Classification of simple Harish-Chandra modules over the N=1 Ramond algebra
aa r X i v : . [ m a t h . R T ] J u l Classification of simple Harish-Chandra modules over the N = 1 Ramond algebra Yan-an Cai, Dong Liu and Rencai L¨u
Abstract
In this paper, we give a new approach to classify all Harish-Chandra modules for the N = 1 Ramond algebra s based on the so called A -cover theory developed in [1]. Keywords:
Virasoro algebra, N = 1 Ramond algebra, cuspidal module, A -cover
1. Introduction
Superconformal algebras have a long history in mathematical physics. The simplestexamples, after the Virasoro algebra itself (corresponding to N = 0) are the N = 1 super-conformal algebras: the Neveu-Schwarz algebra and the Ramond algebra. These infinitedimensional Lie superalgebras are also called the super-Virasoro algebras as they can beregarded as natural super generalizations of the Virasoro algebra. Weight modules for thesuper-Viraoro algebras have been extensively investigated (cf. [4, 6, 7]), for more relatedresults we refer the reader to [5, 8–11, 13–15, 17, 18, 20] and references therein. It is animportant and challenging problem to give complete classifications of Harish-Chandramodules (simple weight modules with finite dimensional weight spaces) for superconfor-mal algebras. In [3], all simple unitary weight modules with finite dimensional weightspaces over the N = 1 superconformal algebra were classified, which includes highest andlowest weight modules. Recently simple weight modules with finite dimensional weightspaces over the N = 2 superconformal algebra were classified in [12]. With the theory ofthe A -cover in [1] for the Virasoro algebra, [21] completed such classification for the Liesuperalgebra W m,n (also see [2]). A complete classification for the N = 1 superconformalalgebra was given by Su in [19]. However, the complicated computations in the proofsmake it extremely difficult to follow. In this paper, we give a new approach to classify allHarish-Chandra modules for the N = 1 Ramond algebra s based on the A -cover theory.This paper is arranged as follows. In Section 2, we recall some notations and collectknown facts about the N = 1 Ramond algebra s . In Section 3, we classify all simple cusp-idal modules for s . With this classification we get the main result about the calssficationof Harish-Chandra modules over s in Section 4.Throughout this paper, we denote by Z , Z + , N , C and C ∗ the sets of all integers, non-negative integers, positive integers, complex numbers, and nonzero complex numbers,respectively. All vector spaces and algebras in this paper are over C . We denote by U ( a )the universal enveloping algebra of the Lie superalgebra a over C . Also, we denote by δ i,j the Kronecker delta. Preprint submitted to July 10, 2020 . Preliminaries
In this section, we collect some basic definitions and results for our study.A vector superspace V is a vector space endowed with a Z -gradation V = V ¯0 ⊕ V ¯1 .The parity of a homogeneous element v ∈ V ¯ i is denoted by | v | = ¯ i ∈ Z . Throughoutthis paper, when we write | v | for an element v ∈ V , we will always assume that v is ahomogeneous element.The N = 1 Ramond algebra s is the Lie superalgebra with basis { L n , G n , C | n ∈ Z } and brackets [ L m , L n ] = ( n − m ) L m + n + δ m + n,
112 ( n − n ) C, [ L m , G p ] = ( p − m G p + m , [ G p , G q ] = − L p + q + δ p + q, p − C. The even part of s is spanned by { L n , C | n ∈ Z } , and is isomorphic to the Virasoroalgebra, the universal central extension of the Witt algebra w . The odd part of s isspanned by { G n | n ∈ Z } . Let ¯ s be the quotient algebra s / C C .Let A = C [ t ± ] ⊗ Λ(1), where Λ(1) is the Grassmann algebra in one variable ξ . A is Z -graded with | t | = ¯0 and | ξ | = ¯1. A is an ¯ s -module with L n ◦ x = t n +1 ∂ t ( x ) + n t n ξ∂ ξ ( x ) ,G n ◦ x = t n +1 ξ∂ t ( x ) − t n ∂ ξ ( x ) , where n ∈ Z , x ∈ A, ∂ t = ∂∂t , ∂ ξ = ∂∂ξ . So, we have Lie superalgebra e s = ¯ s ⋉ A with A anabelian Lie superalgebra and [ x, y ] = x ◦ y, x ∈ s , y ∈ A .On the other hand, ¯ s has a natural A -module structure t i L n := L n + i , t i G n := G n + i , ξL n = 12 G n , ξG n = 0 , ∀ n, i ∈ Z . (2.1)And ¯ s is an e s -module with adjoint ¯ s -actions and A acting as (2.1):[ 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 ] = [ L m , G n ] − ( n − m ) ξL m + n = 14 mG m + n = m t m ξL n , [ L m , t i G n ] − t i [ L m , G n ] = [ L m , G n + i ] − ( n − m t i G m + n = iG m + n + i = it m + i G n , [ L m , ξG n ] − ξ [ L m , G n ] = 0 , [ G m , t i L n ] − t i [ G m , L n ] = [ G m , L n + i ] + ( m − n t i G m + n = i G m + n + i = it m + i ξL n , [ G m , t i G n ] − t i [ G m , G n ] = [ G m , G n + i ] + 2 t i L m + n = 0 , [ G m , ξL n ] + ξ [ G m , L n ] = 12 [ G m , G n ] = − L m + n = − t m L n , [ G m , ξG n ] + ξ [ G m , G n ] = − ξL m + n = − G m + n = − t m G n . A ¯ s -module is an e s -module with A acting associatively. Let U = U ( e s ) and I bethe left ideal of U generated by t i · t j − t i + j , t − , t i · ξ − t i ξ and ξ · ξ for all i, j ∈ Z .Then it is clear I is an ideal of U . Let U = U/I . Then the category of A ¯ s -modules isnaturally equivalent to the category of U -modules.Let g be any of e s , s , ¯ s . A g -module M is called a weight module if the action 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 . The support of M is Supp( M ) := { λ ∈ C | M λ = 0 } . A weight g -module is called cuspidal or uniformlybounded if the dimension of weight spaces of M is uniformly bounded, that is there is N ∈ N such that dim M λ < N for all λ ∈ Supp( M ). Clearly, if M is simple, thenSupp( M ) ⊆ λ + Z for some λ ∈ C .Let 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 ([21, 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) Any B ⊗ B ′ -submodule of M ⊗ M ′ is of the form N ⊗ M ′ for some B -submodule N of M ;(b) Any simple quotient of the B ⊗ B ′ -module M ⊗ M ′ is isomorphic to some M ⊗ M ′ for some simple quotient M of M .(c) M ⊗ M ′ is a simple B ⊗ B ′ -module if and only if M is a simple B -module.(d) 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 . Let K be the Weyl superalgebra A [ ∂ t , ∂ ξ ]. All simple weight K -modules are classifiedin [21]. Lemma 2.2 ([21, Lemma 3.5]) . Any simple weight K -module is isomorphic to some A ( λ ) for some λ ∈ C up to a parity-change, here A ( λ ) ∼ = K /I λ with I λ the left ideal of K generated by t∂ t − λ, ∂ ξ . Also, the following results about ( t − s ⊂ ¯ s follow from (2.1) directly. Lemma 2.3.
Let k, ℓ ∈ Z + . Then for all i, j ∈ Z , [( t − k L i , ( t − ℓ L j ] = ( ℓ − k + j − i )( t − k + ℓ L i + j + ( ℓ − k )( t − k + ℓ − L i + j , [( t − k L i , ( t − ℓ G j ] = ( j − i t − k + ℓ G i + j + ( ℓ − k t − k + ℓ − G i + j +1 , [( t − k G i , ( t − ℓ G j ] = − t − k + ℓ L i + j . Lemma 2.4.
For k ∈ N , let a k = ( t − k ¯ s . Then1. a is a Lie subsuperalgebra of ¯ s ;2. a k is an ideal of a and a / a is a two dimensional Lie superalgebra with bosonicbasis X and femionic basis Y and nontrivial brackets [ X, Y ] = Y .3. The ideal generated by { ( t − k L m | m ∈ Z } is a k . Lemma 2.5.
Let L = C X + C Y be the Lie superalgebra with | X | = ¯0 , | Y | = ¯1 and [ X, Y ] = Y, [ Y, Y ] = 0 . Then any simple finite dimensional L -module is one dimensionalwith X.v = bv, Y.v = 0 for some b ∈ C . Lemma 2.6 ([16, Theorem 2.1], Engel’s Theorem for Lie superalgebras) . Let V be afinite dimensional module for the Lie superalgebra L = L ¯0 ⊕ L ¯1 such that the elements of L ¯0 and L ¯1 respectively are nilpotent endomorphisms of V . Then there exists a nonzeroelement v ∈ V such that xv = 0 for all x ∈ L .
3. Cuspidal modules
For m ∈ Z \ { } , let X m := t − m · L m + m t − m ξ · G m − L ,Y m := t − m · G m − t − m ξ · L m − G + 2 ξ · L ∈ U .
And let T be the subspace of U spanned by { X m , Y m | m ∈ Z \ { }} . Then we have Proposition 3.1. [ T , G ] = [ T , A ] = 0 .2. T is a Lie subsuperalgebra of U . Moreover, T is isomorphic to the Lie superalgebra ( t − s .Proof. The first statement follows from[ G , X m ] = [ G , t − m ] · L m + t − m · [ G , L m ] + m G , t − m ξ ] · G m − t − m ξ · [ G , G m ])= − mt − m ξ · L m + m t − m · G m + m − t − m · G m + 2 t − m ξ · L m )= 0 , [ G , Y m ] = [ G , t − m ] · G m + t − m [ G , G m ] − G , t − m ξ ] · L m − t − m ξ · [ G , L m ]) − [ G , G ] + 2[ G , ξ ] · L = − mt − m ξ · G m − t − m · L m − − t − m · L m − m t − m ξ · G m ) + 2 L − L = 0 , [ t n , X m ] = t − m [ t n , L m ] + m t − m ξ [ t n , G m ] + [ L , t n ] = − nt n + nt n = 0 , t n , Y m ] = t − m [ t n , G m ] − t − m ξ [ t n , L m ] − [ t n , G ] + 2 ξ [ t n , L ] = − nt n ξ + 2 nt n ξ + nt n ξ − nt n ξ = 0 , [ X m , ξ ] = t − m [ L m , ξ ] + m t − m ξ [ G m , ξ ] − [ L , ξ ] = m t − m ξ − m t − m ξ = 0 , [ Y m , ξ ] = t − m [ G m , ξ ] − t − m ξ [ L m , ξ ] − [ G , ξ ] + 2 ξ [ L , ξ ] = − . And the second statement follows from[ X m , X n ] =[ t − m · L m + m t − m ξ · G m − L , t − n · L n + n t − n ξ · G n − L ]= t − m [ L m , t − n ] · L n − t − n [ L n , t − m ] · L m + t − m − n · [ L m , L n ]+ n (cid:0) t − m [ L m , t − n ξ ] · G n − t − n ξ [ G n , t − m ] · L m + t − m − n ξ · [ L m , G n ] (cid:1) − t − m · [ L m , L ] + [ L , t − m ] · L m + m (cid:0) t − m ξ [ G m , t − n ] · L n − t − n [ L n , t − m ξ ] · G m + t − m − n ξ · [ G m , L n ] (cid:1) + mn (cid:0) t − m ξ · [ G m , t − n ξ ] · G n − t − n ξ [ G n , t − m ξ ] · G m (cid:1) − [ L , t − n ] · L n − t − n · [ L , L n ] − n (cid:0) [ L , t − n ξ ] · G n + t − n ξ · [ L , G n ] (cid:1) = − nt − n · L n + mt − m · L m + ( n − m ) t − m − n · L m + n + n (cid:0) ( m − n ) t − n ξ · G n + ( n − m t − m − n ξ · G m + n (cid:1) + m (cid:0) ( m − n t − m ξ · G m − ( m − n t − m − n ξ · G m + n (cid:1) + mn − t − n ξ · G n + t − m ξ · G m )= − nX n + mX m + ( n − m ) X m + n , [ X m , Y n ] =[ t − m · L m + m t − m ξ · G m − L , t − n · G n − t − n ξ · L n − G + 2 ξ · L ]= t − m [ L m , t − n ] · G n − t − n [ G n , t − m ] · L m + t − m − n · [ L m , G n ] − (cid:0) t − m [ L m , t − n ξ ] · L n − t − n ξ [ L n , t − m ] · L m + t − m − n ξ · [ L m , L n ] (cid:1) − [ t − m , G ] · L m − t − m · [ L m .G ] + 2 (cid:0) t − m [ L m , ξ ] · L − ξ [ L , t − m ] · L m + t − n ξ · [ L m , L ] (cid:1) + m (cid:0) t − m ξ [ G m , t − n ] · G n − t − n [ G n , t − m ξ ] · G m + t − m − n ξ · [ G m , G n ] (cid:1) − m (cid:0) t − m ξ [ G m , t − n ξ ] · L n − t − n ξ [ L n , t − m ξ ] · G m (cid:1) − m (cid:0) t − m ξ · [ G m , G ] − [ G , t − m ξ ] · G m (cid:1) + m (cid:0) t − m ξ [ G m , ξ ] · L − ξ [ L , t − m ξ ] · G m (cid:1) − [ L , t − n ] · G n − t − n · [ L , G n ] + 2[ L , t − n ξ ] · L n + 2 t − n ξ · [ L , L n ]= − nt − n · G n + mt − m ξ · L m + ( n − m t − m − n · G m + n − (cid:0) ( m − n ) t − n ξ · L n + mt − m ξ · L m + ( n − m ) t − m − n ξ · L m + n (cid:1) − mt − m ξ · L m + m t − m · G m + 2( m ξ · L + mt − m ξ · L m − mt − m ξ · L m )5 m t − m · G m − t − m − n ξ · L m + n ) + mt − n ξ · L n − m − t − m ξ · L m + t − m · G m ) − mξ · L + nt − n · G n − nt − n · G n − nt − n ξ · L n + 2 nt − n ξ · L n = − nY n + m Y m + ( n − m Y m + n , [ t − m · G m − t − m ξ · L m , t − n · G n − t − n ξ · L n ]= t − m [ G m , t − n ] · G n + t − n [ G n , t − m ] · G m + t − m − n · [ G m , G n ] − (cid:0) t − m [ G m , t − n ξ ] · L n + t − n ξ [ L n , t − m ] · G m − t − m − n ξ · [ G m , L n ] (cid:1) − (cid:0) t − m ξ [ L m , t − n ] · G n + t − n [ G n , t − m ξ ] · L m + t − m − n ξ · [ L m , G n ] (cid:1) + 4 (cid:0) t − m ξ [ L m , t − n ξ ] · L n + t − n ξ [ L n , t − n ξ ] · L m (cid:1) = − nt − n ξ · G n − mt − m ξ · G m − t − m − n · L m + n − (cid:0) − t − n · L n − mt − m ξ · G m + ( m − n t − m − n ξ · G m + n (cid:1) − (cid:0) − nt − n ξ · G n − t − m ξ · L m + ( n − m t − m − n ξ · G m + n ) (cid:1) =2( X n + X m − X m + n + L ) , [ Y m , Y n ] =2( X n + X m − X n + m ) . Moreover, T is isomorphic to ( t − s via ϕ : T 7→ ( t − s ; X m L m − L , Y m G m − G . Proposition 3.2.
We have the associative superalgebra isomorphism U ∼ = K ⊗ U ( T ) .Proof. Note that U ( T ) is an associative subalgebra of U and the map τ : A [ G ] → K with τ | A = Id A , τ ( G ) = ξt∂ t − ∂ ξ is a homomorphism of associative superalgebras.Define the map ι : A [ G ] ⊗ U ( T ) → U by ι ( t i ξ j G k ⊗ y ) = t i ξ j · G k · y + I, ∀ i ∈ Z , j =0 , , k ∈ Z + , y ∈ U ( T ). Then the restrictions of ι on A [ G ] and U ( T ) are well-definedhomomorphisms of associative superalgebras. Also, note that [ T , A ] = [ T , G ] = 0, ι isa well defined homomorphism of associative superalgebras. From ι ( t m ⊗ X m − m t m ξ ⊗ Y m + t m L ⊗ − m t m ξG ⊗
1) = L m ,ι ( t m ⊗ Y m + 2 t m ξ ⊗ X m + t m G ⊗
1) = G m , we can see that ι is surjective.By PBW theorem we know that U has a basis consisting monomials in variables { L m , G m | m ∈ Z \{ }} over A [ G ]. Therefore U has an A [ G ]-basis consisting monomialsin the variables { t − m · L m − L , t − m · G m − G | m ∈ Z \ { }} . So ι is injective and hencean isomorphism.For any ( t − s -module V , we have the A ¯ s -module Γ( λ, V ) = ( A ( λ ) ⊗ V ) ϕ , where ϕ : U ι − −−→ K ⊗ U ( T ) ⊗ ϕ −−−→ K ⊗ U (( t − s ). More precisely, Γ( λ, V ) = A ⊗ V withactions t i ξ r . ( y ⊗ u ) := t i ξ r y ⊗ u,L m . ( y ⊗ u ) := t m y ⊗ ( L m − L ) .u − ( − | y | m t m ξy ⊗ ( G m − G ) .u t m ( λy + t∂ t ( y )) ⊗ u + m t m ξ∂ ξ ( y ) ⊗ u,G m . ( y ⊗ u ) :=( − | y | t m y ⊗ ( G m − G ) .u + 2 t m ξy ⊗ ( L m − L ) .u + t m ξ ( λy + t∂ t ( y )) ⊗ u − t m ∂ ξ ( y ) ⊗ u. Lemma 3.3.
1. For any λ ∈ C and any simple ( t − s -module V , Γ( λ, V ) is a simpleweight A ¯ s -module.2. Any simple weight A ¯ s -module M is isomorphic to some Γ( λ, V ) for some λ ∈ Supp( M ) and some simple ( t − s -module V .Proof. The first statement follows from Lemma 2.1 and Lemma 2.2. For the secondstatement, let M be any simple weight A ¯ s -module with λ ∈ Supp( M ). Then M ϕ − isa simple K ⊗ U (( t − s )-module. Fix a nonzero homogeneous element v ∈ ( M ϕ − ) λ ,then C [ ∂ ξ ] v is a finite dimensional supersubspace with ∂ ξ acting nilpotently. So thereexists a nonzero element v ′ in C [ ∂ ξ ] v with I λ v ′ = 0. Clearly, K v ′ is isomorphic to A ( λ ) orΠ( A ( λ )). Hence by Lemma 2.1 and Lemma 2.2, there exists a simple U (( t − s )-module N such that M ϕ − ∼ = A ( λ ) ⊗ N or M ϕ − ∼ = Π( A ( λ )) ⊗ N ∼ = A ( λ ) ⊗ Π( N T ).Thus, to classify all simple weight A ¯ s -modules, it suffices to classify all simple ( t − s -modules. In particular, to classify all simple cuspidal A ¯ s -modules, it suffices to classifyall finite dimensional simple ( t − s -modules. Lemma 3.4.
1. Let V be any finite dimensional ( t − s -module. Then there exists k ∈ N such that a k V = 0 .2. Let V be any simple finite dimensional simple ( t − s -module. Then a V = 0 . Inparticular, dim V = 1 .Proof.
1. Since V is a finite dimensional ( t − w -module, there exists k ∈ N suchthat ( t − k w V = 0. So the first statement follows from Lemma 2.4.2. Consider the finite dimensional Lie superalgebra g = a / ann V , then V is a finitedimensional g ¯0 -module and a , ¯0 + ann( V ) acts nilpotently on V . Since [ x, x ] ∈ a , ¯0 for all x ∈ a , ¯1 , every element in a , ¯1 + ann( V ) acts nilpotently on V . Hence, byLemma 2.6, there is nonzero v ∈ V annihilated by a + ann( V ). And therefore a V = 0, which means V is a simple finite dimensional module for a / a . Corollary 3.5.
Any simple cuspidal A ¯ s -module is isomorphic to some Γ( λ, b ) = A ⊗ C u with λ, b ∈ C defined as follows: t i ξ r . ( y ⊗ u ) = t i ξ r y ⊗ u,L m . ( t i ξ r ⊗ u ) = ( λ + i + m ( b + 12 δ ¯1 , ¯ r )) t m + i ξ r ⊗ u,G m . ( t i ⊗ u ) = ( λ + i + 2 mb ) t m + i ξ ⊗ u,G m . ( t i ξ ⊗ u ) = − t m + i ⊗ u, where i, m ∈ Z , r = 0 , , y ∈ A . A -cover c M of a cuspidal ¯ s -module M . Consider ¯ s asthe adjoint ¯ s -module. Then the tensor product ¯ s -module ¯ s ⊗ M is an A ¯ s -module by x · ( y ⊗ b ) := ( xy ) ⊗ v, ∀ x ∈ A, y ∈ ¯ s , v ∈ M. Let K ( M ) = { P i x i ⊗ v i ∈ ¯ s ⊗ M | P i ( ax i ) v i = 0 , ∀ a ∈ A } . Then K ( M ) is an A ¯ s -submodule of ¯ s ⊗ M . And hence we have the A ¯ s -module c M = (¯ s ⊗ M ) /K ( M ),called the cover of M when ¯ s M = M , as in [1]. Clearly, the linear map π : c M → ¯ s M ; x ⊗ v + K ( M ) xv is an ¯ s -module epimorphism.Recall that in [1], the authors show that every cuspidal W -module is annihilated bythe operators Ω ( m ) k,s for m large enough. Lemma 3.6 ([1, Corollary 3.7]) . For every ℓ ∈ N there exists m ∈ N such that forall k, s ∈ Z the differentiators Ω ( m ) k,s = m P i =0 ( − i (cid:0) mi (cid:1) L k − i L s + i annihilate every cuspidal W -module with a composition series of length ℓ . Let M be a cuspidal ¯ s -module. Then M is a cuspidal W -module and hence there exists m ∈ N such that Ω ( m ) k,p M = 0 , ∀ k, p ∈ Z . Therefore, [Ω ( m ) k,p , G j ] M = 0 , ∀ j, k, p ∈ Z , s ∈ S .Thus, 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 − ]=[ 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 )8 32 m X i =0 ( − i (cid:18) m + 2 i (cid:19) G k − i + j +1 L p + i − . That is, we have
Lemma 3.7.
Let M be a cuspidal ¯ s -module. Then there exists m ∈ N such that for all j, p ∈ Z the operators Ω ( m ) j,p = m P i =0 ( − i (cid:0) mi (cid:1) G j − i L p + i annihilate M . Lemma 3.8.
For any cuspidal ¯ s -module M , c M is also cuspidal.Proof. Since c M is an A -module, it suffices to show that one of its weight spaces is finitedimensional. Fix a weight α + p, p ∈ Z and let us prove that c M α + p = span { L p − k ⊗ M α + k , G p − k ⊗ M α + k | k ∈ Z } is finite dimensional. Assume that α = 0 when α + Z = Z .From Lemma 3.6 and Lemma 3.7, there exists m ∈ N , 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, p ∈ Z , 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 ) . (3.1)We are going to prove by induction on | q | for q ∈ Z that for all u ∈ M α + q , L p − q ⊗ u, G p − q ⊗ u ∈ X | k |≤ m (cid:16) L p − k ⊗ M α + k + G p − k ⊗ M α + k (cid:17) + K ( M ) . We only need to prove this claim for | q | > m , and we may assume that q < − m , theproof for q > m is similar. Since L acts on M α + q with a nonzero scalar, we can write u = L v for some v ∈ M α + q . Then by (3.1) and induction hypothesis, we have L p − q ⊗ L v = m X i =0 ( − i (cid:18) mi (cid:19) L p − q − i ⊗ L i v − m X i =1 ( − i (cid:18) mi (cid:19) L p − q − i ⊗ L i v ∈ X | k |≤ m (cid:16) L p − k ⊗ M α + k + G p − k ⊗ M α + k (cid:17) + K ( M ) ,G p − q ⊗ L v = m X i =0 ( − i (cid:18) mi (cid:19) G p − q − i ⊗ L i v − m X i =1 ( − i (cid:18) mi (cid:19) G p − q − i ⊗ L i v ∈ X | k |≤ m (cid:16) L p − k ⊗ M α + k + G p − k ⊗ M α + k (cid:17) + K ( M ) . Now we can classify all simple cuspidal ¯ s -modules. Theorem 3.9.
Any nontrivial simple cuspidal ¯ s -module is isomorphic to a simple quo-tient of Γ( λ, b ) for some λ, b ∈ C . roof. Let M be any nontrivial simple cuspidal ¯ s -module. Then ¯ s M = M and there is anepimorphism π : c M → M . From Lemma 3.8, c M is cuspidal. Hence c M has a compositionseries of A ¯ s -submodules: 0 = c M (0) ⊂ c M (1) ⊂ · · · ⊂ c M ( s ) = c M with c M ( i ) / c M ( i − being simple A ¯ s -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 ¯ s -epimorphism from the simple A ¯ s -module c M ( k ) / c M ( k − to M . Now theorem followsfrom Corollary 3.5.
4. Main results
In this section, we will classify all simple weight s -modules with finite dimensionalweight spaces. First of all, from the representation theory of Virasoro algebra, we knowthat C acts trivially on any simple cuspidal s -module, and hence the category of simplecuspidal s -modules is naturally equivalent to the category of simple cuspidal ¯ s -modules.Thus, it remains to classify all all simple weight s -modules with finite dimensional weightspaces which is not cuspidal. From now on, we will assume M is such an s -module. Let λ ∈ supp( M ).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.
Suppose M is a simple weight s -module with finite dimensional weightspaces which is not cuspidal, then M is a highest (or lowest) weight module.Proof. Since M is not cuspidal, then there is a k ∈ Z such that dim M − k + λ > M λ +dim M λ +1 ). Without lost of generality, we may assume that k ∈ N . Then there exists anonzero element w ∈ M − k + λ such that L k w = L k +1 w = G k w = G k +1 w = 0. Therefore, L i w = G i w = 0 for all i ≥ k , since [ s i , s j ] = s i + j .It is easy to see that M ′ = { v ∈ M | dim s + v < ∞} is a nonzero submodule of M ,here s + = P n ∈ N ( C L n + C G n ). Hence M = M ′ . So, Lemma 4.1 tells us that supp( M ) isupper bounded, that is M is a highest weight module.Combining with Lemma 4.2 and Theorem 3.9, we can get the following result, whichwas given in [19] by much complicated calculations. Theorem 4.3.
Let V be a simple s -module with finite dimensional weight spaces. Then V is a highest weight module, a lowest weight module, or a simple quotient of Γ( λ, b ) forsome λ, b ∈ C (which is called a module of the intermediate series). Acknowledgement:
Y. Cai is partially supported by NSF of China (Grant 11801390).D. Liu is partially supported by NSF of China (Grant 1197131511871249). R. L¨u ispartially supported by NSF of China (Grant 11471233, 11771122, 11971440).10 eferences [1] Y. Billig, V. Futorny.
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