Let ∆ denote a root system of g with respect to h. Let ∆+ ◦ ◦ and ∆− be a sets of positive and negative roots of g respectively. Denote by 3 ◦ α ,··· ,α and α∨,··· ,α∨ a sets of simple roots and simple coroots of g. Let 1 r 1 r ◦ ◦ ◦ ◦ ◦ g = n+ ⊕ h ⊕ n− be a triangular decomposition of g. Let e and f be the i i Chevalley generators of g◦. Let Q◦ = ⊕Zα and P◦ = {λ ∈ h◦∗ : λ(α∨) ∈ Z} be i i ◦ ◦ ◦∗ the root and weight lattice of g respectively. Set P = {λ ∈ h : λ(α∨) ≥ 0}, + i ◦ the set of dominant integral weights of g. ◦ ◦ Recall that a g-module V is said to be integrable if it is h-diagonalisable and all the Chevalley generators e and f , 1 ≤ i ≤ r, act locally nilpotently i i on V. For commutative associative algebra with unit A, consider the Lie ◦ algebra algebra g ⊗ A. We recall the definition of local Weyl module for ◦ g⊗A [8, 1]. ◦ Definition 2.1. Let ψ : h⊗A → C be a linear map such that ψ |◦= λ, I a h ◦ cofinite ideal of A. Then W(ψ,I) is called a local Weyl module for g⊗A if there exists a nonzero v ∈ W(ψ,I) such that ◦ ◦ U(g⊗A)v = W(ψ,I),(n+ ⊗A)v = 0,(h⊗1)v = λ(h)v ψ |◦ = 0,(fi ⊗1)λ(α∨i)+1v = 0, for i = 1,··· ,r. h⊗I It is shown in [8](Proposition 4) that the local Weyl modules exists and can be obtained as quotient of global Weyl module. 4 ∼ Let g= g′⊗A⊕h′′ is a Lie algebra with the following bracket operations: [X ⊗a,Y ⊗b] = [X,Y]⊗ab, [h,X ⊗a] = [h,X]⊗a, [h,h′] = 0, ∼ ∼ ∼+ where X,Y ∈ g′, h,h′ ∈ h′′ and a,b ∈ A. Let h:= h′ ⊗ A⊕h′′ and g=N ∼ ∼− ∼ ∼+ ⊕ h ⊕ N be a triangular decomposition of g, where N = N+ ⊗ A and ∼− N = N− ⊗A. ∼ Let ψ :h→ C be a linear map. ∼ Definition 2.2. A module V of g is called highest weight module (of highest weight ψ) if V is generated by a highest weight vector v such that ∼+ (1) N v = 0. ∼ ∼∗ (2) h v = ψ(h)v for h ∈ h,ψ ∈h . ∼+ ∼ ∼+ Let C be the one dimensional representation of N ⊕ h where N acts ∼ ∼ trivially and h acts via h.1 = ψ(h)1 for ∀h ∈h. Define the induced module ∼ M(ψ) = U(g) O C. ∼+ ∼ U(N ⊕h) Then M(ψ) is highest weight module and has a unique irreducible quotient denoted by V(ψ). 5 3 The modules M(ψ,I) and its tensor decom- position Let α ,...,α be a set of simple roots of g and ∆+ a set of corresponding 1 l l l positive roots. Let Q = MZαi be root lattice of g and Q+ = MZ≥0αi. Let i=1 i=1 λ ∈ h∗ be a dominant integral weight of g. Consider α ∈ ∆+ and assume α = n α . Define an usual ordering on ∆+ by α ≤ β for α,β ∈ ∆+ if P i i β −α ∈ Q . + Let I be a cofinite ideal of A. Let {I ,α ∈ ∆+} be a sequence of cofinite α ideals of A such that I ⊆ I and α (1) α ≤ β ⇒ I ⊆ I . β α (2) I I ⊆ I if α+β ∈ ∆+. α β α+β For β ∈ ∆+ let X be a root vector corresponding to the root −β. For −β a cofinite ideal I of A set X I = X ⊗I. −β −β ∼ Let ψ :h→ C be a linear map such that ψ |h′⊗I= 0, ψ |h= λ ∈ h∗ and λ is dominant integral. ∼ Definition 3.1. WewilldenotebyM(ψ,{I ,α ∈ ∆+})thehighestweight g- α modulewithhighestweightψ andhighestweightvectorv suchthat(X I )v = −β β 0 for all β ∈ ∆+. We will show now the existence of modules M(ψ,{I ,α ∈ ∆+}). Let α M(ψ) be the Verma module with a highest weight ψ and a highest weight 6 vector v. We will prove that the module generated by X I v is a proper −α α submodule of M(ψ) for all α ∈ ∆ . We use induction on the height of + α. First recall that there is a cofinite ideal I such that ψ |h′⊗I= 0 and by definition I ⊆ I and I ⊆ I if β ≤ α. α α β Let us consider X I v for a simple root α . We will prove that X av −αi αi i −αi generates a proper submodule of M(ψ) where a ∈ I . Indeed we have αi X bX av = 0 for any simple root α 6= α and b ∈ A. Let N be the g- α −αi i i submodule generated by X I v. Then X bX av = h bav = 0 as I ⊆ I −αi αi αi −αi i αi and ψ |h′⊗I= 0. So M(ψ)/Ni 6= 0 and the induction starts. Let β ∈ ∆ and ht(β) = n. Let N be the submodule generated by + X I v where htν < n. Then by induction, N is a proper submodule Pν∈∆+ −ν ν of M(ψ). Now consider X bX av where α is a simple root, b ∈ A and αi −β i a ∈ I . But X bX av = X bav. Since ht(β − α ) < n and I is an β αi −β −β+αi i β ideal of A, we have ba ∈ I ⊆ I . Hence, we see that X bav ∈ N. β β−αi −β+αi Therefore, X av is a highest vector of M(ψ)/N, and hence generates a −β proper submodule. Lemma 3.2. Letγ ,...,γ ,β ∈ ∆+. ThenB = X In+1X a ...X a v = 1 n −β β −γ1 1 −γn n 0, for a ,...a ∈ A and each γ ≤ β. 1 n i Proof. We prove the statement by induction on n. For n = 0 the lemma follows from the definition of the module. We have B = X a X In+1X a ...X a v+[X ,X ]In+1X a ...X a v. −γ1 1 −β β −γ2 2 −γn n −β −γ1 β −γ2 2 −γn n 7 The first term is zero by induction on n. Repeating the same argument n times for the second term we end up with: B = [...[X ,X ],X ],...,X ]In+1v . Assume −β −γ1 −γ2 −γn β [...[X ,X ],X ],...,X ] 6= 0. −β −γ1 −γ2 −γn Then it is a nonzero multiple of X ) and β + γ is a root. As each −(β+Pγi P i γ ≤ β we have I ⊆ I . i β γi Thus In+1 ⊆ I I I ...I ⊆ I . β β γ1 γ2 γn β+Pγi Since X )I v = 0, −(β+Pγi β+Pγi it completes the proof of the lemma. Proposition 3.3. M(ψ,{I ,α ∈ ∆+}) has finite dimensional weight spaces α with respect to h. Proof. Follows from Lemma 3.2 . We now construct a special sequence of cofinite ideals I ,α ∈ ∆+. Let I α ∼ be any cofinite ideal of A. Let ψ :h→ C be a linear map such that ψ |h′⊗I=0 and ψ | = γ a dominant integral weight. Let recall that for α ∈ ∆ with h + α = l m α , define N = l m λ(α∨). Pi=1 i i λ,α Pi=1 i i 8 Let I = INλ,α. Now if α ≤ β then it implies that , N ≤ N andhence α λ,α λ,β I ⊆ I . Supposeα,β ∈ ∆+ suchthatα+β ∈ ∆+. ThenclearlyI I = I . β α α β α+β For this special sequence of ideals I , define M(ψ,I) := M(ψ,{I ,α ∈ ∆+}). α α Let I and J be coprime cofinite ideals of A. Consider linear maps ∼ ψ1,ψ2 :h7→ C such that ψ1 |h′⊗I= 0, ψ2 |h′⊗J= 0, ψ |h= λ and ψ2 |h= µ. Further assume that λ and µ are dominant integral weights. Let M(ψ ,I) 1 and M(ψ ,J) be the corresponding highest weight modules. Now define the 2 following new sequence ofcofinite ideals ofA. Let K = INλ,α∩JNµ,α ⊆ I∩J. α It is easy to check that: (1) If α ≤ β, α,β ∈ ∆+ then K ⊆ K . β α (2) K K ⊆ K if α,β,α+β ∈ ∆+. α β α+β Let ψ = ψ1 +ψ2, so that ψ |h′⊗(I∩J)= 0, ψ |h= λ+µ. Then we have Theorem 3.4. As a g-module e M(ψ +ψ ,{K ,α ∈ ∆+}) ∼= M(ψ,I)⊗M(ψ ,J). 1 2 α 2 The following is standard but we include the proof for convenience of the reader. Lemma 3.5. Let I and J be the coprime cofinite ideals of A. Then a) A = In +Jm, for all n,m ∈ Z . ≥1 b) A/(In ∩Jm) ∼= A/In ⊕A/Jm. 9 Proof. a) As I and J are coprime there exist f ∈ I and g ∈ J such that f + g = 1. Considering the expression (f + g)m+n+1 = 1 , we see that the left hand side is the sum of two elements of Im and In. b) is clear from a) and the Chinese reminder theorem. Assume dimA/INλ,α = m α dimA/JNµ,α = n α dimA/INλ,α ∩JNµ,α = k α then m +n = k α α α by the above lemma. Proof of Theorem 3.4. Let a ,...,a be a C-basis of A/INλ,α. Let a1,a2,..., be a C-basis 1,α mα,α α α ∼− of INλ,α. Then clearly N has the following C-basis: {X a ,1 ≤ i ≤ m ,α ∈ ∆+}∪ −α i,α α {X ai ,i ∈ N, α ∈ ∆+}. −α α ∼− Let U , Uλ be the subspaces of U(N ) spanned by the ordered products of λ the first set and the second set respectively. Then by the PBW theorem we ∼− have U(N ) = U Uλ. λ 10