Classification of irreducible integrable representations of loop toroidal Lie algebras
aa r X i v : . [ m a t h . R T ] J u l CLASSIFICATION OF IRREDUCIBLE INTEGRABLEREPRESENTATIONS OF LOOP TOROIDAL LIE ALGEBRAS
PRIYANSHU CHAKRABORTY, PUNITA BATRA
Abstract.
The aim of this paper is to classify irreducible integrable representa-tions of loop toroidal Lie algebras, τ ( B ) = τ ⊗ B ⊕ D , where τ is a toroidal Liealgebra, B is a commutative, associative, finitely generated unital algebra over C and D is the space spanned by degree derivations d , d , ..., d n over C . When a partof center acts non-trivially on modules, we prove modules come from direct sum offinitely many copies of affine Kac-Moody Lie algebras and in the case when centeracts trivialy on modules, modules come from direct sum of finitely many copies ofsimple Lie algebras. Introduction
The importance of toroidal Lie algebras are well known in the theory of bothmathematics and physics. Toroidal Lie algebra can be viewed as a universal centralextension of the multiloop algebra ˙ g ⊗ A n +1 , where ˙ g is a finite dimensional simpleLie algebra over C and A n +1 = C [ t ± , t ± , ..., t ± n ] be the laurent polynomial ring over C in n + 1 commuting variables, see [1, 15, 16, 18]. Representations of toroidal Liealgebras are also well studied, see [1, 5, 9, 21]. For example, the study of irreduciblemodules for toroidal Lie algebras with finite dimensional weight spaces reduces to thestudy of irreducible modules for loop affine Lie algebras, see [5]. Moreover , now adays several authors have studied the representations of loop algebras for finite di-mensional simple Lie algebras [10], Kac-Moody algebras [19], Virasoro algebras [20].In the current paper we will study the representations of loop toroidal Lie algebras.Let us consider the toroidal Lie algebra τ = ˙ g ⊗ A n +1 ⊕ Z , Z is the universal centralextension of ˙ g ⊗ A n +1 and define the loop toroidal Lie algebra τ ( B ) = τ ⊗ B ⊕ D ,where B is a commutative, associative, finitely generated unital algebra over C and D is the derivation space spanned by d , d , ..., d n over C , Lie algebra structure on τ ( B ) is given in section 2. It is well known that center of toroidal Lie algebra (withderivation space) is spanned by K , K , ..., K n , see [1], Section 2 for details. In theloop toroidal case center is spanned by K ⊗ B, K ⊗ B, ..., K n ⊗ B. For the toroidalcase, S.Eswara Rao has been classified irreducible integrable modules for τ with finitedimensional weight spaces on the assumption that K ′ i s , ≤ ≤ n acting on modulestrivially and non-trivially, [1, 9]. In this paper we will classify the irreducible inte-grable modules for τ ( B ) with finite dimensional weight spaces, when K ⊗ , ..., K n ⊗ acts non-trivially and trivially on modules.The paper is organised as follows. In section 2 we start with recalling definition oftoroidal Lie algebra, current Kac-Moody Lie algebra [2] and define loop toroidal Liealgebra. Then we will define root systems, Weyl group, integrable module for τ ( B ) and produce a natural triangular decomposition of τ ( B ) as, τ ( B ) = τ ( B ) − ⊕ τ ( B ) ⊕ τ ( B ) + . In this section we prove an equivalent condition on integrability of τ ( B ) modules and τ modules.Let V be an irreducible integrable module for τ ( B ) with finite dimensional weightspaces. In section 3, after twisting the action of τ ( B ) by an automorphism we assumethat only K ⊗ acts non-trivially and all other K i ⊗ for ≤ i ≤ n acts triviallyon V (when we consider non-trivial action). On this assumption we prove that thereexists a non-zero vector v such that τ ( B ) + v = 0 , Proposition 3.1, via the methodsused in [1, 11]. Then we construct the top space T = { v ∈ V : τ ( B ) + v = 0 } , anirreducible module for τ ( B ) and we prove that weight spaces of T are uniformlybounded, Theorem 3.2. In Proposition 3.3 we prove that most of the center acts triv-ially on our module, using results of Heisenberg algebra from [4]. Finally in Theorem3.4 we will reduce our module V to an irreducible integrable highest weight module V for current Kac-Moody Lie algebra, G = g ′ af ⊗ A n ⊗ B ⊕ C d , where g ′ af is an affineLie algebra, and classify the module V using results of S.Eswara.Rao, P. Batra [2].In section 4, we construct a module V ⊗ A n for the algebra e G = G ⊕ D, D is thespace spanned by the derivations d , d , ..., d n . We prove that V ⊗ A n is a completelyreducible e G module and V is isomorphic to one irreducible component of V ⊗ A n as e G module.In section 5, we assume that all K i ⊗ , for ≤ i ≤ n acting on V trivially andprove that whole Z ⊗ B acts on V trivially, there was a similar results for toroidalcase in [1]. Then our modules becomes a module for the algebra ˙ g ⊗ A n +1 ⊗ B ⊕ D .After that we proceed like [9].Let ψ be a Z n +1 graded morphism from the universal enveloping algebra U ( ˙ h ⊗ A n +1 ⊗ B ) to A n +1 , where ˙ h be a fixed cartan sub-algebra of ˙ g . Then we defineuniversal highest weight module M ( ψ ) for e τ = ˙ g ⊗ A n +1 ⊗ B ⊕ D . Let V ( ψ ) be theunique irreducible quotient of M ( ψ ) . In Theorem 5.1 we prove that any irreducibleintegrable module for τ ( B ) is isomorphic to V ( ψ ) . For each ψ we associate a map ψ from U ( ˙ h ⊗ A n +1 ⊗ B ) to C and construct the non-graded irreducible highestweight module V ( ψ ) for τ = ˙ g ⊗ A n +1 ⊗ B . We prove that V ( ψ ) is isomorphicto an irreducible component of V ( ψ ) ⊗ A n +1 as e τ module. After that we provea Proposition, 5.6 related to loop algebra ˙ g ⊗ S for any commutative, associative,finitely generated unital algebra S over C . Using the help of Proposition 5.6 wehave prove that V ( ψ ) is finite dimensional. Finally we have prove that V ( ψ ) is afinite dimensional irreducible module for direct sum of finitely many simple finitedimensional Lie algebras, Theorem 5.3, using result of [10].2. Notations and Preliminaries
Through out the paper all the vector spaces, algebras, tensor products are takenover the field of complex numbers C . Let Z , N , Z + denote the sets of integers, naturalnumbers and non-negative integers respectively.Let A = A n +1 = C [ t ± , ..., t ± n ] and A n = C [ t ± , ..., t ± n ] denotes the Laurent polyno-mial rings in n + 1 ( t , t , ..., t n ) and n ( t , t , ..., t n ) commuting variables respectively.For m = ( m , .., m n ) ∈ Z n , we denote t m = t m ..t m n n and ( m , m ) = ( m , m , ..., m n ) ∈ Z n +1 . For any lie algebra G , let U ( G ) denote the Universal enveloping algebra of G .Let ˙ g be a finite dimensional simple Lie algebra over C and ˙ h be a Cartan sub-algebra of ˙ g . Let ˙ g = ˙ g − ⊕ ˙ h ⊕ ˙ g + be the root space decomposition of ˙ g . Let ˙∆ be theroot system of ˙ g , π = { α , α , ..α d } be the simple roots of ˙∆ , ˙∆ + ( ˙∆ − ) be the set ofpositive (negative) roots of ˙∆ and ˙Ω be the Weyl group of ˙ g . Let β be the maximalroot of ˙∆ . For each root α ∈ ˙∆ , let α ∨ ∈ ˙ h be such that α ( α ∨ ) = 2 . Let(,) be thenormal non-degenerate symmetric bilinear form on ˙ g .Let ˙ P + = { λ ∈ ˙ h ∗ : λ ( α ∨ i ) ≥ , ∀ ≤ i ≤ d } , we call elements of ˙ P + as dominant integral weight.Let us define λ ≤ µ in ˙ h ∗ iff µ − λ = d X i =1 n i α i , n i ∈ Z + .We call λ ∈ ˙ P + as miniscule weight if µ ∈ ˙ P + and µ ≤ λ imply λ = µ. Aweight λ is miniscule iff ˙Ω orbit of λ is saturated with highest weight λ. Each cosetof root lattice in weight lattice contains exactly one miniscule weight and hence thenumbers of miniscule weight of a simple Lie algebra is equal to the determinant ofCartan matrix, see [12], problem 13. Now it immediately follow from Chapter VI,21.3, Proposition that every finite dimensional module for simple Lie algebra containsatleast one miniscule weight as its weight.Let us consider the multi-loop algebra ˙ g ⊗ A with the usual Lie bracket. Now werecall the definition of toroidal Lie algebra.Let Ω A be the vector space spanned by the symbol t k t k K i , ≤ i ≤ n, k ∈ Z , k ∈ Z n and dA be the subspace of Ω A spanned by n X i =0 k i t k t k K i . Let Z = Ω A /dA .Let τ = ˙ g ⊗ A ⊕ Z , Lie algebra structure on τ is defined by1. (cid:2) x ⊗ t k t k , y ⊗ t r t r (cid:3) = [ x, y ] ⊗ t k + r t k + r + ( x, y ) d ( t k t k ) t r t r ,where d ( t k t k ) t r t r = n X i =0 k i t k + r t k + r K i .2. Z is central in ˙ g ⊗ A ⊕ Z . τ is called the toroidal Lie algebra. It is well known that τ is the universal centralextension of ˙ g ⊗ A ; for more details see [15, 16]. We consider the toroidal Lie algebraas τ = ˙ g ⊗ A ⊕ Z ⊕ D , where D is the derivation space spanned by d , d , ..., d n .Bracket operation is define as [ d i , x ⊗ t k t k ] = k i x ⊗ t k t k , [ d i , t k t k K j ] = k i t k t k K j , [ d i , d j ] = 0 , for ≤ i, j ≤ n. see [1] for details.Recall the definition of current Kac-Moody Lie algebra from [2].Let B be an associative, commutative, finitely generated unital algebra over C . Let g be a Kac-Moody lie algebra and h be a cartan sub-algebra of g . Let g ′ be the derived algebra of g . Let h ′ = h ∩ g ′ and h ′′ be the complementary subspace of h ′ in h . Let g ′ = g ′− ⊕ h ′ ⊕ g ′ + be the standard triangular decomposition of g ′ . Let g B = g ′ ⊗ B ⊕ h ′′ and define a Lie algebra structure on g B by [ X ⊗ b, Y ⊗ b ′ ] = [ X, Y ] ⊗ bb ′ , [ X ⊗ b, h ] = [ X, h ] ⊗ b, for all X, Y ∈ g , b, b ′ ∈ B, h ∈ h ′′ , this algebra is called current Kac-Moody Liealgebra.We now define loop toroidal Lie algebra. Let τ ( B ) = ˙ g ⊗ A ⊗ B ⊕ Z ⊗ B ⊕ D, where D is the space spanned by d , d , ..d n and B is an associative, commutative,finitely generated algebra over C . Define a Lie algebra structure on τ ( B ) by1. [ x ⊗ t k t k ⊗ b, y ⊗ t r t r ⊗ b ′ ] = [ x, y ] t k + r t k + r ⊗ bb ′ + ( x, y ) d ( t k t k ) t r t r ⊗ bb ′ ,where d ( t k t k ) t r t r = n X i =0 k i t k + r t k + r K i .2. Z ⊗ B is central in ˙ g ⊗ A ⊗ B ⊕ Z ⊗ B .3. [ d i , x ⊗ t k t k ⊗ b ] = k i x ⊗ t k t k ⊗ b , for ≤ i ≤ n, ∀ b ∈ B. [ d i , t k t k K j ⊗ b ] = k i t k t k K j ⊗ b , for ≤ i ≤ n, ∀ b ∈ B. [ d i , d j ] = 0 , for ≤ i, j ≤ n. We call τ ( B ) as loop toroidal Lie algebra.Let H = ˙ h ⊕ ( ⊕ ni =0 C K i ) ⊕ ( ⊕ ni =0 C d i ) , where we have identified ˙ h ⊗ ⊗ , K i ⊗ with ˙ h , K i respectively.Then H is an abelian Lie sub-algebra of τ ( B ) . Let δ i , ω i ∈ H ∗ be such that δ i ( ˙ h ) = 0 , δ i ( K j ) = 0 , δ i ( d j ) = δ ij , ≤ i, j ≤ n,ω i ( ˙ h ) = 0 , ω i ( K j ) = δ ij , ω i ( d j ) = 0 , ≤ i, j ≤ n. For m = ( m , .., m n ) ∈ Z n , let δ m = n X i =1 m i δ i . Then τ ( B ) has root space decomposi-tion with respect to H as τ ( B ) = τ ( B ) ⊕ ( M α ∈ ∆ τ ( B ) α ) , where ∆ = ˙∆ ∪ { α + m δ + δ m : α ∈ ˙∆ ∪ { } , m ∈ Z n , m ∈ Z , ( m , m ) = (0 , } and τ ( B ) α + m δ + δ m = ˙ g α ⊗ t m t m ⊗ B,τ ( B ) m δ + δ m = ˙ h ⊗ t m t m ⊗ B ⊕ ( n M i =0 C t m t m K i ) ⊗ B, τ ( B ) = ˙ h ⊗ B ⊕ n M i =0 C K i ⊗ B ⊕ n M i =0 C d i . Let τ ( B ) + = ˙ g + ⊗ C [ t ± , ..., t ± n ] ⊗ B ⊕ ˙ g ⊗ t C [ t , t ± , ..., t ± n ] ⊗ B ⊕ n M i =0 t C [ t , t ± , ..., t ± n ] K i ⊗ B,τ ( B ) − = ˙ g − ⊗ C [ t ± , ..., t ± n ] ⊗ B ⊕ ˙ g ⊗ t − C [ t − , t ± , ..., t ± n ] ⊗ B ⊕ n M i =0 t − C [ t − , t ± , ..., t ± n ] K i ⊗ B,τ ( B ) = ˙ h ⊗ C [ t ± , ..., t ± n ] ⊗ B ⊕ n M i =0 C [ t ± , ..., t ± n ] K i ⊗ B ⊕ ( n M i =0 C d i ) . Then τ ( B ) has a natural triangular decomposition as τ ( B ) = τ ( B ) − ⊕ τ ( B ) ⊕ τ ( B ) + . Now extend α ∈ ˙∆ to an element in H ∗ by defining α ( K i ) = 0 , α ( d i ) = 0 for ≤ i ≤ n and the non degenerate symmetric bilinear form of ˙ h to a bilinear form on H ∗ by ( α i , δ k ) = ( α i , ω k ) = 0 , ≤ i ≤ d, ≤ k ≤ n, ( δ k , δ p ) = ( ω k , ω p ) = 0 , ( ω k , δ p ) = δ kp , ≤ k, p ≤ n. We call γ = α + m δ + δ m ∈ ∆ is a real root of ∆ if ( γ, γ ) = 0 and denote the set ofreal roots as ∆ re . For each root γ ∈ ∆ re , define γ ∨ = α ∨ + 2( α, α ) n X i =0 m i K i , where α ∨ ∈ ˙ h such that α ( α ∨ ) = 2 , also check that γ ( γ ∨ ) = 2 .For a real root γ ∈ ∆ define a reflection r γ on H ∗ by r γ ( λ ) = λ − λ ( γ ∨ ) γ, λ ∈ H ∗ . Let Ω be the Weyl group generated by the reflections corresponding to real roots of ∆ . Definition 2.1.
A module V for τ ( B ) is said to be integrable if1. V is a weight module, i.e V = M λ ∈ H ∗ V λ ,where V λ = { v ∈ V : hv = λ ( h ) v, ∀ h ∈ H } , V λ ’s are called weight spaces of V ofweight λ
2. For all α ∈ ˙∆ , ( m , m ) ∈ Z n +1 , b ∈ B and v ∈ V there exists an integer k = k ( α, ( m , m ) , b, v ) such that ( X α ⊗ t m t m ⊗ b ) k .v = 0 , where X α be the root vectorof the root α ∈ ˙∆ . Lemma 2.1.
Let V be a module for τ ( B ) with finite dimensional weight spaces withrespect to H . Then V is integrable for τ ( B ) if and only if it is integrable for τ ( seethe definition of integrability for τ in [1] ) . Proof.
Integrability of τ immediately follows from Definition 2.1 and the definitionof integrable module for τ .Conversely assume that V is τ integrable. Note that weight spaces of V as τ and τ ( B ) modules are same.Consider V ′ = M k ∈ N V λ + kγ , where λ is a fixed weight and γ = α + m δ + δ m is a fixedreal root. Claim: V ′ is finite dimensional.Suppose not, let k be the smallest positive integer such that V λ + k γ = 0 , let v ′ ( =0) ∈ V λ + k γ . Then there exists a natural number r such ( X α ⊗ t m t m ) r .v ′ = 0 but ( X α ⊗ t m t m ) r − .v ′ = 0 . Consider v = ( X α ⊗ t m t m ) r − .v ′ ∈ V λ +( k + r − γ = V λ + n γ ( say ) , then ( X α ⊗ t m t m ) .v = 0 . Since V ′ is not finite dimensional, then there exists a n > n such that v ∈ V λ + n γ and ( X α ⊗ t m t m ) .v = 0 . Continuing this we can construct an increasingsequence { n i } of natural numbers and v i ∈ V λ + n i γ such that ( X α ⊗ t m t m ) .v i = 0 . Consider the sl copy of the toroidal Lie algebra S γ = span < X α ⊗ t m t m , X − α ⊗ t − m t − m , γ ∨ > and the modules U ( S γ ) v i for i ≥ . Due to integrability and bychoices of v i , U ( S γ ) v i ’s are finite dimensional highest weight modules. Let V i be theirreducible component of U ( S γ ) v i containing the highest weight vector v i with highestweight λ ( γ ∨ ) + 2 n i . Then the sum of V i is direct.Consider N ∈ Z such that λ ( γ ∨ ) + 2 N ≥ , this is possible since λ ( γ ∨ ) ∈ Z , Lemma2.3 [1]. Then V λ + N γ is not finite dimensional, since sum of V i ’s is direct and sequence { n i } is increasing. Hence the claim.Therefore, there are only finitely many weights of V as τ ( B ) module of the form λ + kγ such that k ∈ N . Thus V is τ ( B ) integrable. (cid:3) Lemma 2.2. If V is an irreducible weight module for τ ( B ) with finite dimensionalweight spaces with respect to H , then K i ⊗ b acts as scaler, for all ≤ i ≤ n , b ∈ B .Proof. Note that each K i ⊗ b leaves each finite dimensional weight spaces invariantand they commutes with τ ( B ) . Since V λ is finite dimensional so K i ⊗ b has an eigenvector, say v with eigen value l ∈ C . Now, K i ⊗ b − lI : V → V is a homomorphismof τ ( B ) modules with non zero kernel , hence K i ⊗ b = lI . (cid:3) Lemma 2.3.
Let V is an irreducible integrable module for τ ( B ) with finite dimen-sional weight spaces with respect to H . Then,1. P ( V ) is Ω -invariant , P ( V ) is the set of all λ ∈ H ∗ such that V λ = 0 .2. dimV λ = dimV ω λ , ∀ ω ∈ Ω , λ ∈ P ( V ) .3. If α ∈ ∆ re , λ ∈ P ( V ) then λ ( α ∨ ) ∈ Z .4. If α ∈ ∆ re , λ ∈ P ( V ) , λ ( α ∨ ) > then λ − α ∈ P ( V ) .5. λ ( K i ⊗ ∈ Z , for all ≤ i ≤ n .Proof. Use Lemma 2.3 [1], Note that for 1 to 4 only integrability is required and useLemma 2.1. 5 follows from Lemma 2.2 and Lemma 2.3 [1]. (cid:3) Reduction to a module for current Kac-Moody Lie algebra,when a part of center acts non-trivially
In this section we will reduce our module to a module for current Kac-Moody Liealgebra, when K ′ i s, ≤ i ≤ n acts non-trivially on our modules. Let GL ( n + 1 , Z ) be the set of all n + 1 × n + 1 matrices with determinant ± .There is a natural action of GL ( n + 1 , Z ) on Z n +1 . For every P ∈ GL ( n + 1 , Z ) define P : τ ( B ) → τ ( B ) by P ( X ⊗ t m ⊗ b ) = X ⊗ t P m ⊗ b,P ( d ( t m ) t r ⊗ b ) = d ( t P m ) t P r ⊗ b, for all b in B and for all m ∈ Z n +1 .Let ( d ′ , d ′ , .., d ′ n ) = ( P T ) − ( d , d , .., d n ) and define P ( d i ) = d ′ i .It is easy to check that P is an automorphism of τ ( B ) . If P = ( p ij ) ≤ i,j ≤ n , then P ( t m K j ⊗ b ) = n X i =0 p ij t P m K i ⊗ b . Hence taking m = 0 we get P ( K j ⊗ b ) = n X i =0 p ij K i ⊗ b. (3.1) Definition 3.1.
A linear map z : V → V is said to be a central operator of degree ( m , m ) for τ ( B ) if z commutes with ˙ g ⊗ A ⊗ B ⊕ Z ⊗ B and d i z − zd i = m i z for all ≤ i ≤ n . Lemma 3.1. ( Lemma 1.7,1.8 [5] ) Let V be an irreducible module for τ ( B ) with finitedimensional weight spaces with respect to H . Then,1. Let z be a central operator of degree ( m , m ) such that zu = 0 for some u ∈ V then zw = 0 for all non-zero w ∈ V .2. Let z be a non zero central operator of degree ( m , m ) then there exists a operator T of degree ( − m , − m ) such that zT = T z = Id on V .3. Up to a scaler multiple there is at most one non zero central operator of a givendegree. Theorem 3.1.
Let V be an irreducible module for τ ( B ) with finite dimensionalweight spaces with respect to H and let L = { ( r , r ) ∈ Z n +1 : t r t r K i ⊗ b = 0 on V for some ≤ i ≤ n, b ∈ B } . Let S be the subgroup of Z n +1 generated by L and letrank of S is k. Then upto an automorphism of τ ( B )
1. there exists non zero central operators z i for all i ≥ n − k + 1 such that degree of z i is equal to (0 , , ..., l i , , .. . k < n + 1 .3. t r t r K i ⊗ b = 0 on V , for all r j , n − k + 1 ≤ i ≤ n and for all b ∈ B. t r t r K i ⊗ b = 0 on V , for all r j = 0 , ≤ j ≤ n − k and for all ≤ i ≤ n , b ∈ B.
5. there exists a proper sub-module W of V for ˙ g ⊗ A ⊗ B ⊕ Z ⊗ B ⊕ D k such that V /W has finite dimensional weight spaces with respect to h ⊕ n M i =0 C K i ⊕ D k , where D k is the space spanned by d , d , .., d n − k .Further We can take the maximal proper sub-module W of V for ˙ g ⊗ A ⊗ B ⊕ Z ⊗ B ⊕ D k such that V /W is irreducible and has finite dimensional weight spaces with respectto h ⊕ n M i =0 C K i ⊕ D k , where D k is the space spanned by d , d , .., d n − k . Proof.
Proof of this theorem follows from same lineup as [1], Theorem 4.5. and [5],Theorem 2.5 . (cid:3)
Now assume that K i ⊗ acting on V as c i for ≤ i ≤ n and not all c i arezero. Then we know from Theorem 3.1 that c n = c n − = .. = c n − k +1 = 0 upto anautomorphism of τ ( B ) , if the rank of S is k . Now consider the matrix P = (cid:18) P I (cid:19) where I is the identity matrix of order k × k and P is a matrix in GL ( n − k + 1 , Z ) such that P ( c , c , .., c n − k +1 ) = ( c ′ , , , .. , and c ′ is a positive integer. Thus by 3.1we can assume that only K ⊗ acting as a positive integer and all other K i ⊗ ’sacts trivially on V upto an automorphism of τ ( B ) . Proposition 3.1.
Let V be an irreducible integrable module for τ ( B ) with finitedimensional weight spaces with respect to H . Also assume that K acts by a positiveinteger c and K i acts trivially on V , for ≤ i ≤ n . Then there exists a non zerovector v in V such that τ ( B ) + .v = 0 . The Proof of Proposition 3.1 reffers to method used in [1, 11].
Lemma 3.2.
Let M be a module for any Lie algebra L . Then the following statementsare equivalent:1. M is the sum of a family of irreducible submodules.2. M is the direct sum of a family of irreducible submodules.3. Every submodule of M is a direct summand.Proof. See [17], Theorem 3.5. (cid:3)
Lemma 3.3.
Let V be an module for τ ( B ) satisfying the conditions of Proposition3.1, then for given any λ ∈ P ( V ) there exists a λ ∈ P ( V ) such that λ + η P ( V ) with ( λ + η )( d i ) = λ ( d i ) for all η ∈ ˙Γ + − { } , ˙Γ + denote the set of non negativeintegral linear combination of α i ’s, ≤ i ≤ d .Proof. Let λ ∈ P ( V ) , λ ( d i ) = g i and g = ( g , g , .., g n ) .Also let V g = { v ∈ V : d i .v = g i v } and P g ( V ) = { λ ∈ P ( V ) : V λ ⊆ V g } .Let λ ∈ P g ( V ) , then λ = λ + c ω + d X i =1 g i d i , where λ = λ | ˙ h , since all K ′ i s acts as zeroexcept K .Note that V g is an integrable module for ˙ g with finite dimensional weight spaces withrespect to ˙ h . Due to integrability U ( ˙ g ) v is a finite dimensional module for ˙ g for all v ∈ P g ( V ) , by PBW theorem.Now V g = X v ∈ V g U ( ˙ g ) v and since each U ( ˙ g ) v is finite dimensional module for ˙ g , so each U ( ˙ g ) v breaks into direct sum of irreducible submodules for ˙ g . Hence exists a subset F of ˙ h ∗ such that V g = M λ ∈ F ⊂ h ∗ V ( λ ) , by Lemma 3.2.Now we prove that F is a finite set. Let λ , λ , .., λ t are the miniscule weight of ˙ g .Then each V ( λ ) contains at least one λ i as its weight. Hence if F is an infinite setthen one weight space of V g will be infinite dimensional. Hence F is finite, so V g isfinite dimensional, which imply that P g ( V ) is a finite set.Let λ be the maximal weight with respect to the ordering ≤ in P g V . Now for any two weight ν, µ in P g ( V ) , ν − µ = ν − µ = d X i =1 m i α i , m i ∈ Z , since V isan irreducible module. Hence λ + η P g ( V ) for all η ∈ ˙Γ + − { } , which imply that λ + η P ( V ) for all η ∈ ˙Γ + − { } . (cid:3) Define a new ordering in H ∗ by λ ≤ µ iff µ − λ = d X i =0 n i α i , n i ∈ Z + , where α = δ − β .Review of affine Lie algebra. Consider the affine Lie algebra g af = ˙ g ⊗ C [ t ± ] ⊕ C K ⊕ C d . Let ˙∆ + af = ˙∆ + ∪ { α + m δ : α ∈ ˙∆ ∪ { } , m ∈ N } and ˙∆ − af = ˙∆ − ∪{ α + m δ : α ∈ ˙∆ ∪ { } , − m ∈ N } are the sets of positive and negative roots of g af respectively. Let ˙∆ + ,reaf ( ˙∆ − ,reaf ) be the set of positive (negative) real roots of the affineLie algebra. Let g af ± = M α ∈ ˙∆ ± af ( g af ) α and h af = ˙ h ⊕ C K ⊕ C d be a cartan subalgebraof g af , then g af = g af − ⊕ h af ⊕ g af + be the triangular decomposition of g af . Lemma 3.4.
Let V be an module for τ ( B ) satisfying the conditions of proposition3.1 with the condition that c > . Suppose also that for all λ ∈ P ( V ) there exists < η such that λ + η ∈ P ( V ) . Then, there exists infinitely many λ i ∈ P ( V ) for ∀ i ∈ N such that1. there exists a weight vector u i of weight λ i such that g af + u i = 0 .2. λ i ( d j ) = λ k ( d j ) for all i, k ∈ N , ≤ j ≤ n .3. λ i ( d ) − λ i − ( d ) ∈ N for all i ≥
4. there exists a common weight in V ( λ i ) , where V ( λ i ) is the irreducible highestweight module for g af generated by u i , for all i and each V ( λ i ) ⊆ V .Proof. Proof is similar to Lemma 2.8, [1]. Note that Integrability of τ ( B ) imply theintegrability of τ , Lemma 2.1 and the property of irreducibility used in Lemma 2.8,[1] provide the irreducibility of τ ( B ) . (cid:3) Proof. (Proof of proposition 3.1)
Claim: λ + η / ∈ P ( V ) for all < η and for some λ ∈ P ( V ) .If not, then for all λ ∈ P ( V ) , there exists < η such that λ + η ∈ P ( V ) , hence byLemma 3.3 there exists a common weight for V ( λ i ) , for infinitely many i . Since λ i ’sare distinct V ( λ i ) ′ s are non-isomorphic, hence the dimension of the common weightspace is infinite, a contradiction, Hence the claim. In particular λ + α / ∈ P ( V ) for all α ∈ ˙∆ + ,reaf , for some λ ∈ P ( V ) , hence ( λ, α ) ≥ , for all α ∈ ˙∆ + ,reaf by Lemma 2.3.Claim 1: V λ + α + δ m = 0 for all m ∈ Z n , for all α ∈ ˙∆ + ,reaf and for some λ ∈ P ( V ) . If not, then there exists some m ∈ Z n and some α ∈ ˙∆ + ,reaf such that V λ + α + δ m = 0 .Then we prove that V µ + β + δ r = 0 , for all r ∈ Z n and β ∈ ˙∆ + ,reaf , where µ = λ + α + δ m .If not, then V µ + β + δ r = 0 for some n ∈ Z n and some β ∈ ˙∆ + ,reaf Case 1: Let ( α + β, α ) > , then ( µ + β + δ n , β + δ n + δ m ) = ( µ, β ) + ( β, β ) =( λ, β ) + ( α + β, β ) > , since ( λ, β ) ≥ . Hence λ + α ∈ P ( V ) , a contradiction.Case 2: Let ( α + β, β ) > Similar to case 1. Since the symmetric bilinear form on h ∗ af is positive semi-definite, hence only pos-sibility is,Case 3: Let ( α + β, α + β ) = 0 , then α + β = kδ for some k ∈ Z + − { } .Now ( λ, α ) = λ ( α ∨ ) = λ ( − β + kδ ) ∨ = λ ( − β ∨ + k ( β,β ) K ) = − ( λ, β ) + k ( β,β ) c . So ( λ, α ) + ( λ, β ) > . Hence either ( λ, α ) > or ( λ, β ) > , i.e either λ + α ∈ P ( V ) or λ + β ∈ P ( V ) , a contradiction. Hence the claim.Now, if V λ + m δ + δ m = 0 for all m ∈ Z n and m ∈ N , then τ ( B ) + .V λ = 0 . If not, thenthere exits some m ∈ N and δ m ∈ Z m such that V λ + m δ + δ m = 0 then V µ + n δ + δ n = 0 for all n ∈ N and n ∈ Z n , where µ = λ + m δ + δ m . If not, then ( λ + m δ + n δ + δ m + δ n , α + n δ + δ m + δ n ) = ( λ, α ) + n ( λ, δ ) = ( λ, α ) + n c > ,for all α ∈ ˙∆ + , so V λ − α + m δ =0 , by Lemma 2.3, a contradiction. Hence τ ( B ) + .V µ = 0 .This completes the proof. (cid:3) Remark 3.1.
We call a weight vector v ∈ V as highest weight vector for τ ( B ) if τ ( B ) + .v = 0 . From the proof of Proposition 3.1 it follows that highest weight isdominant integral for g af , i.e λ ( α ∨ i ) ≥ for ≤ i ≤ n, if λ is highest weight. Let T = { v ∈ V : τ ( B ) + v = 0 } , then T = 0 by Proposition 3.1. Clearly T is amodule for τ ( B ) with finite dimensional weight spaces. We have, V = U ( τ ( B )) T = U ( τ ( B ) − ) T , as V is irreducible. Now it follows by weight argument that T is anirreducible module for τ ( B ) .Let v λ ∈ T be a weight vector of V . Then the weights of T are lies in the set { λ + δ m : δ m = n X i =1 m i δ i , m = ( m , m , .., m n ) ∈ Z n } , since T is irreducible τ ( B ) module.Now we are going to prove the weight spaces of T are uniformly bounded, ideasfrom [7, 3].Let λ be a weight of T , then λ = λ | ˙ h is dominant integral by Remark 3.1.Now for ≤ i ≤ n consider the loop algebra G i = ˙ g ⊗ C [ t ± i ] ⊕ C d i and the lattice M i generated by { t i,h : h ∈ Z [ ˙Ω( β ∨ )] } , where ˙Ω is the weyl group of ˙ g and t i,h ( λ ) = λ − λ ( h ) δ i . Note that t i,h − ( λ ) = λ + λ ( h ) δ i . Let r λ = min { λ ( h ) : λ ( h ) > , h ∈ Z [ ˙Ω( β ∨ )] } . This r λ exists and r λ ∈ N , since Z [ ˙Ω( β ∨ )] is contained in the Z linearspan of α i ∨ , ≤ i ≤ d . Let Ω i be the Weyl group of G i . It is well known that Weylgroup Ω i is semi direct product of M i and ˙Ω . See [13] for more details. Lemma 3.5.
Consider λ + rδ i , r ∈ Z , then there exists a w i ∈ Ω i such that w i ( λ + rδ i ) = λ + rδ i , ≤ r < r λ = R i ( say ) .Proof. Note that t i,hk ( λ ) = λ − kλ ( h ) δ i , for all k ∈ N and t i,h − k ( λ ) = λ + kλ ( h ) δ i ,for all k ∈ N . Hence apply t i,h or t i,h − sufficient number of times, according as r ispositive or negative. (cid:3) Now each Ω i ⊂ Ω , so we get there exists w i ∈ Ω such that w i ( λ + rδ i ) = λ + rδ i , ≤ r < R i , for ≤ i ≤ n . Hence it follows from Lemma 3.5, Corollary 3.1.
Consider λ + δ r , r ∈ Z n then there exists a w ∈ Ω such that w ( λ + δ r ) = λ + n X i =1 s i δ i , ≤ s i < R i for ≤ i ≤ n . Now consider the set S = { µ : µ = λ + n X i =1 s i δ i , ≤ s i ≤ R i } . This is a finite setand let N = M ax { dimT µ : µ ∈ S } . Then it follows from Corollary 3.1 and Lemma2.3, Theorem 3.2.
The dimensions of the weight spaces of T are uniformly bounded by N .Now we are going to prove that most of the center acts trivially on our module.Before that we recall some facts about Heisenbarg algebra from [4].Consider the Heisenberg algebra e H = { h ( n ) , c : n ∈ Z } with the bracket define by [ h ( n ) , h ( m )] = nδ n + m, c , and c is central in e H . Let e H + = M n> h ( n ) and e H − = M n< h ( n ) .Since e H is Z -graded so U ( e H ) is also Z -graded. Let a ∈ C ∗ and let C v a be a onedimensional module for e H ǫ ⊕ C c such that e H ǫ .v a = 0 and cv a = av a for ǫ ∈ { , − } .Consider the e H module e H ǫ ( a ) = U ( e H ) O U ( e H ǫ ) ⊕ C c C v a , Then e H ǫ ( a ) is also Z -graded via gradation of U ( e H ) . Let e H ǫ ( a ) = M i ∈ Z e H ǫ ( a ) i , then dim e H ǫ ( a ) i = P ( | i | ) , P is the partion function, Proposition 4.1, [4]. Proposition 3.2.
Let V be a Z -graded finitely generated module for e H , where centeracts non-trivially and at least one component of V is finite dimensional. Then1. V is completely reducible.2. Each irreducible component of V is isomorphic to e H ǫ ( a ) , for some a, ǫ upto somegrade shift.Proof. Follows from [4] Proposition 4.5 and 4.3. (cid:3)
Proposition 3.3.
Let V be a module for τ ( B ) satisfying conditions of Proposition3.1, then K i ⊗ b acts on V trivially, for ≤ i ≤ n and for all b ∈ B. Proof.
Fix ≤ i ≤ n and b ∈ B and assume that K i ⊗ b acts on V non-trivially.Consider the Heisenberg algebra e H = span { h ⊗ t li ⊗ b, h ′ ⊗ t − li ⊗ , K i ⊗ b, l ∈ N } with the bracket [ h ⊗ t ni ⊗ b, h ′ ⊗ t − mi ⊗
1] = ( h, h ′ ) nδ n − m, K i ⊗ b such that ( h, h ′ ) = 0 , h, h ′ ∈ ˙ h (this is possible due to the non degenerate bilinearform on ˙ h ).Let M = U ( e H ) v λ , where v λ is a weight vector in T , then M = M r ∈ Z M r , where M r = M ∩ T λ − rδ i , hence dimM r ’s are uniformly bounded by Theorem 3.2. Thus M is a finite generated Z -graded module for e H with finite dimensional component, so by Proposition 3.2 M is completely reducible and each components are isomorphic to e H ǫ ( a ) for some a ∈ C ∗ and ǫ ∈ { , − } . But dimensions of components of e H ǫ ( a ) arenot uniformly bounded , a contradiction with the fact that dimM r ’s are uniformlybounded. (cid:3) Theorem 3.3.
Let V be an irreducible integrable module for τ ( B ) with finite di-mensional weight spaces with respect to H and some K i acting on V non trivially,then rank of S = n Proof.
Let us assume k < n . We know that after twisting the action of τ ( B ) by anautomorphism we can assume that all K i ’s acting on V trivially and K act as apositive integer. Hence by Proposition 3.1, there exists a weight vector v ∈ V , sayweight λ such that τ ( B ) + v = 0 . In particular h ⊗ t m t rn − k ⊗ bv = 0 for all h ∈ ˙ h , m > , r ∈ Z and b ∈ B . For h ( = 0) ∈ ˙ h consider the following set of vectors S ′ = { h ⊗ t − m t r − dn − k ⊗ b.h ⊗ t − ( m +1)0 t dn − k ⊗ b ′ .v : d ∈ Z } , for some fixed m > , r ∈ Z , b, b ′ ∈ B. Claim:
The set S ′ is linearly independent. Consider X d ∈ Z a d h ⊗ t − m t r − dn − k ⊗ b.h ⊗ t − ( m +1)0 t dn − k ⊗ b ′ .v = 0 . (3.2)consider h ′ ∈ h such that ( h, h ′ ) = 0 and applying h ′ ⊗ t m +10 t sn − k ⊗ for any s ∈ Z on (3 . we have X d ∈ Z a d h ⊗ t − m t r − dn − k ⊗ b.h ′ ⊗ t m +10 t sn − k ⊗ .h ⊗ t − ( m +1)0 t dn − k ⊗ b ′ .v +( h, h ′ ) X d ∈ Z a d ( m + 1) t t r + s − dn − k K ⊗ b.h ⊗ t − ( m +1)0 t dn − k ⊗ b ′ .v +( h, h ′ ) X d ∈ Z a d st t r + s − dn − k K n − k ⊗ b.h ⊗ t − ( m +1)0 t dn − k ⊗ b ′ .v = 0 Now 2nd and 3rd term of the above sum is zero by Theorem 3.1 and the 1st termimply X d ∈ Z a d h ⊗ t − m t r − dn − k ⊗ b.h ⊗ t − ( m +1)0 t dn − k ⊗ b ′ .h ′ ⊗ t m +10 t sn − k ⊗ .v +( h, h ′ ) X d ∈ Z a d h ⊗ t − m t r − dn − k ⊗ b. ( m + 1) t s + dn − k K ⊗ b ′ .v +( h, h ′ ) X d ∈ Z a d h ⊗ t − m t r − dn − k ⊗ b.st s + dn − k K n − k ⊗ b ′ .v = 0 . v is a highest weight vector and 2nd , 3rdterms are zero when s + d = 0 by Proposition 3.3. Now if s + d = 0 , then the sum isequal to ( h, h ′ ) a − s h ⊗ t − m t r + sn − k ⊗ b. ( m + 1) K ⊗ b ′ .v +( h, h ′ ) a − s h ⊗ t − m t r + sn − k ⊗ b.sK n − k ⊗ b ′ .v = 0 . Now since k < n , the last term is zero by Proposition 3.3 , hence only remaining termis a − s h ⊗ t − m t r + sn − k ⊗ b. ( m + 1) K ⊗ b ′ .v = 0 . Since not all K ⊗ b acting on V trivially and they acts as scaler, so to prove a − s = 0 ,it is sufficient to prove that h ⊗ t − m t r + sn − k ⊗ b acting on V non trivially for some b.Let h ⊗ t − m t r + sn − k ⊗ b.v = 0 , for all b, then h ′ ⊗ t m t − ( r + s ) n − k ⊗ .h ⊗ t − m t r + sn − k ⊗ b.v = 0 which imply ( h, h ′ ) K ⊗ b.v − ( h, h ′ ) K n − k ⊗ b.v = 0 for all b.Thus K ⊗ b = 0 for all b, by Proposition 3.3, a contradiction, Hence the claim.Hence we have proved that dimension of the weight space V λ − (2 m +1) δ + rδ n − k is infinite,a contradiction. (cid:3) Method of the proof is refer to [1].By Theorem 3.1 and 3.3, after twisting the module by an automorphism we canassume that t r t r K i ⊗ b = 0 on V for all ≤ i ≤ n and all r , r, b,t r t r K ⊗ b = 0 on V if r = 0 , for all r, b. Moreover, V = V /W is an irreducible module for ˙ g ⊗ A ⊗ B ⊕ Z ⊗ B ⊕ C d .Let g af = ˙ g ⊗ C [ t ± ] ⊕ C K ⊕ C d be the affine Lie algebra with the well knownLie bracket. Now Consider the current Kac-Moody Lie algebra g ′ af ⊗ A n ⊗ B ⊕ C d and define an algebra homomorphism φ : ˙ g ⊗ A ⊗ B ⊕ Z ⊗ B ⊕ C d g ′ af ⊗ A n ⊗ B ⊕ C d X ⊗ t r t r ⊗ b ( X ⊗ t r ) ⊗ t r ⊗ b,t r t r K i ⊗ b , ≤ i ≤ n,t r t r K ⊗ b , if r = 0 ,t r t r K ⊗ b K ⊗ t r ⊗ b, if r = 0 ,d d , for all X ∈ ˙ g , b ∈ B and r = ( r , r , ..., r n ) ∈ Z n , r ∈ Z . Clearly this is a surjective Lie algebra homomorphism, whose Kernel acts triviallyon our module, hence V is actually an irreducible module for g ′ af ⊗ A n ⊗ B ⊕ C d with finite dimensional weight spaces with respect to h ⊕ C K ⊕ C d . Since V isintegrable, so V is integrable for the algebra g ′ af ⊗ A n ⊗ B ⊕ C d .Recall the definition of highest weight module for current Kac-Moody Lie algebrafrom [2]. Definition 3.2.
A module W for current Kac-Moody algebra g B is said to be a highestweight module if there exists a vector w ∈ W such that1. U ( g B ) w = W, g ′ + ⊗ B.w = 0 ,
3. There exists a map η : h ′ ⊗ B ⊕ h ′′ C such that h.w = η ( h ) w for all h ∈ h ′ ⊗ B ⊕ h ′′ . Now we will prove that V is highest weight module for the current Kac-Moodyalgebra G = g ′ af ⊗ A n ⊗ B ⊕ C d .Note that h ⊗ t r ⊗ b, K ⊗ t r ⊗ b leaves each finite dimensional weight spaces of V invariant and commutes with g ′ af ⊗ A n ⊗ B ⊕ C d , hence acts as scaler on V ( sameproof like Lemma 2.2).Therefore V is an irreducible integrable highest weight module for g ′ af ⊗ A n ⊗ B ⊕ C d with finite dimensional weight spaces. Now it follows from Theorem 3.4 [2], Theorem 3.4.
Let V be a module for τ ( B ) satisfying conditions of Theorem 3.3,then upto an automorphism of τ ( B ) there exists an irreducible integrable highestweight module V for the current Kac Moody algebra g ′ af ⊗ A n ⊗ B ⊕ C d with finitedimensional weight spaces. Moreover V is an irreducible module for k M i =1 g ′ af ⊕ C d ,isomorphic to k O i =1 V ( λ i ) , for some k ∈ N ; where λ i ’s are dominant integral weightand each V ( λ i ) is irreducible module for g ′ af .4. Classification when a part of center acts non-trivially
In this section we are going to recover our original module V for τ ( B ) . Main ideasare from [5] and [6].Let e G = g ′ af ⊗ A n ⊗ B ⊕ C d ⊕ D , D is the space spanned by d , d , ..., d n and definea Lie algebra structure on e G by natural way. Then G = M m ∈ Z n G m via the action of D .Now define a e G module action on ( V ⊗ A n , ρ ( α )) for any α = ( α , α , .., α n ) ∈ C n by X r . ( v ⊗ t s ) = ( X r .v ) ⊗ t r + s , X r ∈ G r ,d . ( v ⊗ t s ) = ( d .v ) ⊗ t s ,d i . ( v ⊗ t s ) = ( α i + s i )( v ⊗ t s ) , for ≤ i ≤ n .It is easy to see that this action define a module structure for any α ∈ C n but wefix α depending on situation.Fix a highest weight vector v of V for the rest of the section. Lemma 4.1.
Any non zero e G sub-module of V ⊗ A n contains v ⊗ t s for some s ∈ Z n .Proof. Let W be a non zero e G sub-module of V ⊗ A n . Consider the the map φ : V ⊗ A n V defined by φ ( w ⊗ t m ) = w . It is easy to check that φ is a surjective G module map (but not e G ). Claim: φ ( W ) = V , since φ is surjective and V is irreducible G module so it issufficient to show φ ( W ) = 0 .Let V = M λ ∈ h ∗ V λ with respect to the Cartan sub-algebra h = span { h , K , d } thenweight spaces of V ⊗ A n are ( V ⊗ A n ) λ + δ r = V λ ⊗ t r with respect to the Cartan,span { h , K , d , D } . Hence W is also a weight module, so contains a vector the form w ⊗ t m for some weight vector w ∈ V , therefore φ ( W ) = 0 .Now v ∈ V so there exists a vector w ∈ W such that φ ( w ) = v . Since φ is a G module map w and v has same weight as G module. But { v ⊗ t r : r ∈ Z n } are theonly weights in V ⊗ A n of same weight that of v . Hence w = X finite a i v ⊗ t r i . Now W is a Z n graded sub-module of V ⊗ A n , hence v ⊗ t r i ∈ W for all i. (cid:3) Remark 4.1.
Consider the e G sub-module U ( e G ) v ⊗ t r and U ( e G ) v ⊗ t s and definea map φ : U ( e G ) v ⊗ t r U ( e G ) v ⊗ t s by φ ( w ⊗ t k ) = w ⊗ t k + s − r . then φ is a G module isomorphism ( but not e G module morphism ) . Now we define grade shiftisomorphism between U ( e G ) v ⊗ t r and U ( e G ) v ⊗ t s . Let D acts on v ⊗ t r by a scaler α = ( α , α , ..., α n ) ∈ C n , i.e d i .v ⊗ t r = α i v ⊗ t r and acts on v ⊗ t s by a scaler β = ( β , β , ..., β n ) . Let C be a one dimensional module for e G such that G. C = 0 and D. C as a scaler β − α . Then clearly U ( e G ) v ⊗ t r ≃ U ( e G ) v ⊗ t s ⊗ C as e G module. wecall this as grade shift isomorphism. Remark 4.2.
Let us consider the irreducible integrable module V for τ ( B ) with finitedimensional weight spaces. Also let V = M r ∈ Z n V r , where V r = { v ∈ V : d i .v =( λ ( d i ) + r i ) v for ≤ i ≤ n } for the weight λ of V . Define a map ψ : V V ⊗ A n by ψ ( v r ) = v ⊗ t r if v r ∈ V r . Then ψ is a e G module map for the fixed choice of α = ( λ ( α ) , λ ( α ) , ..., λ ( α n )) ∈ C n . In fact ψ is non zero, since ψ ( v ) = v ⊗ forany v ∈ V λ . Thus ψ is injective, since V is irreducible. Hence ψ ( V ) is an irreduciblesubmodule of V ⊗ A n . Lemma 4.2. U ( e G ) v ⊗ t r are irreducible e G module for all r ∈ Z n . In fact anyirreducible submodule of V ⊗ A n is of the above form.Proof. By Remark 4.2 there exists an irreducible submodule for V ⊗ A n . Let W beany irreducible submodule of V ⊗ A n , then W contains v ⊗ t r for some r ∈ Z n , byLemma 4.1. Hence U ( e G ) v ⊗ t r = W . Hence by Remark 4.1 U ( e G ) v ⊗ t r irreduciblefor all r ∈ Z n . (cid:3) Lemma 4.3.
Each z in Z ⊗ B acts as scaler on V . Further if z in Z ⊗ B acts asnon zero on V then it acts as non zero scaler on V Proof.
Follows from lemma 3.1 [5]. (cid:3)
From Theorem 3.1 and 3.3 we have n non zero central operator z i of degree l i =(0 , , ..., l i , .., . Let z i acts on V by l ′ i = 0 by Lemma 4.3, then we have z i ( w ⊗ t m ) = l ′ i ( w ⊗ t m + l i ) , this imply that z i is an invertible central operator on V ⊗ A n . Let Γ = l Z + l Z + ..... + l n Z Then for all s = ( l s , .., l n s n ) ∈ Γ there exists an invertible central operator z s = Q ni =1 z s i i on V ⊗ A n . Note that z s ( U ( e G ) v ⊗ t r ) ⊆ ( U ( e G ) v ⊗ t r + s ) , hence z s ( U ( e G ) v ⊗ t r ) =( U ( e G ) v ⊗ t r + s ) . Let < S > be the space spanned by all central operators of the form z s for all s ∈ Γ . Let U ( G ) = M m ∈ Z n U ( G ) m , where U ( G ) m = { X ∈ U ( G ) : [ d i , X ] = m i X, ≤ i ≤ n } . Proposition 4.1. V ⊗ A n = X s ∈ Z n ( U ( e G ) v ⊗ t s ) = M s ∈ F ( U ( e G ) v ⊗ t s ) , where F ⊆{ ( s , s , .., s n ) : 0 ≤ s i < l i } as e G module.2. V ≃ ( U ( e G ) v ⊗ upto grade shift as e G module.Proof.
1. Let w ⊗ t s ∈ V ⊗ A n , since V is irreducible there exist X ∈ U ( G ) such that X.v = w . Let X = P X r , X r ∈ U ( G ) r , then P X r . ( v ⊗ t s − r ) = ( P X r .v ) ⊗ t s = w ⊗ t s , hence V ⊗ A n = X s ∈ Z n ( U ( e G ) v ⊗ t s ) .Now since z s ( U ( e G ) v ⊗ t r ) = U ( e G ) v ⊗ t r + s for all s ∈ Γ , hence V ⊗ A n = X ≤ s i ≤ l i ( U ( e G ) v ⊗ t s i ) as U ( e G ) ⊕ < S > module. Now use Lemma 3.2, then V ⊗ A n = M s ∈ F U ( e G )) v ⊗ t s as U ( e G ) ⊕ < S > module, where F ⊆ { ( s , s , .., s n ) : 0 ≤ s i < l i } . But all U ( e G )) v ⊗ t s i are irreducible e G modules, hence the result.2. By Remark 4.2, ψ ( V ) is irreducible and V ⊗ A n is direct sum of finitely manyirreducible e G module, so ψ ( V ) = ( U ( e G ) v ⊗ t s ) for some s ∈ F, by Lemma 3.2. Hencethe result, by Remark 4.1. (cid:3) Theorem 4.1.
Let V be an irreducible integrable representation of τ ( B ) with finitedimensional weight spaces with respect to H such that some of K i ’s acts non triviallyon V . Then upto an automorphism of τ ( B ) , V is isomorphic to an irreducible com-ponent of ( V ⊗ A n , ρ ( α )) for some fixed α ∈ C n , namely ( U ( e G ) v ⊗ as e G moduleupto some grade shift, where V ≃ k O i =1 V ( λ i ) as G module for some k ∈ N , and each V ( λ i ) ’s are irreducible modules for derived affine Lie algebra, with dominant integralweight λ i , for ≤ i ≤ k. Classification when a part of center acts trivially
In this section we will classify irreducible integrable modules for τ ( B ) , where K ′ i s, ≤ i ≤ n acts trivially on modules. Proposition 5.1.
Let V be an irreducible integrable module for τ ( B ) with finitedimensional weight spaces with respect to H . Suppose all K i (0 ≤ i ≤ n ) actingtrivially on V . Then there exists weight vectors v , w such that ˙ g + ⊗ A ⊗ B.v = 0 and ˙ g − ⊗ A ⊗ B.w = 0 . Proof.
See Proposition 2.12 [1], same proof will work. (cid:3)
Proposition 5.2.
Let V be a module for τ ( B ) satisfying conditions of the Proposition5.1, then Z ⊗ B acts on V trivially.Proof. Let t r t r K i ⊗ b acting on V non trivially for some r ∈ Z and r ∈ Z n . Thenby Theorem 3.1, i ≤ n − k and r j = 0 for ≤ j ≤ n − k .Fixed some i ≤ n − k , ( r , r ) ∈ Z n +1 , b ∈ B and h, h ′ ∈ ˙ h such that ( h, h ′ ) = 0 Let us construct the Heisenberg algebra e H = span { h ⊗ t r t r t ki ⊗ b, h ′ ⊗ t − ki , t r t r K i ⊗ b,k ∈ Z } with a Lie bracket defined by (cid:2) h ⊗ t r t r t ki ⊗ b, h ′ ⊗ t − li (cid:3) = k ( h, h ′ ) t r t r K i ⊗ bδ k + l, . By Proposition 5.1 there exists a weight vector v λ of weight λ such that ˙ g + ⊗ A ⊗ B.v λ =0 . Now by Theorem 3.1 there is a proper sub-module W of V such that v λ W . Con-sider M = U ( e H ) v λ , v λ ∈ V = V /W . Then M is a Z graded e H module, in fact M = M k ∈ Z M ∩ V λ + r δ + δ r + kδ i , hence each components of M are finite dimensional,since weight spaces of V are so.Now since λ is dominant by Corollary 3.1, there exists a ω ∈ Ω such that ω ( λ + kδ i ) = λ + kδ i , where ≤ k < R for some R ∈ Z . Hence by Lemma 2.3(2) { dimV λ + r δ + δ r + kδ i : k ∈ Z } is a finite set, hence dimensions of components of M are uniformly bounded, which is not possible, follows from the proof of Proposition3.3. (cid:3) Fix some notations for rest of the section.Let τ = ˙ g ⊗ A ⊗ B , e τ = τ ⊕ D , h = ˙ h ⊗ A ⊗ B and e h = h ⊕ D , then U ( τ ) and U ( h ) are both Z n +1 graded algebras.Let ψ : U ( h ) A be a Z n +1 graded algebra homomorphism and A ψ = Image of ψ We make A ψ into a e h module by defining h ⊗ t r ⊗ b.t m = ψ ( h ⊗ t r ⊗ b ) t m ,d i .t m = m i t m , for all h ∈ ˙ h , b ∈ B , r ∈ Z n +1 . Lemma 5.1. ( Lemma 1.2 , [8] ) A ψ is an irreducible e h module iff every homogeneouselements of A ψ is invertible in A ψ . Let ψ be as above and let ˙ g + ⊗ A ⊗ B acting on A ψ trivially. Now construct theinduced e τ module M ( ψ ) = U ( e τ ) O U (˙ g + ⊗ A ⊗ B ⊕ e h ) A ψ . By standard arguments we have Proposition 5.3.
1. As a e h module M ( ψ ) is a weight module.2. M ( ψ ) is a free ˙ g − ⊗ A ⊗ B module and as a vector space M ( ψ ) ≃ U ( ˙ g − ⊗ A ⊗ B ) ⊗ C .3. M ( ψ ) has a unique irreducible quotient , say V ( ψ ) . A module V for e τ is said to be a graded highest weight module if there exists aweight vector v such that(1) U ( e τ ) .v = V. (2) ˙ g + ⊗ A ⊗ B.v = 0 .(3) U ( e h ) v ≃ A ψ as a e h module for some Z n +1 graded morphism ψ . A module W is said to be a non graded highest weight module for τ if there existsa weight vector w such that(1) U ( τ ) .w = W. (2) ˙ g + ⊗ A ⊗ B.w = 0 . (3) there exists ψ ∈ h ∗ such that h.w = ψ ( h ) w for all h ∈ h ∗ .Let ψ ∈ h ∗ and let C ψ be a one dimensional vector space. Let ˙ g + ⊗ A ⊗ B actingon C ψ trivially and h acting by ψ . Now construct the induced module for τ , M ( ψ ) = U ( τ ) O U (˙ g + ⊗ A ⊗ B ⊕ h ) C ψ . By standard argument we have Proposition 5.4.
1. As a h module M ( ψ ) is a weight module.2. M ( ψ ) is a free ˙ g − ⊗ A ⊗ B module and as a vector space M ( ψ ) ≃ U ( ˙ g − ⊗ A ⊗ B ) ⊗ C .3. M ( ψ ) has a unique irreducible quotient , say V ( ψ ) . Let ψ : U ( h ) A ψ be a Z n +1 morphism and let E (1) : A ψ C be a linear mapsuch that E (1) t r = 1 . Let ψ = E (1) ψ and V ( ψ ) be the irreducible τ module. Wewill make V ( ψ ) ⊗ A into a e τ module by x ⊗ t r ⊗ b. ( v ⊗ t s ) = ( x ⊗ t r ⊗ b.v ) ⊗ t r + s ,d i . ( v ⊗ t s ) = s i ( v ⊗ t s ) , for all r, s ∈ Z n +1 , b ∈ B . Proposition 5.5.
Let ψ , ψ be as above and A ψ be an irreducible e h module. Let G ⊂ Z n +1 and { t m : m ∈ G } denote the set of coset representative of A/A ψ . Let v ∈ V ( ψ ) be the highest weight vector. Then1. V ( ψ ) ⊗ A = M m ∈ G U ( v ⊗ t m ) as a e τ module , where U ( v ⊗ t m ) is the e τ sub modulegenerated by v ⊗ t m .2. Each U ( v ⊗ t m ) is irreducible e τ module.3. V ( ψ ) ≃ U ( v ⊗ as e τ module.4. V ( ψ ) has finite dimensional weight spaces as e h module iff V ( ψ ) has finite di-mensional weight spaces as h module .proof follows from same lineup of [9] , Proposition 1.8, 1.9 and Lemma 1.10. Lemma 5.2.
Let I and J be two co-finite ideal of a commutative, associative, finitelygenerated algebra S over C . Then IJ is a co-finite ideal of S . Hence product of finitenumbers of co-finite ideal of S is co-finite.Proof. To prove this we use the fact that, a finitely generated algebra over C isartinian iff it is a finite dimensional over C , [14] (Chapter 8, problem 3). Now sincedim S/I and dim
S/J both are finite so both are Artinian ring.Consider the morphism f : S/IJ S/I × S/J by f ( a + IJ ) = ( a + I, a + J ) , then f ( S/IJ ) is finitely generated as an algebra over C and is of finite dimensional. Hence f ( S/IJ ) is an Artinian ring. Now consider an exact sequence S/I S/IJ f ( S/IJ ) , since both S/I and f ( S/IJ ) are Artinian, hence S/IJ is so. (cid:3)
Proposition 5.6.
Let V be an irreducible module for ˙ g ⊗ S with finite dimensionalweight spaces with respect to ˙ h , where S is a commutative, associative, finitely gen-erated unital algebra over C . Then there exists a co-finite ideal I of S such that V isa module for a finite dimensional algebra ˙ g ⊗ S/I .Proof.
Let V λ be a weight spaces of V . Then V λ is invariant under the commutativesub-algebra ˙ h ⊗ S of ˙ g ⊗ S . Hence by Lie’s theorem there exists a non zero vector v ∈ V λ such that h ⊗ s.v = µ ( h ⊗ s ) v , for some µ ∈ ( ˙ h ⊗ S ) ∗ , for all h ∈ ˙ h , s ∈ S .Fix a root α of ˙ g and Let X α be the root vector corresponding to the root α .Let I α = { s ∈ S : X α ⊗ s.v = 0 } Claim: I α is an ideal of S .Let s ∈ I α and t ∈ S , then X α ⊗ s.v = 0 . Now consider h ∈ ˙ h such that α ( h ) = 0 .Then, h ⊗ t.X α ⊗ s.v = 0 ⇒ [ h ⊗ t, X α ⊗ s ] .v + X α ⊗ s.h ⊗ t.v = 0 ⇒ [ h ⊗ t, X α ⊗ s ] .v + µ ( h ⊗ t ) X α ⊗ s.v = 0 ⇒ [ h, X α ] ⊗ st.v = 0 i.e α ( h ) X α ⊗ st.v = 0 , hence st ∈ I α , since α ( h ) = 0 . Claim : dim
S/I α is finite.Let dim V λ + α ≤ k . Now consider any n ( > k ) non zero elements in S/I α , say s + I α , s + I α , ..., s n + I α .Consider the vectors X α ⊗ s .v, X α ⊗ s .v, ...., X α ⊗ s n .v in V λ + α . Since dim V λ + α is ≤ k so, there exists c , c , ..., c n ∈ C not all zero such that X α ⊗ n X i =1 c i s i .v = 0 , hence n X i =1 c i s i ∈ I α . So any set of n non zero vectors in S/I α are linearly dependent.Hence dim S/I α ≤ k. Now consider I = Y α ∈ ˙∆ I α , then S/I is finite dimensional, by Lemma 5.2.
Claim: ˙ g ⊗ I.v = 0
Clearly ˙ g ± ⊗ I.v = 0 , since I ⊆ I α for all α ∈ ˙∆ . Again, h α ⊗ I.v = [ X α , X − α ] ⊗ I.v = 0 for all α ∈ ˙∆ . Hence the claim.Now consider W = { v ∈ V : ˙ g ⊗ I.v = 0 } , which is a non zero sub-module of V ,hence by irreducibility W = V . Thus our module is a module for ˙ g ⊗ S/I . (cid:3) Lemma 5.3. ( Lemma 5.1, [1] ) Any Z n graded simple, commutative algebra of whicheach graded component is finite dimensional is isomorphic to a subalgebra of A n suchthat each homogeneous elements is invertible. Theorem 5.1.
Let V be an irreducible integrable module for τ ( B ) with finite di-mensional weight spaces with respect to H , then V ≃ V ( ψ ) for some Z n +1 gradedmorphism ψ : U ( h ) A . More over V ( ψ ) is finite dimensional module for τ , where ψ = E (1) ψ . Proof.
By Proposition 5.1, V is actually a module for e τ . Now consider the e h module W = U ( e h ) v for some weight vector v of weight λ . Then W = U ( e h ) v = U ( h ) v = M m ∈ Z n +1 V λ + δ m ∩ W. Thus each graded components of W is finite dimensional. Define φ : U ( h ) W by X r X r v for X r ∈ U ( h ) r , where U ( h ) r is the r component of U ( h ) . Clearly φ is a Z n +1 graded surjective h module morphism. Thus U ( h ) /kerφ ≃ W . Nowirreducibility of V imply that W is irreducible e h module. Hence W is a commuta-tive, associative graded simple algebra, since U ( h ) is commutative. Now it followsfrom Lemma 5.3 that W ≃ A ψ for some Z n +1 graded morphism ψ such that eachhomogeneous elements of A ψ is invertible in A ψ . Thus W ≃ A ψ as e h module byLemma 5.1, Hence V ≃ V ( ψ ) Clearly V ( ψ ) is an irreducible integrable module for τ . Again by Proposition 5.5 V ( ψ ) has finite dimensional weight spaces with respect to h . Now by Proposi-tion 5.6 there exists a co-finite ideal I of A ⊗ B such that V ( ψ ) is a module for ˙ g ⊗ ( A ⊗ B/I ) . Now consider the highest weight vector v ∈ V ( ψ ) then by PBWtheorem, V ( ψ ) = U ( ˙ g − ⊗ ( A ⊗ B/I )) v. But each vector of ˙ g − ⊗ ( A ⊗ B/I ) acts locallynilpotently on v . Hence V ( ψ ) is finite dimensional. (cid:3) For toroidal case this theorem is due to S.Eswara Rao [9].Let S be a finitely generated commutative, associative unital algebra over C . Let M axS denote the set of all maximal ideals of S and Supp ( M axS, P + ) denote the setof all finitely supported functions from M axS to ˙ P + . Theorem 5.2.
Let S be an associative, commutative, finitely generated unital alge-bra over C . Then any finite dimensional irreducible modules for ˙ g ⊗ S reduces themodules for M N − copies ˙ g ( N = | Supp ( M axS, P + ) | ) isomorphic to O χ ( M ) =0 V ( χ ( M )) , forsome χ ∈ Supp ( M axS, P + ) , Proof.
Proposition in section 6 , [10]. (cid:3)
Theorem 5.3.
Let V be an irreducible integrable representation of τ ( B ) with finitedimensional weight spaces with respect to H such that all K i ’s acts trivially on V . Then there exists a Z n +1 graded morphism ψ : U ( h ) A such that V ≃ V ( ψ ) .Further V ( ψ ) is isomorphic to an irreducible component of V ( ψ ) ⊗ A as e τ module and V ( ψ ) is a finite dimensional irreducible module for M N − copies ˙ g ( N = | Supp ( M axA ⊗ B, P + ) | ) which is isomorphic to O χ ( M ) =0 V ( χ ( M )) , for some χ ∈ Supp ( M axA ⊗ B, P + ) . Proof.
Follows from Theorem 5.2, 5.1 and Proposition 5.5 (cid:3) Acknowledgments:
We would like to thank Professor S. Eswara Rao for somehelpful discussion.
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