A class of super Heisenberg-Virasoro algebras
aa r X i v : . [ m a t h . R T ] S e p A class of super Heisenberg-Virasoro algebras
Haibo Chen ∗ , Xiansheng Dai and Yanyong Hong Abstract:
In this paper, a class of super Heisenberg-Virasoro algebras is introduced on the baseof conformal modules of Lie conformal superalgebras. Then we construct a class of simple su-per Heisenberg-Virasoro modules, which is induced from simple modules of the finite-dimensionalsolvable Lie superalgebras. These modules are isomorphic to simple restricted super Heisenberg-Virasoro modules, and include the highest weight modules, Whittaker modules and high orderWhittaker modules.
Key words: super Heisenberg-Virasoro algebra, Lie conformal superalgebra, simple module.
Mathematics Subject Classification (2010):
Throughout the present paper, we denote by C , C ∗ , Z , Z + and N the sets of complex numbers,nonzero complex numbers, integers, nonnegative integers and positive integers, respectively.All vector superspaces (resp. superalgebras, supermodules) and spaces (resp. algebras,modules) are considered to be over C .The concept of Lie conformal (super)algebras was introduced by Kac, which encodes anaxiomatic description of the operator product expansion (or rather its Fourier transform)of chiral fields in conformal field theory (see [18, 20]). The theory of Lie conformal (su-per)algebras gives us a powerful tool for the study of infinite-dimensional Lie (super)algebrassatisfying the locality property (see [19]).As an important infinite-dimensional Lie algebra, the twisted Heisenberg-Virasoro algebra H is the universal central extension of the Lie algebra H := (cid:26) f ( t ) ddt + g ( t ) (cid:12)(cid:12)(cid:12) f ( t ) , g ( t ) ∈ C [ t, t − ] (cid:27) of differential operators of order at most one, which was studied by Arbarello et al. in [1].They established a connection between the second cohomology of certain moduli spaces ofcurves and the second cohomology of the Lie algebra of differential operators of order atmost one.The highest weight modules and Whittaker modules are two classes of important modulesin representation theory. In [1], the authors precisely determined the determinant formula ofthe Shapovalov form for the Verma modules, and showed any simple highest weight modulefor H is isomorphic to the tensor product of a simple module for the Virasoro algebra anda simple module for the infinite-dimensional Heisenberg algebra. In [5], Billig obtainedthe character formula for simple highest weight modules under some conditions of trivial ∗ Corresponding author: H. Chen ([email protected]). H were completely classified in [26], which turn out to be modules ofintermediate series or highest (lowest) weight modules. A simpler and more conceptual proofof the classification of irreducible Harish-Chandra modules over H were presented in [21].The Whittaker modules were first introduced for sl (2) by Arnal and Pinczon in [4]. Mean-while, the Whittaker modules of finite-dimensional complex semisimple Lie algebras weredefined in [17]. From then on, these non-weight modules have been studied subsequentlyin a variety of different settings, especially for affine Lie algebra and Lie superalgebra (see,e.g. [2, 6–8, 12, 23, 24]). Moreover, Whittaker modules have been investigated in the frame-work of vertex operator algebra theory (see [2, 3, 16, 32]). Whittaker modules for H wereinvestigated in [25]. In [9], Chen and Guo constructed a large class of simple modules for thetwisted Heisenberg-Virasoro algebra. In particular, the known simple modules such as thehighest weight modules and Whittaker modules were included. Furthermore, the high orderWhittaker modules of the twisted Heisenberg-Virasoro algebra were clearly defined in [28]by using generalized oscillator representations.Recently, the representation theory of super Virasoro algebra was widely researched (seee.g. [13, 24, 34]), which inspires us to study some super cases of twisted Heisenberg-Virasoroalgebra. Now we present the definition of super Heisenberg-Virasoro algebra . It is an infinite-dimensional Lie superalgebra S = M m ∈ Z C L m ⊕ M m ∈ Z C G m + ⊕ M m ∈ Z C I m ⊕ C C, which satisfies the following super-brackets[ L m , L n ] = ( n − m ) L m + n + m − m δ m + n, C, [ L m , I n ] = nI m + n , [ L m , G r ] = rG m + r , [ G r , G s ] = 2 I r + s , [ I m , I n ] = [ I m , G r ] = 0for m, n ∈ Z , r, s ∈ Z + . By definition, we have the following decomposition: S = S ¯0 ⊕ S ¯1 , where S ¯0 = span C { L m , I m , C | m ∈ Z } , S ¯1 = span C { G r | r ∈ Z + } . Notice that theeven part S ¯0 is isomorphic to twisted Heisenberg-Virasoro algebra with some trivial centerelements. The super Heisenberg-Virasoro algebra has a Z -grading by the eigenvalues of theadjoint action of L . Then S possesses the following triangular decomposition: S = S + ⊕ S ⊕ S − , S + = span C { L m , I m , G r | m ∈ N , r ∈ Z + + } , S − = span C { L m , I m , G r | − m ∈ N , − r ∈ Z + + } , S = span C { L , I , C } .The actions of elements in the positive part of the algebra are locally finite, which isthe same property of highest weight modules and Whittaker modules. Mazorchuk and Zhaoin [31] proposed a very general construction of simple Virasoro modules, which generalizesand includes both highest weight modules and various versions of Whittaker modules. Thisconstruction enabled them to classify all simple Virasoro modules that are locally finite over apositive part. Motivated by this, new simple modules over the Virasoro algebra and its some(super)extended cases have been investigated (see [9–11, 24, 27, 29]). The high order Whit-taker modules were studied in [22] by generalized some known results on Whittaker modulesof the Virasoro algebra. They obtained concrete bases for all irreducible Whittaker modules(instead of a quotient of modules). Observe that simple highest weight modules, simpleWhittaker modules and simple high order Whittaker modules over the super Heisenberg-Virasoro algebra are restricted modules. This makes us to study restricted modules for thesuper Heisenberg-Virasoro algebra.The rest of this paper is organized as follows.In Section 2, we make some preparations for later use, which are associated with the Lieconformal superalgebra and Lie superalgebra.In Section 3, a class of superalgebras of twisted Heisenberg-Virasoro is introduced by thepoint of view of Lie conformal superalgebra.In Section 4, a class of simple restricted S -modules is constructed, which generalizes andincludes the highest weight modules, Whittaker modules and high order Whittaker modules.In Section 5, we give a characterization of simple restricted modules for super Heisenberg-Virasoro algebra. More precisely, it reduces the problem of classification of simple restricted S -modules to classification of simple modules over a class of finite-dimensional solvable Liesuperalgebras.At last, we show some examples of restricted S -modules, such as the highest weightmodules, Whittaker modules and high order Whittaker modules. In this section, we will make some preparations for later use.
Some definitions and results related to Lie conformal superalgebras and conformal modulesare recalled (see [15, 18, 20]).A Lie conformal superalgebra is called finite if it is finitely generated as a C [ ∂ ]-module,or else it is called infinite . 3 efinition 2.1. (1) A Lie conformal superalgebra S = S ¯0 ⊕ S ¯1 is a Z -graded C [ ∂ ] -moduleendowed with a λ -bracket [ a λ b ] which defines a linear map S α ⊗ S β → C [ λ ] ⊗ S α + β , where α, β ∈ Z and λ is an indeterminate, and satisfy the following axioms: [ ∂a λ b ] = − λ [ a λ b ] , [ a λ ∂b ] = ( ∂ + λ )[ a λ b ] , [ a λ b ] = − ( − | a || b | [ b − λ − ∂ a ] , [ a λ [ b µ c ]] = [[ a λ b ] λ + µ c ] + ( − | a || b | [ b µ [ a λ c ]] for a ∈ S α , b ∈ S β , c ∈ S γ and α, β, γ ∈ Z . (2) A conformal module M = M ¯0 ⊕ M ¯1 over a Lie conformal superalgebra S is a Z -graded C [ ∂ ] -module endowed with a λ -action S α ⊗ M β → C [ λ ] ⊗ M α + β such that ( ∂a ) λ v = − λa λ v, a λ ( ∂v ) = ( ∂ + λ ) a λ v,a λ ( b µ v ) − ( − | a || b | b µ ( a λ v ) = [ a λ b ] λ + µ v for all a ∈ S α , b ∈ S β , v ∈ M γ and α, β, γ ∈ Z . In particular, there is an important Lie superalgebra related to a Lie conformal super-algebra. Let S be a Lie conformal superalgebra. Assume that Lie( S ) is the quotient of thevector space with basis a n ( a ∈ S, n ∈ Z ) by the subspace spanned over C by elements:( αa ) n − αa n , ( a + b ) n − a n − b n , ( ∂a ) n + na n − , where a, b ∈ S, α ∈ C , n ∈ Z . The operation on Lie( S ) is given as follows:[ a m , b n ] = X j ∈ N (cid:18) mj (cid:19) ( a ( j ) b ) m + n − j , (2.1)where a ( j ) b is called the j -th product , given by [ a λ b ] = P ∞ n =0 λ n n ! ( a ( n ) b ). Then Lie( S ) is a Liesuperalgebra and it is called a formal distribution Lie superalgebra of S (see [18]). Let V = V ¯0 ⊕ V ¯1 be a Z -graded vector space. Then any element v ∈ V ¯0 is said to be evenand any element v ∈ V ¯1 is said to be odd. Define | v | = 0 if v is even and | v | = 1 if v is odd.Elements in V ¯0 or V ¯1 are called homogeneous. Throughout the present paper, all elementsin superalgebras and modules are homogenous unless specified.Assume that G is a Lie superalgebra. A G -module is a Z -graded vector space V togetherwith a bilinear map G × V → V , denoted ( x, v ) xv such that x ( yv ) − ( − | x || y | y ( xv ) = [ x, y ] v G ¯ i V ¯ j ⊆ V ¯ i +¯ j for all x, y ∈ G , v ∈ V . Thus there is a parity-change functor Π on the category of G -modulesto itself. In other words, for any module V = V ¯0 ⊕ V ¯1 , we have a new module Π( V ) withthe same underlining space with the parity exchanged, i.e., Π( V ¯0 ) = V ¯1 and Π( V ¯1 ) = V ¯0 .We use U ( G ) to denote the universal enveloping algebra. All modules for Lie superalgebrasconsidered in this paper are Z -graded and all irreducible modules are non-trivial. Definition 2.2.
Let V be a module of a Lie superalgebra G and x ∈ G .(1) If for any v ∈ V there exists n ∈ Z + such that x n v = 0, then we call that the actionof x on V is locally nilpotent . Similarly, the action of G on V is locally nilpotent if for any v ∈ V there exists n ∈ Z + such that G n v = 0.(2) If for any v ∈ V we have dim( P n ∈ Z + C x n v ) < + ∞ , then we call that the action of x on V is locally finite . Similarly, the action of G on V is locally finite if for any v ∈ V wehave dim( P n ∈ Z + G n v ) < + ∞ .We note that the action of x on V is locally nilpotent implies that the action of x on V is locally finite. If G is a finitely generated Lie superalgebra, then we obtain that the actionof G on V is locally nilpotent implies that the action of G on V is locally finite. S In this section, we construct a class of super Heisenberg-Virasoro algebras in the point ofview of Lie conformal superalgebra.The Heisenberg-Virasoro Lie conformal algebra v is a free C [ ∂ ]-module generated by L and I satisfying [ L λ L ] = ( ∂ + 2 λ ) L, [ L λ I ] = ( ∂ + λ ) I, [ I λ I ] = 0 . It is well known that all free non-trivial modules of Heisenberg-Virasoro Lie conformalalgebra of rank one over C [ ∂ ] are as follows (see [33]): V ( a, b, c ) = C [ ∂ ] v, L λ v = ( ∂ + aλ + b ) v, I λ v = cv, where a, b, c ∈ C . The module V ( a, b, c ) is irreducible if and only if ( a, c ) = (0 , Z -graded C [ ∂ ]-module S ( φ, ϕ, a, b, c ) = S ¯0 ⊕ S ¯1 with S ¯0 = C [ ∂ ] L ⊕ C [ ∂ ] I, S ¯1 = C [ ∂ ] G and satisfying[ L λ G ] = ( ∂ + aλ + b ) G, [ I λ G ] = cG, [ G λ G ] = φ ( ∂, λ ) L + ϕ ( ∂, λ ) I, (3.1)where φ ( ∂, λ ) , ϕ ( ∂, λ ) ∈ C [ ∂, λ ] without φ ( ∂, λ ) = ϕ ( ∂, λ ) = 0.5 emma 3.1. Let a, b, c ∈ C . Then the Z -graded C [ ∂ ] -module S ( φ, ϕ, a, b, c ) becomes a Lieconformal superalgebra if and only if a = 1 , φ ( ∂, λ ) = b = c = 0 , ϕ ( ∂, λ ) = ∆ ∈ C ∗ .Proof. First we prove the necessity. Assume that S ( φ, ϕ, a, b, c ) is a Lie conformal superal-gebra. Using the Jacobi identity for triple ( I, G, G ), we have c ( φ ( ∂, λ + µ ) + φ ( ∂, µ )) = 0 , (3.2) λφ ( ∂ + λ, µ ) = c ( ϕ ( ∂, λ + µ ) + ϕ ( ∂, µ )) . (3.3)First consider c = 0. Letting λ = 0 in (3.2) and (3.3), we have φ ( ∂, µ ) = ϕ ( ∂, µ ) = 0. Nowwe consider c = 0, by (3.3), one has φ ( ∂, λ ) = 0. Then the third relation of (3.1) can berewritten as [ G λ G ] = ϕ ( ∂, λ ) I . By the Jacobi identity for triple ( L, G, G ), we get( ∂ + λ ) ϕ ( ∂ + λ, µ ) = ( − µ + ( a − λ + b ) ϕ ( ∂, λ + µ ) + ( ∂ + λa + µ + b ) ϕ ( ∂, µ ) . (3.4)If b = 0, putting λ = 0 into (3.4), we get ϕ ( ∂, µ ) = 0. This contradicts the hypothesis.Consider b = 0. Setting ∂ = 0 in (3.4), one can see that µ ϕ (0 , µ + λ ) − ϕ (0 , µ ) λ = aϕ (0 , µ ) + ( a − ϕ (0 , λ + µ ) − ϕ ( λ, µ ) . (3.5)Choosing λ →
0, we check that µ ddµ ϕ (0 , µ ) = 2( a − ϕ (0 , µ ). If 2( a − / ∈ Z + , one has ϕ (0 , µ ) = 0. Inserting this into (3.5), we get ϕ ( ∂, λ ) = 0. Then consider 2( a − ∈ Z + , wehave ϕ (0 , µ ) = ∆ µ a − for ∆ ∈ C ∗ . Putting this into (3.5), we obtain ϕ ( λ, µ ) = a ∆ µ a − + ( a − λ + µ ) a − − µ ∆ ( λ + µ ) a − − µ a − λ Letting ∂ = − λ in (3.4), it is easy to get(( a − λ − µ ) ϕ ( − λ, λ + µ ) + (( a − λ + µ ) ϕ ( − λ, µ ) = 0 . (3.6)Taking µ = ( a − λ in (3.6), we obtain ( a − λϕ ( − λ, ( a − λ ) = 0, which forces a = 1.Therefore, ϕ ( ∂, λ ) = ∆ ∈ C ∗ .In summary, by the definition of Lie conformal superalgebra, we get a = 1, b = c = 0, φ ( ∂, λ ) = 0 and ϕ ( ∂, λ ) = ∆ ∈ C ∗ .Based on Definition 2.1, the sufficiency is clear.Up to isomorphism, we may assume that ∆ = 2 in Lemma 3.1. We can define a classof finite Lie conformal superalgebras s = s ¯0 ⊕ s ¯1 with s ¯0 = C [ ∂ ] L ⊕ C [ ∂ ] I, s ¯1 = C [ ∂ ] G andsatisfying the following non-trivial λ -brackets[ L λ L ] = ( ∂ + 2 λ ) L, [ L λ I ] = ( ∂ + λ ) I, [ L λ G ] = ( ∂ + λ ) G, [ G λ G ] = 2 I. In the rest of this section, a class of infinite-dimensional Lie superalgebras associatedwith twisted Heisenberg-Virasoro algebras is presented.6 emma 3.2.
A formal distribution Lie superalgebra of s is given by Lie( s ) = n L m , I n , G r | m, n ∈ Z , r ∈ Z + 12 o with non-vanishing relations: [ L m , L n ] = ( n − m ) L m + n , [ L m , I n ] = nI m + n , [ L m , G r ] = rG m + r , [ G r , G s ] = 2 I r + s , where m, n ∈ Z , r, s ∈ Z + .Proof. By the definition of the j -th product in (2.1), we conclude that L (0) L = ∂L, L (1) L = 2 L, L (0) I = ∂I, L (1) I = I,L (0) G = ∂G, L (1) G = G, G (0) G = 2 I,L ( i ) L = L ( i ) I = L ( i ) G = G ( i − G = I ( i − I = G ( i − I = 0 , ∀ i ≥ . It is easy to check that[ L ( m ) , L ( n ) ] = ( m − n ) L ( m + n − , [ L ( m ) , I ( n ) ] = nI ( m + n − , [ L ( m ) , G ( n ) ] = nG ( m + n − , [ G ( m ) , G ( n ) ] = 2 I ( m + n ) . (3.7)Then the lemma is proved by making the shift L m → L ( m +1) , I m → I ( m ) , G m := G ( m + ) → G ( m ) in (3.7) for m ∈ Z .Note that the central extensions of the formal distribution Lie superalgebra Lie( s ) isexactly the super Heisenberg-Virasoro algebra S . S -modules In this section, we will study a class of simple restricted modules of super Heisenberg-Virasoroalgebra, which includes the highest weight modules and Whittaker modules, and so on.
We denote by M the set of all infinite vectors of the form i := ( . . . , i , i ) with entries in Z + , satisfying the condition that the number of nonzero entries is finite and c M = { i ∈ M | i k = 0 , , ∀ k ∈ Z + } . Write = ( . . . , , ∈ M (or c M ). For i ∈ Z + , denote7 i = ( . . . , , , , . . . , ∈ M (or c M ), where 1 is in the i ’th position from the right. For any i ∈ M , j ∈ c M , we write w ( i ) = X s ∈ N s · i s and w ( j ) = X s ∈ N ( s −
12 ) · j s , which are both non-negative integers. For any = i ∈ M (or c M ), assume that p is thesmallest integer such that i p = 0, and define i ′ = i − ǫ p . Definition 4.1.
Denote by ≻ the reverse lexicographical total order on M (or c M ), definedas follows: for any i , j ∈ M (or c M ) j ≻ i ⇔ there exists r ∈ Z + such that ( j s = i s , ∀ ≤ s < r ) and j r > i r . Then induce a principal total order on M × c M × M , still denoted by ≻ : ( i , j , k ) ≻ ( l , m , n )if and only if one of the following conditions is satisfied:(i) ( k , w ( k )) ≻ ( n , w ( n ));(ii) k = n and ( j , w ( j )) ≻ ( m , w ( m ));(iii) k = n , j = m and ( i , w ( i )) ≻ ( l , w ( l )) for all i , k , l , n ∈ M , j , m ∈ c M . Set B = M i ≥ C L i ⊕ M i ≥ C G i − ⊕ M i ≥ C I i ⊕ C C. Letting V be a simple B -module, we get the induced S -moduleInd( V ) = U ( S ) ⊗ U ( B ) V. Considering simple modules of the superalgebra S or one of its subalgebras containing thecentral elements I and C , we always assume that the actions of I and C are scalars c and c , respectively.For i , k ∈ M , j ∈ c M , we denote I i G j L k = · · · I i − I i − · · · G j − G j − · · · L k − L k − ∈ U ( S ) . From the PBW Theorem (see [14]) and G i + = I i +1 for i ∈ Z , every element of Ind( V ) canbe uniquely written as follow X i , k ∈ M , j ∈ c M I i G j L k v i , j , k , (4.1)where all v i , j , k ∈ V and only finitely many of them are nonzero. For any v ∈ Ind( V ) as in(4.1), we denote by supp( v ) the set of all ( i , j , k ) ∈ M × c M × M such that v i , j , k = 0. For a8onzero v ∈ Ind( V ), we write deg( v ) the maximal element in supp( v ) by the principal totalorder on M × c M × M , which is called the degree of v . Note that deg( v ) is defined only for v = 0. Definition 4.2. An S -module W is called restricted in the sense that for any w ∈ W , L i w = G i − w = I i w = 0 for i sufficiently large. The purpose of this subsection is to obtain the simple restricted S -modules. Theorem 4.3.
Let V be a simple B -module. Assume that there exists z ∈ Z + satisfying thefollowing two conditions: (i) I z on V is injective, (ii) L i V = I j V = 0 for all i > z and j > z ,then we get (1) G k − V = 0 for all k > z ; (2) Ind( V ) is a simple S -module.Proof. (1) Take any k ≥ z . From G k + V = I k +1 V = 0, we check that Q = G k + V is aproper subspace of V . For t ∈ Z + , we get I t Q = I t G k + V = G k + I t V ⊂ Q, G t + Q = G t + G k + V = − G k + G t + V + I k + t +1 V ⊂ Q. For t > , k ≥ z , one get G k + t + V = 2( L k + t G V − G L k + t V ) = 0. Then we have L t Q = L t G k + V = G k + L t V + ( k + ) G k + t + V ⊂ Q, which shows Q is a proper submoduleof V . By the simplicity of V , we obtain Q = G k + V = 0 for k ≥ z .(2) To prove this, we first give the following claim. Claim 1.
For any v ∈ Ind( V ) \ V , let deg( v ) = ( i , j , k ) for i , k ∈ M , j ∈ c M , ˜ i = min { s : i s = 0 } if i = , ˜ j = min { s : j s = 0 } if j = and ˜ k = min { s : k s = 0 } if k = . Then wehave (a) if k = , then ˜ k > and deg( I ˜ k + z v ) = ( i , j , k ′ ) ; (b) if k = , j = , then ˜ j > and deg( G ˜ j + z − v ) = ( i , j ′ , ) ; (c) if j = k = , i = , then ˜ i > and deg( L ˜ i + z v ) = ( i ′ , , ) . We assume that v is of the form in (4.1), i.e., v = X x , z ∈ M , y ∈ c M I x G y L z v x , y , z . v x , y , z with I ˜ k + z I x G y L z v x , y , z = 0 . It is easy to see that I ˜ k + z I x G y L z v x , y , z = I x G y [ I ˜ k + z , L z ] v x , y , z . Clearly, I z v x , y , z = 0. Consider the following two cases.If z = k , one can getdeg( I ˜ k + z I x G y L z v x , y , z ) = ( x , y , k ′ ) (cid:22) ( i , j , k ′ ) , where the equality holds if and only if x = i , y = j .Suppose ( z , w ( z )) ≺ ( k , w ( k )) and denotedeg( I ˜ k + z I x G y L z v x , y , z ) = ( x , y , z ) ∈ M × c M × M . If w ( z ) < w ( k ), then we obtain w ( z ) ≤ w ( z ) − ˜ k < w ( k ) − ˜ k = w ( k ′ ), which implies( x , y , z ) ≺ ( i , j , k ′ ). Then suppose w ( z ) = w ( k ) and z ≺ k . Let ˜ z := min { s : z s = 0 } > z > ˜ k , one can see that w ( z ) = w ( z ) − ˜ z < w ( z ) − ˜ k = w ( k ′ ). If ˜ z = ˜ k , by thesimilar method, we deduce that ( x , y , z ) = ( x , y , z ′ ). Then by z ′ ≺ k ′ , one can see thatdeg( I ˜ k + z I x G y L z v x , y , z ) = ( x , y , z ) ≺ ( i , j , k ′ ) in both cases.Combining all the arguments above we conclude that deg( I ˜ k + z v ) = ( i , j , k ′ ).(b) Now we consider v x , y , with G ˜ j + z − I x G y v x , y , = 0 . According to I z v x , y , = 0 for any( x , y , ) ∈ supp( v ), we check that G ˜ j + z − I x G y v x , y , = I x [ G ˜ j + z − , G y ] v x , y , . If y = j , one can get thatdeg( G ˜ j + z − I x G y v x , y , ) = ( x , y ′ , ) (cid:22) ( i , j ′ , ) , where the equality holds if and only if x = i .Suppose ( y , w ( y )) ≺ ( j , w ( j )). Then we havedeg( G ˜ j + z − I x G y v x , y , ) = ( x , y , ) ∈ M × c M × M . If w ( y ) < w ( j ), then we have w ( y ) ≤ w ( y ) − ˜ j < w ( j ) − ˜ j = w ( j ′ ), which gives ( x , y , ) ≺ ( i , j ′ , ).Then we suppose w ( y ) = w ( j ) and y ≺ j . Let ˜ y := min { s : y s = 0 } >
0. If ˜ y > ˜ j ,we obtain w ( y ) = w ( y ) − ˜ y < w ( j ) − ˜ j = w ( j ′ ). If ˜ y = ˜ j , we can similarly check that( x , y , ) = ( x , y ′ , ). Now from y ′ ≺ j ′ , we have deg( G ˜ j + z − I x G y v x , y , ) = ( x , y , ) ≺ ( i , j ′ , ). Therefore, we conclude that deg( G ˜ j + z − v ) = ( i , j ′ , ).10c) Consider v x , , with L ˜ i + z I x v x , , = 0 . By the similar arguments in (b), we have theresults. Claim 1 holds.Using the Claim repeatedly, from any 0 = v ∈ Ind( V ) we can reach a nonzero elementin U ( S ) v ∩ V = 0, which gives the simplicity of Ind( V ). This completes the proof oftheorem. Remark 4.4.
We note that the induced module Ind( V ) is a simple restricted S -module. S -modules For x ∈ Z + , we denote S ( x ) = M i>x C L i ⊕ M i>x C G i − ⊕ M i>x C I i . Note that S (0) = S + . Proposition 5.1.
Let P be a simple S -module. Then the following conditions are equivalent: (1) There exists z ∈ Z + such that the actions of L i , G i − , I i , i ≥ z on P are locally finite. (2) There exists z ∈ Z + such that the actions of L i , G i − , I i , i ≥ z on P are locally nilpotent. (3) There exists z ∈ Z + such that P is a locally finite S ( z ) -module. (4) There exists z ∈ Z + such that P is a locally nilpotent S ( z ) -module. (5) There exists a simple B -module V satisfying the conditions in Theorem 4.3 such that P ∼ = Ind( V ) .Proof. We first prove (1) ⇒ (5). Suppose that P is a simple S -module. Then there exists z ∈ Z + such that the actions of L i , G i − and I i for all i ≥ z on P are locally finite. Takea simple H -submodule P ′ of P . It is clear that L i , I i , i ≥ z are locally finite on P ′ . Write H + = L m> L m ⊕ L m> I m . By Theorem 6 of [9], we know that there exist z ′ ∈ Z + ( z ′ ≥ z )and a simple H + -module S such that P ′ ∼ = Ind( S ) as twisted Heisenberg-Virasoro modulesand L m S = I m S = 0 for all m ≥ z ′ . Then we can choose 0 = v ∈ S such that L m v = I m v = 0for all m ≥ z ′ .Choose j ∈ Z + with j > z ′ and we denote N G = X m ∈ Z + C L mz ′ G j − v = U ( C L z ′ ) G j − v, which are all finite-dimensional. From the definition of S and any m ∈ Z + , we check that G j + mz ′ − v ∈ N G ⇒ G j +( m +1) z ′ − v ∈ N G , j > z ′ . According to induction on m ∈ Z + , we have G j + mz ′ − v ∈ N G . As a matter offact, P m ∈ Z + C G j + mz ′ − v are finite-dimensional for j > z ′ . Hence, X i ∈ Z + C G z ′ + i + v = C G z ′ + v + z ′ X j = z ′ +1 (cid:0) X m ∈ Z + C G j + mz ′ + v (cid:1) , are all finite-dimensional. Now we can choose l ∈ Z + such that X i ∈ Z + C G z ′ + i + v = l X i =0 C G z ′ + i + v. Then write V ′ = P x ,...,x l ∈{ , } C G x z ′ + · · · G x l z ′ + l + v . It is easy to see that V ′ is a (finite-dimensional) S ( z ′ ) -module.It follows that we can take a minimal n ∈ Z + such that( L m + a L m +1 + · · · + a n L m + n ) V ′ = 0 (5.1)for some m ≥ z ′ and a i ∈ C , i = 1 , . . . , n . Applying L m to (5.1), one get( a [ L m , L m +1 ] + · · · + a n [ L m , L m + n ]) V ′ = 0 , which shows n = 0, that is, L m V ′ = 0. Therefore, for m ≥ z ′ , we have0 = G i − L m V ′ = [ G i − , L m ] V ′ + L m ( G i − V ′ ) = − ( i −
12 ) G m + i − V ′ , namely, G m + i − V ′ = 0 for all i > z ′ . Similarly, we conclude that I m + i V ′ = L m + i V ′ = 0 forall i > z ′ .For any α ∈ Z , we consider the vector space N α = { v ∈ P | L i v = G i − v = I i v = 0 for any i > α } . Clearly, N α = 0 for sufficiently large α ∈ Z + . Thus we can find a smallest nonnegativeinteger β such that V := N β = 0. Using r > β and s ≥
1, it follows from r + s − > β and r + s − > β that we can easily check that L r ( G s − v ) = ( s −
12 ) G r + s − v = 0 ,G r − ( G s − v ) = 2 I r + s − v = 0 , I r ( G s − v ) = 0 . Then G s − v ∈ V for all s ≥
1. Similarly, we can also obtain I e v ∈ V and L e v ∈ V for all e ∈ Z + . Therefore, V is a B -module. 12f z ≥
1, by the definition of V , we get that the action of I z on V is injective. Since P issimple and generated by V , there exists a canonical surjective map π : Ind( V ) → P, π (1 ⊗ v ) = v, ∀ v ∈ V. Next, we only need to prove that π is also injective, i.e., π as the canonical map is bijective.Denote K = ker( π ). Obviously, K ∩ V = 0. If K = 0, we can choose 0 = v ∈ K \ V such that deg( v ) = ( i , j , k ) is minimal possible. Note that K is an S -submodule of Ind( V ).According to Claim 1 in the proof of Theorem 4.3, we can create a new vector u ∈ K withdeg( u ) ≺ ( i , j , k ), which leads to a contradiction. This forces K = 0, i.e., P ∼ = Ind( V ). Thenwe see that V is a simple B -module.If z = 0, by the assumption c = 0 and similar arguments as above, we can deduce thesame result (also see Theorem 1 (d) of [31]).Moreover, (5) ⇒ (3) ⇒ (1), (5) ⇒ (4) ⇒ (2) and (2) ⇒ (1) are clear. The propositionholds. Lemma 5.2.
Let S be a simple S -module. Then S is a restricted S -module if and only ifthere exists t ∈ Z + such that the actions of L i , G i − , I i , i ≥ t on S are locally nilpotent.Proof. We first prove the “if” part. Suppose 0 = s ∈ S , there exists t ∈ Z + such that L i s = G i − s = I i s = 0 for all i ≥ t . Every element b s in S can be uniquely written as follow b s = X i , k ∈ M , j ∈ c M I i G j L k s Thus, for i ≥ t , there exists k ∈ Z + sufficiently large such that L ki S = G ki − S = I ki S = 0 . Now we prove the “only if” part. By Proposition 5.1, we obtain that there exists a simple B -module V satisfying the conditions in Theorem 4.3 such that S ∼ = Ind( V ). Then for any s ∈ S , by (4.1), we can write s = X i , k ∈ M , j ∈ c M I i G j L k v i , j , k , where v i , j , k ∈ V . Denote σ = max { w ( i ) + w ( j ) + w ( k ) | i , k ∈ M , j ∈ c M } . Choosing i ≫ σ ,we check that L i s = 0. By the similar method, we have G i − s = I i s = 0 for i sufficientlylarge, completing the proof.From Proposition 5.1 and Lemma 5.2, one can immediately obtain the following results. Theorem 5.3.
Let W be a simple restricted S -module. Then W ∼ = Ind( V ) , where V is asimple B z -module, and B z = B / S ( z ) is a quotient algebra for some z ∈ Z + . Examples
For i ∈ Z + , we denote T ( i ) = M n ≥ i C L n ⊕ M n ≥ i C G n + ⊕ M n ≥ i C I n . Note that T (0) = S ⊕ S + . In the following, we present some examples of S -module Ind( V ) . For h, c , c ∈ C , let V h be a 1-dimensional vector space over C spanned by v h , namely, V h = C v h . Regard V h as a S -module such that L v = hv, I v = c v, Cv = c v . Then V h is a T (0) -module by setting S + · V h = 0. The Verma module V ( h, c , c ) over superHeisenberg-Virasoro algebra can be defined by V ( h, c , c ) = U ( S ) ⊗ U ( S ⊕ S + ) V h , and has the highest weight ( h, c , c ). It is straightforward to check that V ( h, c , c ) is asimple S -module if c = 0. These simple modules is exactly the highest weight modules inTheorem 4.3. We first recall the definition of the classical Whittaker modules.
Definition 6.1.
Assume that φ : T (1) → C is a Lie superalgebra homomorphism. For c , c ∈ C , an S -module W is called a Whittaker module of type ( φ, c , c ) if (1) W is generated by a homogeneous vector u , (2) xu = φ ( x ) u for any x ∈ T (1) , (2) I u = c u , Cu = c u ,where u is called a Whittaker vector of W . Lemma 6.2.
Let V be the finite-dimensional simple T (1) -module. Then dim( V ) = 1 .Proof. Regarding V as a finite-dimensional Vir + -module, there exists m ≫ L m V = 0, where Vir + = L i ≥ C L i . For n ∈ N and m ≫
0, it is easy to get0 = [ L m , I n ] V = nI m + n V and 0 = [ L m , G n + ] V = ( n + 12 ) G m + n + V. Then for sufficiently large m ∈ Z + , we know that V can be viewed as a module of quotientalgebra T (1) /T ( m ) . By Lemma 1.33 of [14], we immediately obtain dim( V ) = 1.14 emark 6.3. If T (1) contains G , the Lemma 1.33 of [14] in above proof will be declaredinvalid. Let φ be a Lie superalgebra homomorphism φ : T (1) → C . Then φ ( L i ) = φ ( I j ) = φ ( G j + ) = 0 for i ≥ , j ≥
2. Let C v be a 1-dimensional T (1) -module with xv = φ ( x ) v forall x ∈ T (1) and I v = c v, Cv = c v for some c , c ∈ C . The Whittaker module W ( φ, c , c )is given by W ( φ, c , c ) = U ( S ) ⊗ U ( T (1) ) C v. We note that W ( φ, c , c ) is simple if φ ( I ) = 0 and the Whittaker vector spanned by { v, G v } over C .Assume that φ : T (1) → C is a non-trivial Lie superalgebra homomorphism. Let t φ = C v ⊕ C w be a 2-dimensional vector space with xv = φ ( x ) v, G v = w, I v = c v, Cv = c v for all x ∈ T (1) . Clearly, if φ ( I ) = 0, t φ is a simple S + -module and dim( t φ ) = 1 + 1. Nowwe consider the induced module V φ = U ( T (0) ) ⊗ U ( T (1) ) t φ . It is easy to see that V φ is a simple T (0) -module if φ ( I ) = 0. Then the simple induced S -module Ind( V φ ) in Theorem 4.3 is so-called Whittaker module. Now we show the example of high order Whittaker modules of the super Heisenberg-Virasoroalgebra. This is a generalization of Whittaker modules.For i ∈ Z + , we denote P ( i ) = M n ≥ i C L n ⊕ M n ≥ i C G n − ⊕ M n ≥ i C I n . Let φ k be a Lie superalgebra homomorphism φ k : T ( k ) → C for some k ∈ Z + . Then φ k ( L i ) = φ k ( I j ) = φ k ( G j + ) = 0 for i ≥ k + 1 , j ≥ k . Let C v be a 1-dimensional T ( k ) -module with xv = φ k ( x ) v for all x ∈ T ( k ) and I v = c v, Cv = c v for some c , c ∈ C . Thehigher order Whittaker module W ( φ k , c , c ) is defined by W ( φ k , c , c ) = U ( S ) ⊗ U ( T ( k ) ) C v. It is easy to get that W ( φ k , c , c ) is simple if φ k ( I k − ) = 0 and the Whittaker vector spannedby { v, G k − v } over C . 15et φ k be a Lie superalgebra homomorphism φ k : T ( k ) → C for some k ∈ Z + . Assumethat t φ k = C v ⊕ C w is a 2-dimensional vector space with xv = φ ( x ) v, G k − v = w, I v = c v, Cv = c v for all x ∈ T ( k ) . If φ ( I k − ) = 0, t φ k is a simple T ( k ) -module and dim( t φ k ) = 1 + 1. Considerthe induced module V φ k = U ( P ( k ) ) ⊗ U ( T ( k ) ) t φ . Clearly, V φ k is a simple T ( k ) -module if φ ( I k − ) = 0 and dim( V φ k ) = 1 + 1. The correspondingsimple S -module Ind( V φ k ) in Theorem 4.3 is just the high order Whittaker module. Acknowledgements
This work was supported by the National Natural Science Foundation of China (No.11801369,11871421), the China Scholarship Council (No.202008310011), the Zhejiang Provincial Nat-ural Science Foundation of China (No.LY20A010022) and the Scientific Research Foundationof Hangzhou Normal University (No.2019QDL012).
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