Classification of simple Harish-Chandra modules over the Neveu-Schwarz algebra and its contact subalgebra
aa r X i v : . [ m a t h . R T ] S e p Classification of simple Harish-Chandra modules over theNeveu-Schwarz algebra and its contact subalgebra
Yan-an Cai, Rencai L¨u
Abstract
In this paper, we classify all simple jet modules for the Neveu-Schwarz algebra b k andits contact subalgebra k + . Based on these results, we give a classification of simpleHarish-Chandra modules for b k and k + . Keywords: the contact subalgebra, Neveu-Schwarz algebra, Harish-Chandra modules,Jet modules
1. Introduction
We denote by Z , Z + , N , C and C ∗ the sets of all integers, non-negative integers, posi-tive integers, complex numbers, and nonzero complex numbers, respectively. All vectorspaces and algebras in this paper are over C . We denote by U ( L ) the universal envelop-ing algebra of the Lie (super)algebra L over C . Also, we denote by δ i,j the Kroneckerdelta. Throughout this paper, by subalgebras, submodules for Lie superalgebras we meansubsuperalgebras and subsupermodules respectively.Superconformal algebras have been widely studied in mathematical physics. Thesimplest examples are the Virasoro algebra, the central extension of the Witt algebra W ,and its subalgebra W + . An important class of modules for the Virasoro algebra and W + are the so-called quasifinite modules (or Harish-Chandra modules), the weight moduleswith finite dimensional weight spaces, which were classified by Mathieu in [18].After the Virasoro algebra (corresponding to N = 0), we have the N = 1 supercon-formal algebras, also known as the super-Virasoro algebras: the Neveu-Schwarz algebraand the Ramond algebra. Weight modules for the super-Viraoro algebras have beenextensively investigated (cf. [5, 8, 9]), for more related results we refer the reader to[3, 6, 10–13, 16–20, 22] and references therein.Based on the classification of simple jet modules introduced by Y. Billig in [1] (seealso [6]), the authors gave a complete classification of simple Harish-Chandra modules forLie algebra of vector fields on a torus with the so-called A cover theory in [2]. Recently,with the study of jet modules, a classification of simple Harish-Chandra modules forgeneralized Heisenberg-Virasoro, the Witt superalgebras, etc., were given, see [14, 15, 24]and references therein.A complete classification for the N = 1 superconformal algebra was given in [21].However, the complicated computations in the proofs maike it difficult to follow. Re-cently, with the theory of the A -cover, a new approach to classify all simple Harish- Preprint submitted to October 2, 2020 handra modules for the N = 1 Ramond algebra s was given in [4]. However, it is moredifficult to classify such modules for the Neveu-Schwarz algebra.A superversion of W and W + are the Lie superalgebras of contact vector fields onthe supercircle S | and the superspace C | , say k (also known as the centerless Neveu-Schwarz algebra) and k + . In this paper, we classify simple jet modules for k + . Asa consequence, we give a classification of simple jet modules for k . Then based onthese results, with the A cover theory we classify simple Harish-Chandra modules forthe Neveu-Schwarz algebra b k and its contact subalgebra k + , which is a super version ofMathieu’s results on the Virasoro algebra and its subalgebra W + in [18]. This also givesa new approach for Su’s results in [21].The paper is organized as follows. In Section 2, we collect some basic results forour study. Simple jet modules are classified in Section 3. Finally, we classify all simpleHarish-Chandra modules for the Neveu-Schwarz algebra and its contract subalgebra inSection 4.
2. Preliminaries
The Neveu-Schwarz algebra b k is the Lie superalgebra over C with a basis { L n , G r , C | n ∈ Z , r ∈ Z + } satisfying the commutation relations | L n | = ¯0 , | G r | = ¯1 , | C | = ¯0;[ L m , L n ] = ( n − m ) L m + n + δ m + n, m − m C ;[ L m , G r ] = ( r − m G m + r ;[ G r , G s ] = − L r + s + 13 δ r + s, ( r −
14 ) C. b k is a Z -graded Lie superalgebra with b k i = C L i + δ i, C C, ∀ i ∈ Z and b k j = C G j , ∀ j ∈ + Z .Let k = b k / C C . Then k is also a Z -graded Lie superalgebra. For p ∈ Z , set k ≥ p = P i ≥ p k i .Then k ≥− and k ≥ p ( p ∈ Z + ) are Lie supersubalgebra of k . In particular, k + = k ≥− isalso a subalgebra of b k , called the contact subalgebra of b k , and k ≥ p is an ideal of k ≥ forany p ∈ Z + .Let A = C [ t, t − ] ⊗ Λ(1), where Λ(1) is the Grassmann algebra in one variable ξ .This algebra is Z -graded with | t | = ¯0 , | ξ | = ¯1. A is naturally a k -module: for all x ∈ A, i ∈ Z , m ∈ Z + , L i ◦ x = t i +1 ∂ t ( x ) + 12 ( i + 1) t i ξ∂ ξ ( x ) ,G m ◦ x = t m + ξ∂ t ( x ) − t m + ∂ ξ ( x ) , here ∂ t = ∂∂t , ∂ ξ = ∂∂ξ . Hence, we have the extended Neveu-Schwarz algebras ˜ k = k ⋉ A, ˜ k + = k + ⋉ A and ˜ k ++ = k + ⋉ A + , with A and A + being abelian Lie superalgebras,where A + = C [ t ] ⊗ Λ(1). 2n the other hand, k ( k + , respectively) has a natural module structure over theabelian superalgebra A ( A + , respectively): t i L j = L i + j , t i G m = G m + i , ξL j = 12 G j + , ξG m = 0 . (2.1)And k ( k + , respectively) is a ˜ k (˜ k ++ ) module with adjoint k ( k + , respectively) actions and A ( A + ) acting as (2.1). To see this, we only need to verify v ( ax ) − ( − | v || a | a ( vx ) = [ v, a ] x = ( v ◦ a ) x, for all homogeneous x, v ∈ k ( k + , respectively) and a ∈ A ( A + , respectively). In fact, wehave[ L m , t i L n ] − t i [ L m , L n ] = [ L m , L n + i ] − ( n − m ) t i L m + n = iL m + n + i = it m + i L n , [ L m , ξL n ] − ξ [ L m , L n ] = 12 [ L m , G n + ] − ( n − m ) ξL m + n = m + 14 G m + n + = m + 12 t m ξL n , [ L m , t i G r ] − t i [ L m , G r ] = [ L m , G r + i ] − ( r − m t i G m + r = iG m + r + i = it m + i G r , [ L m , ξG r ] − ξ [ L m , G r ] = 0 , [ G r , t i L n ] − t i [ G r , L n ] = [ G r , L n + i ] + ( r − n t i G r + n = i G r + n + i = it r + i − ξL n , [ G r , t i G s ] − t i [ G r , G s ] = [ G r , G s + i ] + 2 t i L r + s = 0 , [ G r , ξL n ] + ξ [ G r , L n ] = 12 [ G r , G n + ] − ( r − n ξG r + n = − L r + n + = − t r + L n , [ G r , ξG s ] + ξ [ G r , G s ] = − ξL r + s = − G r + s + = − t r + G s . An A k ( A k + , A + k + , respectively) module is a e k ( e k + , e k ++ , respectively) module with A ( A, A + , respectively) acting associatively. Let U = U ( e K ) , U + = U ( e K + ) , U ++ = U ( e K ++ ) and I ( I + , I ++ , respectively) be the left ideal of U ( U + , U ++ , respectively)generated by t i · t j − t i + j , t − , t i · ξ − t i ξ and ξ · ξ for all i, j ∈ Z ( Z , Z + , respectively).Then it is clear that I ( I + , I ++ respectively) is an ideal of U ( U + , U ++ respectively).Let U = U/I, U + = U + /I + , U ++ = U ++ /I ++ . Then the category of A k ( A k + , A + k + respectively) modules is naturally equivalent to the category of U ( U + , U ++ respectively)modules.Let g be any of k , k + , e k , e k + , e k ++ , b k , k ≥ . A g module M is called a weight module if theaction of L on M is diagonalizable. Let M be a weight g module. Then M = L λ ∈ C M λ ,where M λ = { v ∈ M | L v = λv } , called the weight space of weight λ . Supp( M ) := { λ ∈ C | M λ = 0 } is called the support of M . A cuspidal ( uniformly bounded ) g module is aweight module M such that the dimension of weight spaces is uniformly bounded, thatis there is N ∈ N with dim M λ < N for all λ ∈ Supp( M ). Clearly, if M is simple, thenSupp( M ) ⊆ λ + Z for some λ ∈ C .Let ( g , A ) be ( k , A ) , ( k + , A ) or ( k + , A + ). A jet g module associated to A is a weight A g module with a finite dimensional weight space. Denote the category consisting of alljet g modules associated to A by J ( g , A ). Clearly, any simple module in J ( k , A ) and J ( k + , A ) is cuspidal. 3et σ : L → L ′ be any homomorphism of Lie superalgebras or associative super-algebras, and M be any L ′ -module. Then M become an L -module, denoted by M σ ,under x · v := σ ( x ) v, ∀ x ∈ L, v ∈ M . Denote by T the automorphism of L definedby T ( x ) := ( − | x | x, ∀ x ∈ L . For any L -module M , Π( M ) is the module defined by aparity-change of M .A module M over an associative superalgebra B is called strictly simple if it is asimple module over the associative algebra B (forgetting the Z -gradation).We need the following result on tensor modules over tensor superalgebras. Lemma 2.1 ([23, Lemma 2.1, 2.2]) . Let
B, B ′ be associative superalgebras, and M, M ′ be B, B ′ modules, respectively.1. M ⊗ M ′ ∼ = Π( M ) ⊗ Π( M ′ T ) as B ⊗ B ′ -modules.2. If in addition that B ′ has a countable basis and M ′ is strictly simple, then(a) M ⊗ M ′ is a simple B ⊗ B ′ -module if and only if M is a simple B -module.(b) If V is a simple B ⊗ B ′ -module containing a strictly simple B ′ = C ⊗ B ′ module M ′ , then V ∼ = M ⊗ M ′ for some simple B -module M . The subalgebra of k spanned by { L k | k ∈ Z } is isomorphic to the Witt algebra W .Denote by W + the subalgebra of W spanned by { L k | k ∈ − Z + } . The followingresults for W modules and W + modules will be used. Lemma 2.2.
Let Ω ( m ) k,s = m P i =0 ( − i (cid:0) mi (cid:1) L k − i L s + i .1. ([2, Corollary 3.7]) For every ℓ ∈ N there exists m ∈ N such that for all k, s ∈ Z , Ω ( m ) k,s annihilate every cuspidal W -module with a composition series of length ℓ .2. ([24, Corollary 4.4]). For every r ∈ N there is an m ∈ N such that for all k, s ∈ Z + , Ω ( m ) k + m − ,s − annihilates every cuspidal W + module V with dim V λ ≤ r for all λ ∈ Supp( V ) .
3. Jet modules
In this section, we will classify all simple jet modules in J ( k , A ) , J ( k + , A ) and J ( k + , A + ). First, we have Lemma 3.1.
Let M ∈ J ( k , A ) be simple. Then M is also a simple module in J ( k + , A ) ,and the A k module structure is uniquely determined by the A k + module structure.Proof. Following from Lemma 2.2, on M we have0 = X i =0 2 X j =0 ( − i + j (cid:18) j (cid:19) [ t r + i − j , Ω ( m ) k +1 − i,s − j ] = m +2 X i =0 ( − i (cid:18) m + 2 i (cid:19) t r + k +1 − i · L s − i , r, k, s ∈ Z . So A · n P i =0 ( − i (cid:0) ni (cid:1) t k − i · L s + i M = 0 for all k, s ∈ Z , n ≥ m + 2. Hence,on M we have0 =[ m +3 X i =0 ( − i (cid:18) m + 3 i (cid:19) t k − i · L s − i , G p +1 ] − [ m +3 X i =0 ( − i (cid:18) m + 3 i (cid:19) t k − i · L s + i , G p ]= m +3 X i =0 ( − i (cid:18) m + 3 i (cid:19) ( p + 3 − i − s t k − i · G p + s + i − m +3 X i =0 (cid:18) m + 3 i (cid:19) ( k − i ) t p + k − i + ξ · L s − i − (cid:16) m +3 X i =0 ( − i (cid:18) m + 3 i (cid:19) ( p − i + s t k − i · G p + s + i − m +3 X i =0 (cid:18) m + 3 i (cid:19) ( k − i ) t p + k − i − ξ · L s + i (cid:17) = 32 m +3 X i =0 ( − i (cid:18) m + 3 i (cid:19) t k − i · G p + s + i , where k, s ∈ Z , p ∈ Z + . So A · n P i =0 ( − i (cid:0) ni (cid:1) t k − i · G p − i M = 0 for all n ≥ m + 3 , k ∈ Z , p ∈ Z + . Therefore, the actions of L i ( i ∈ Z ) , G p ( p ∈ Z + ) are determined by theactions of L i ( − ≤ i ≤ m ) , G p ( − ≤ p ≤ m + ). And hence lemma follows.Thus, to determine all simple modules in J ( k , A ), it suffices to determine all simplemodules in J ( k + , A ).For n ∈ Z + , let L ′ n := n +1 X i =0 ( − i +1 (cid:18) n + 1 i (cid:19) t n − i +1 · L i − + n + 12 n X i =0 ( − i (cid:18) ni (cid:19) t n − i ξ · G i − ,G ′ n − := n X i =0 ( − i (cid:18) ni (cid:19) ( t n − i · G i − − t n − i ξ · L i − ) . Then we have n X k =0 ( − k (cid:18) n + 1 k + 1 (cid:19) t n − k · ( L ′ k − k + 12 ξ · G ′ k − ) + t n +1 · L − = n X k =0 k +1 X i =0 ( − k + i +1 (cid:18) n + 1 k + 1 (cid:19)(cid:18) k + 1 i (cid:19) t n − i +1 · L i − + t n +1 · L − = n X j =0 n X k = j ( − k + j (cid:18) n + 1 k + 1 (cid:19)(cid:18) k + 1 j + 1 (cid:19) t n − j · L j + n +1 X j =0 ( − j (cid:18) n + 1 j (cid:19) t n +1 · L − = n X j =0 ( − j (cid:18) n + 1 j + 1 (cid:19) n X k = j ( − k (cid:18) n − jk − j (cid:19) t n − j · L j = L n , (3.1)and n X k =0 ( − k (cid:18) nk (cid:19) t n − k · ( G ′ k − − ξ · L ′ k − ) = n X k =0 k X i =0 ( − k + i (cid:18) nk (cid:19)(cid:18) ki (cid:19) t n − i · G i − n X i =0 n X k = i ( − k + i (cid:18) ni (cid:19)(cid:18) n − ik − i (cid:19) t n − i · G i − = G n − . (3.2)Let T be the supersubspace of U + with a basis B = { L ′ n , G ′ m − | n ∈ Z + , m ∈ N } .Clearly, T is also a supersubspace of U ++ . We have Lemma 3.2. e B = B ∪ { G − , L − } is a basis of the free left A ( A + , respectively)module A · k + ( A + · k + , respectively).2. T = { x ∈ A · k + | [ x, A ] = [ x, G − ] = 0 } = { x ∈ A + · k + | [ x, A + ] = [ x, G − ] = 0 } . Thus, T is a Lie supersubalgebra of U + as well as a Lie supersubalgebra of U ++ .Proof.
1. The A -linear independence of e B is easy to check. And the first statementfollows from (3.1) and (3.2).2. For the second statement, first we have[ G − , L ′ n ]=[ G − , n +1 X i =0 ( − i +1 (cid:18) n + 1 i (cid:19) t n − i +1 · L i − + n + 12 n X i =0 ( − i (cid:18) ni (cid:19) t n − i ξ · G i − ]= n +1 X i =0 ( − i +1 (cid:18) n + 1 i (cid:19) (( n − i + 1) t n − i ξ · L i − + i t n − i +1 · G i − )+ n + 12 n X i =0 ( − i (cid:18) ni (cid:19) ( − t n − i · G i − + 2 t n − i ξ · L i − ) = 0 , [ L ′ n , t k ]= n +1 X i =0 ( − i +1 (cid:18) n + 1 i (cid:19) t n − i +1 · [ L i − , t k ] + n + 12 n X i =0 ( − i (cid:18) ni (cid:19) t n − i ξ · [ G i − , t k ]= n +1 X i =0 ( − i +1 (cid:18) n + 1 i (cid:19) t n − i +1 · kt k + i − + n + 12 n X i =0 ( − i (cid:18) ni (cid:19) t n − i ξ · kt i + k − ξ = 0 , [ L ′ n , t k ξ ]= n +1 X i =0 ( − i +1 (cid:18) n + 1 i (cid:19) t n − i +1 · [ L i − , t k ξ ] + n + 12 n X i =0 ( − i (cid:18) ni (cid:19) t n − i ξ · [ G i − , t k ξ ]= n +1 X i =0 ( − i +1 (cid:18) n + 1 i (cid:19) t n − i +1 · ( kt k + i − ξ + i t k + i ξ ) − n + 12 n X i =0 ( − i (cid:18) ni (cid:19) t n − i ξ · t i + k =0 , [ G − , G ′ n − ]= n X i =0 ( − i (cid:18) ni (cid:19) [ G − , t n − i · G i − − t n − i ξ · L i − ]6 n X i =0 ( − i (cid:18) ni (cid:19) (( n − i ) t n − i − ξ · G i − − t n − i · L i − + 2 t n − i · L i − + it n − i ξ · G i − )= n X j =1 ( − j +1 (cid:18) n − j − (cid:19) nt n − j ξ · G j − + n X i =1 ( − i n (cid:18) n − i − (cid:19) t n − i ξ · G i − = 0 , [ G ′ n − , t k ]= n X i =0 ( − i (cid:18) ni (cid:19) ( t n − i · [ G i − , t k ] − t n − i ξ · [ L i − , t k ])= − k n X i =0 ( − i (cid:18) ni (cid:19) t n + k − ξ = 0 , [ G ′ n − , t k ξ ]= n X i =0 ( − i (cid:18) ni (cid:19) ( t n − i · [ G i − , t k ξ ] − t n − i ξ · [ L i − , t k ξ ])= − n X i =0 ( − i (cid:18) ni (cid:19) t n + k = 0 . That is
T ⊆ T = { x ∈ A · k + | [ x, A ] = [ x, G − ] = 0 } . On the other hand, let x = P f i · x i + x ′ ∈ T with f i ∈ A, x i ∈ B , x ′ ∈ A · { G − , L − } . Then0 = [ G − , x ] = X [ G − , f i ] · x i + [ G − , x ′ ]implies that f i ∈ C and x ′ = a ( G − − ξ · L − ) + bL − for some a, b ∈ C . And[ x, f ] = [ x ′ , f ] = 0 for all f ∈ A implies that a = b = 0, that is x ′ = 0. So T ⊆ T and hence T = T . Lemma 3.3.
We have the associative superalgebra isomorphism ι : K ⊗ U ( T ) → U + , ι ( x ⊗ y ) = x · y, where K = C [ t ± , ξ, ∂ t , ∂ ξ ] is the Weyl superalgebra, and x ∈ K , y ∈ U ( T ) . Moreover,the restriction of ι to K + ⊗ U ( T ) is an isomorphism from K + ⊗ U ( T ) to U ++ , where K + = A + [ ∂ t , ∂ ξ ] .Proof. Consider the map ι ′ : A [ G − ] ⊗ U ( T ) → U + , ι ( x ⊗ y ) = x · y . Since T is aLie supersubalgebra of U + and A [ G − ] is an associative supersubalgebra of U + , therestrictions of ι ′ on A [ G − ] and U ( T ) are well-defined. From Lemma 3.2, in U , ι ′ ( U ( T ))and ι ′ ( A [ G − ]) are super commutative, hence ι ′ is a well-defined homomorphism ofassociative superalgebras. Let L = A ⊗T +( A [ G − ]+ A ) ⊗ C ⊆ A [ G − ] ⊗ U ( T ). Equations(3.1) and (3.2) tell us that ι ′ | L : L → A · k + + A is a Lie superalgebra isomorphism. Thus,we have a Lie superalgebra homomorphism η : e k + ( ι ′ | L ) − −−−−−→ L ⊆ A [ G − ] ⊗ U ( T ) with η ( t n ξ r ) = t n ξ r ⊗ , n ∈ Z , r = 0 ,
1; 7 ( L n ) = n X k =1 ( − k (cid:18) n + 1 k + 1 (cid:19) ( t n − k ⊗ L ′ k − k + 12 t n − k ξ ⊗ G ′ k − ) + ( n + 1) t n ⊗ L ′ − n + 12 t n ξ · G − ⊗ t n +1 · L − ⊗ , n ≥ − η ( G n − ) = n X k =1 ( − k (cid:18) nk (cid:19) ( t n − k ⊗ G ′ k − − t n − k ξ ⊗ L ′ k − ) + t n · G − ⊗ − t n ξ · L − ⊗ . So we have the associative superalgebra homomorphism ˜ η : U + → A [ G − ] ⊗ U ( T ) with˜ η | e K + = η and I + ⊆ Ker˜ η . Therefore, we have the induced associative superalgebrahomomorphism ¯ η : U + → A [ G − ] ⊗ U ( T ) with ¯ η = ι ′− . Hence ι ′ is an isomorphismand lemma follows from the fact that A [ G − ] is isomorphic to K . Lemma 3.4.
We have the Lie superalgebra isomorphism
T ∼ = k ≥ .Proof. Consider the linear map ψ : T → k ≥ defined by ψ ( L ′ n ) → ( − n Ln, ψ ( G ′ n + ) =( − n +1 G n + . Then clearly ψ is an isomorphism of vector superspaces. From (3.1) and(3.2), we have k ≥ ⊆ m · L − + m · G − + A + ·T , where m = A + t and m = m + C ξ . Hence,we have the Lie superalgebra homomorphism ω : mk + ⊆ m · L − + m · G − + A + · T → ( m · L − + m · G − + A + · T ) / ( m · L − + m · G − + m · T ) → ( A + · T ) / ( m · T ) → T .More precisely, we have ω ( L n ) = ( − n L ′ n , ω ( G n + ) = ( − n +1 G n + . So ψ = ω − and ω is a Lie superalgebra isomorphism, so is ψ .The following results on simple weight K modules and simple weight K + = C [ t, ξ, ∂ t , ∂ ξ ]modules, on which t∂ t acts diagonalizable, is needed. Lemma 3.5.
1. Up to a parity change, any simple weight K module is isomorphic tosome strictly simple K module A ( λ ) := t λ C [ t ± , ξ ] for some λ ∈ C .2. Up to a parity change, any simple weight K + module is isomorphic to one of thefollowing strictly simple K + module: t λ C [ t ± , ξ ] , C [ t, ξ ] , C [ t ± , ξ ] / C [ t ] , where λ ∈ C \ Z .Proof. The first statement is just Lemma 3.5 in [23]. For the second statement, it iseasy to check the giving modules are strictly simple K + modules. Now let V be asimple weight K + module with λ ∈ Supp( V ). For a fixed nonzero homogeneous element v ∈ V λ , since V ′ = C [ ∂ ξ ] v is finite dimensional with ∂ ξ acting nilpotently, we can find anonzero homogeneous element v ′ ∈ V ′ such that ∂ ξ v ′ = 0. Then C [ ξ, ∂ ξ ] v ′ = C [ ξ ] v ′ isa strictly simple C [ ξ, ∂ ξ ] module. So lemma follows from Lemma 2.1 and the facts that K + ∼ = C [ t, ∂ t ] ⊗ C [ ξ, ∂ ξ ] and any simple weight C [ t, ∂ t ] module is isomorphic one of thefollowing (cf. [7]): t λ C [ t ± ] , C [ t ] , C [ t ± ] / C [ t ].Now for any k ≥ module, we have the A k + module Γ( λ, V ) := ( A ( λ ) ⊗ V ) ϕ , where ϕ : U + ι − −−→ K ⊗ U ( T ) id ⊗ ψ −−−→ K ⊗ U ( k ≥ ) and the A + k + modules F ( P, V ) = ( P ⊗ V ) ϕ | U ++ P a simple weight K + module. More precisely, Γ( λ, V ) = A ⊗ V ( F ( P, V ) = P ⊗ V ,respectively) with actions x · ( y ⊗ v ) = xy ⊗ v,L n · ( y ⊗ v ) = n X k =1 (cid:18) n + 1 k + 1 (cid:19) ( t n − k y ⊗ L k · v − k + 12 t n − k ξy ⊗ G k − · v )+ ( n + 1) t n y ⊗ L · v + n + 12 t n ξ∂ ξ ( y ) ⊗ v + t n ( t∂ t ( y ) + λy ) ⊗ v,G n + · ( y ⊗ v ) = n +1 X k =1 (cid:18) n + 1 k (cid:19) ( t n +1 − k y ⊗ G k − · v − t n +1 − k ξy ⊗ L k − · v ) − t n ξ ( t∂ t ( y ) + λy ) ⊗ v − t n +1 ∂ ξ ( y ) ⊗ v, where x ∈ A ( A + , respectively), y ∈ A ( P , respectively), n ∈ Z ≥− and v ∈ V . Lemma 3.6.
1. For any λ ∈ C and any simple k ≥ module V , the A k + module Γ( λ, V ) is simple.2. For any simple weight K + module P and any simple k ≥ module V , the A + k + module F ( P, V ) is simple.3. Any simple module in J ( k + , A ) is isomorphic to some Γ( λ, V ) for some λ ∈ C and some simple weight k ≥ module V .4. Any simple module in J ( k + , A + ) is isomorphic to some F ( P, V ) for some simpleweight K + module P and some simple weight k ≥ module V .Proof. Statement 1 and 2 follow from Lemma 3.5 and Lemma 2.1. Now let M ∈ J ( k + , A ) ( J ( k + , A + ), respectively) be simple with dim M λ < ∞ for some λ ∈ Supp( M ).Then M ϕ − is a simple K⊗ U ( k ≥ ) ( K + ⊗ U ( k ≥ ), respectively) module. If M ∈ J ( k + , A ),we can find a common eigenvector v for t∂ t and L such that ( t∂ t − λ ) v = ∂ ξ v = 0, andhence K v ∼ = A ( λ ) or K v ∼ = Π( A ( λ )). From Lemma 2.1, there exists a simple U ( k ≥ )module V such that M ϕ − ∼ = A ( λ ) ⊗ V or M ϕ − ∼ = Π( A ( λ )) ⊗ V ∼ = A ( λ ) ⊗ Π( V ). Thefact that the adjoint action of L on K ⊗ U ( k ≥ ) is diagonalizable tells us that V andΠ( V ) is a simple weight k ≥ module.For Statement 4, we can find a nonzero element u such that ( t∂ t − λ ) · u = ∂ ξ · u = 0( λ / ∈ Z ) or ∂ t · u = ∂ ξ · u = 0 or t · u = ∂ ξ · u = 0, and hence K + u ∼ = t λ C [ t ± , ξ ] or K + u ∼ = C [ t, ξ ]or K + u ∼ = C [ t ± , ξ ] / C [ t ]. And again by Lemma 2.1, M ϕ − = ( K + ⊗ U ( k ≥ )) u ∼ = K + u ⊗ V for some simple U ( k ≥ ) module V . Moreover, since the adjoint actions of t∂ t and L on K + ⊗ U ( k ≥ ) is diagonalizable, L is diagonalizable on M ϕ − , and hence V is a simpleweight k ≥ module. Lemma 3.7.
Suppose V is a simple weight k ≥ module. Then k ≥ V = 0 . Thus, V canbe regarded as a simple weight k ≥ / k ≥ module. Moreover, V = C v is one dimensionalwith L v = bv for some b ∈ C , and L i v = G i − v = 0 , ∀ i ≥ .Proof. Let V be a simple weight k ≥ module and 0 = v ∈ V λ for some λ ∈ Supp( V ).Then V = U ( k ≥ ) v and Supp( V ) ⊆ λ + Z + . On the other hand, k ≥ V is a submodule9f V with support λ + N so that v / ∈ k ≥ v . Therefore, from the simplicity of V , we have k ≥ V = 0. Corollary 3.8.
Let P be a simple weight K ( K + , respectively) module and V be a simpleweight k ≥ / k ≥ module. Then F ( P, V ) is a cuspidal A k + ( A + k + , respectively) module. Remark 3.9.
Clearly, any Γ( λ, V ) is an A + k + module. It is easy to check as Γ( λ, V ) issimple as A + k + module if and only if λ / ∈ Z . And any simple module in J ( k + , A + ) is aquotient of some Γ( λ, V ) . In particular, any simple module in J ( k + , A + ) is cuspidal. Now we can classify all simple modules in J ( k , A ) , J ( k + , A ) and J ( k + , A + ). Theorem 3.10.
1. Any simple module in J ( k , A ) as well as in J ( k + , A ) is isomor-phic to the module Γ( λ, b ) = A for some λ, b ∈ C with actions: for all n, k ∈ Z , x · y = xy, ∀ x, y ∈ A, (3.3) L n · t k = ( λ + k + b ( n + 1)) t n + k , (3.4) L n · t k ξ = ( λ + k + ( n + 1)( b + 12 )) t n + k ξ, (3.5) G n + · t k = − ( k + λ + 2 b ( n + 1)) t n + k ξ, (3.6) G n + · t k ξ = − t n + k +1 . (3.7)
2. Any simple module in J ( k + , A + ) is isomorphic to one of the following(a) Γ( λ, b ) for some λ ∈ C \ Z and b ∈ C ;(b) The A + k + submodule Γ + (0 , b ) := A + of Γ(0 , b ) for some b ∈ C ;(c) The A + k + quotient module Γ − (0 , b ) := Γ(0 , b ) / Γ + (0 , b ) .
4. Main results
In this section, we will classify all simple weight b k modules and k + modules withfinite dimensional weight spaces. First of all, from the representation theory of Virasoroalgebra, we know that C acts trivially on any simple cuspidal b k module, and hence thecategory of simple cuspidal b k modules is naturally equivalent to the category of simplecuspidal k modules.The following result is well-known Lemma 4.1.
Let M be a weight module with finite dimensional weight spaces for theVirasoro algebra with supp( M ) ⊆ λ + Z . If for any v ∈ M , there exists N ( v ) ∈ N suchthat L i v = 0 , ∀ i ≥ N ( v ) , then supp( M ) is upper bounded. Lemma 4.2 tells us that classification of simple cuspidal modules is an important stepin classifying all simple Harish-Chandra modules over b k and k + . Lemma 4.2.
1. Any simple weight b k module with finite dimensional weight spaceswhich is not cuspidal is a highest/lowest weight module.2. Any simple weight k + module with finite dimensional weight spaces is cuspidal. roof.
1. For a simple weight b k module M , fix a λ ∈ Supp( M ). Suppose M is not cus-pidal, then there exists a k ∈ Z such that dim M − k + λ > M λ + dim M λ + +dim M λ +1 ). Without lost of generality, we may assume that k ∈ N . Then thereexists a nonzero element w ∈ M − k + λ such that L k w = L k +1 w = G k + w = 0.Therefore L i w = G i − w = 0 , ∀ i ≥ ( k + 1) .Since M ′ = { v ∈ M | L i v = G i − v = 0 , ∀ i ≫ } is a nonzero submodule of M , weknow M = M ′ . And hence by Lemma 4.1, Supp( M ) is upper bounded, that is M is a highest weight module.2. Let M be a simple weight k + module with finite dimensional weight spaces. Firstwe show that if M is not cuspidal, then M is a highest weight module or a lowestweight module. If the action of G − is not injective, then there exists v ∈ M suchthat G − v = 0, and clearly M is a lowest weight module. So we may assumethat G − acts injectively on M , then we have dim M i − ≥ dim M i . Consider therestricted dual M ∗ := L λ ∈ Supp( M ) Hom C ( M λ , C ). Then M ∗ is a k + module withsupport − Supp( M ) under( x · f )( v ) := − ( − | x || f | f ( xv ) , ∀ x ∈ ( K + ) i , v ∈ M λ + j , f ∈ Hom C ( M λ + i + j , C ) . More precisely, ( M ∗ ) − λ = Hom C ( M λ , C ). Hence, dim( M ∗ ) − λ = dim M λ . Since M is not cuspidal, there exists µ such that dim M µ − > dim M µ and hence the actionof G − on M ∗ is not injective. Therefore, M ∗ is a lowest weight module, that is − Supp( M ) is lower bounded. So Supp( M ) is upper bounded.To complete the proof, it remains to show any simple highest weight k + moduleis cuspidal. This is clear since from the PBW theorem, as vector spaces, M ∼ = C [ G − ] v , which is cuspidal with dim M λ ≤ λ ∈ Supp( M ).Now let ( g , A ) be ( k , A ) or ( k + , A + ) and M be a cuspidal g module. Consider g asthe adjoint g module. Then the tensor product g ⊗ M becomes an A g module under x · ( y ⊗ u ) = ( xy ) ⊗ u, ∀ x ∈ A , y ∈ g , u ∈ M. Denote K ( M ) = { P i x i ⊗ v i ∈ g ⊗ M | P i ( ax i ) v i = 0 , ∀ a ∈ A} . Then it is easy tosee that K ( M ) is an A g submodule of g ⊗ M . And hence we have the A g module c M = ( g ⊗ M ) / K ( M ). Also, we have a g module epimorphism defined by π : c M → g M ; x ⊗ y + K ( M ) xy, ∀ x ∈ g , y ∈ M. c M is called the A -cover of M if g M = M .Let M be a cuspidal k module. Then M is a cuspidal W module and hence thereexists m ∈ N such that for all k, p ∈ Z , Ω ( m ) k,p M = 0. Therefore [Ω mk,p , G j ] M = 0 , ∀ k, p ∈ Z , j ∈ + Z . Then, on M we have0 =[Ω ( m ) k,p − , G j +1 ] − ( m ) k,p , G j ] + [Ω ( m ) k,p +1 , G j − ] − [Ω ( m ) k +1 ,p − , G j ]+ 2[Ω ( m ) k +1 ,p , G j − ] − [Ω ( m ) k +1 ,p +1 , G j − ] 11[ m X i =0 ( − i (cid:18) mi (cid:19) L k − i L p − i , G j +1 ] − m X i =0 ( − i (cid:18) mi (cid:19) L k − i L p + i , G j ]+ [ m X i =0 ( − i (cid:18) mi (cid:19) L k − i L p +1+ i , G j − ] − [ m X i =0 ( − i (cid:18) mi (cid:19) L k +1 − i L p − i , G j ]+ 2[ m X i =0 ( − i (cid:18) mi (cid:19) L k +1 − i L p + i , G j − ] − [ m X i =0 ( − i (cid:18) mi (cid:19) L k +1 − i L p +1+ i , G j − ]= m X i =0 ( − i (cid:18) mi (cid:19)(cid:16) ( j + 1 − k − i G k − i + j +1 L p − i + ( j + 1 − p − i L k − i G p + i + j − j − k − i G k − i + j L p + i − j − p + i L k − i G p + i + j + ( j − − k − i G k − i + j − L p + i +1 + ( j − − p + i + 12 ) L k − i G p + i + j − ( j − k − i + 12 ) G k − i + j +1 L p + i − − ( j − p + i −
12 ) L k − i +1 G p + i + j − + 2( j − − k − i + 12 ) G k − i + j L p + i + 2( j − − p + i L k − i +1 G p + i + j − − ( j − − k − i + 12 ) G k − i + j − L p + i +1 − ( j − − p + i + 12 ) L k +1 − i G p + i + j − (cid:17) = 32 m X i =0 ( − i (cid:18) mi (cid:19) ( G k − i + j +1 L p + i − − G k − i + j L p + i + G k − i + j − L p + i +1 )= 32 m +2 X i =0 ( − i (cid:18) m + 2 i (cid:19) G k − i + j +1 L p + i − . Similarly, replacing k by k + m − k, p ∈ Z + , we know that any cuspidal k + moduleis annihilated by m P i =0 ( − i (cid:0) m +2 i (cid:1) G m + k − i + j L p + i − for some m ∈ N and for all k, p ∈ Z + .That is we have Lemma 4.3.
1. Any cuspidal k module is annihilated by m P i =0 ( − i (cid:0) mi (cid:1) G k − i L p + i forsome m ∈ N and for all k ∈ + Z , p ∈ Z .2. Any cuspidal k + module is annihilated by m P i =0 ( − i (cid:0) mi (cid:1) G m + k − i − L p + i − for some m ∈ N and for all k ∈ + Z + , p ∈ Z + . Lemma 4.4.
Let g be k or k + . For any simple cuspidal g module M , c M is cuspidal.Proof. It is obvious if M it trivial. Now suppose M is nontrivial. Then g M = M and Supp( M ) ⊆ λ + Z . Suppose dim M µ ≤ r, ∀ µ ∈ Supp( M ). For g = k ( k + , re-spectively), from Lemma 2.2 and Lemma 4.3, there exists m ∈ N , such that such that m P i =0 ( − i (cid:0) mi (cid:1) L j − i L p + i v = m P i =0 ( − i (cid:0) mi (cid:1) G j − i + L p + i v = 0 , ∀ j ∈ Z ( Z + + m, respectively) , p ∈ ( Z + , respectively) , v ∈ M . Hence, m X i =0 ( − i (cid:18) mi (cid:19) L j − i ⊗ L p + i v, m X i =0 ( − i (cid:18) mi (cid:19) G j − i + ⊗ L p + i v ∈ K ( M ) . (4.1)Let S = span { L k , G k + | ≤ k ≤ m } . Then dim S = 2( m + 1) and S ⊗ M is a C L submodule of g ⊗ M withdim( S ⊗ M ) µ ≤ m + 1) r, ∀ µ ∈ λ + 12 Z . We will prove that g ⊗ M = S ⊗ V + K ( M ), from which we know c M is cuspidal. Indeed,we will prove by induction that for all u ∈ M µ , L n ⊗ u, G n + ⊗ u ∈ S ⊗ M. We only prove the claim for n > m , the proof for n < L acts on M µ with a nonzero scalar, we can write u = L v for some v ∈ M µ . Then by (4.1) andinduction hypothesis, we have L n ⊗ L v = m X i =0 ( − i (cid:18) mi (cid:19) L n − i ⊗ L i v − m X i =1 ( − i (cid:18) mi (cid:19) L n − i ⊗ L i v ∈ S ⊗ M + K ( M ) ,G n + ⊗ L v = m X i =0 ( − i (cid:18) mi (cid:19) G n − i + ⊗ L i v − m X i =1 ( − i (cid:18) mi (cid:19) G n − i + ⊗ L i v ∈ S ⊗ M + K ( M ) . Now we can classify all simple cuspidal k modules and all simple cuspidal k + modules. Theorem 4.5.
Let ( g , A ) be ( k , A ) or ( k + , A + ) . Then up to a parity change, any non-trivial simple cuspidal g module is isomorphic to a simple quotient of a simple cuspidal A g module. More precisely, we have, up to a parity change,1. any nontrivial simple cuspidal k module is isomorphic to a simple quotient of Γ( λ, b ) for some λ, b ∈ C ;2. any nontrivial simple cuspidal k + module is isomorphic to a simple quotient of Γ( λ, b ) , Γ + (0 , b ) or Γ − (0 , b ) for some λ, b ∈ C .Proof. Let M be any nontrivial simple cuspidal g module. Then g M = M and there is anepimorphism π : c M → M . From Lemma 4.4, c M is cuspidal. Hence c M has a compositionseries of A g submodules: 0 = c M (0) ⊂ c M (1) ⊂ · · · ⊂ c M ( s ) = c M with c M ( i ) / c M ( i − being simple A g modules. Let k be the minimal integer such that π ( c M ( k ) ) = 0. Then we have π ( c M ( k ) ) = M, c M ( k − = 0 since M is simple. So we havean g -epimorphism from the simple A g module c M ( k ) / c M ( k − to M .13t is straightforward to check that as a b k module, Γ( λ, b ) has a unique nontrivialsub-quotient which we denote by Γ ′ ( λ, b ). More precisely, we have Proposition 4.6.
1. As a b k module, Γ( λ, b ) is simple if and only if λ / ∈ Z or λ ∈ Z and b = 0 , .2. As b k modules, Γ( λ , b ) ∼ = Γ( λ , b ) if and only if λ − λ ∈ Z , b = b or λ / ∈ Z , λ − λ ∈ Z , b = , b = 0 .3. Γ ′ (0 , ∼ = Π(Γ ′ (0 , )) , where Γ ′ (0 ,
0) = Γ(0 , / C , Γ ′ (0 , ) = span { t i , t k ξ | i, k ∈ Z , k = − } .4. For any b ∈ C , Γ( λ, b )( λ / ∈ Z ) , Γ + (0 , b ) and Γ − (0 , b ) are simple k + modules. Combining Lemma 4.2, Theorem 4.5 and Proposition 4.6, we can get the followingresult, which is a super version of the classical result for the Virasoro algebra and itssubalgebra W + (cf. [18]). The result of b k was also given in [21] by much complicatedcalculations. Theorem 4.7.
1. Any simple b k module with finite dimensional weight spaces is ahighest weight module, lowest weight module, or is isomorphic to Γ ′ ( λ, b ) , Π(Γ ′ ( λ, b )) for some λ, b ∈ C (which is called a module of the intermediate series).2. Up to a parity change, any simple b k + module with finite dimensional weight spacesis isomorphic to Γ( λ, b ) , Γ + (0 , b ) , or Γ − (0 , b ) for some λ ∈ C \ Z , b ∈ C . Acknowledgement:
Y. Cai is partially supported by NSF of China (Grant 11801390)and High-level Innovation and Entrepreneurship Talents Introduction Program of JiangsuProvince of China. R. L¨u is partially supported by NSF of China (Grant 11471233,11771122, 11971440).
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