Irreducible tensor product modules over the affine-Virasoro algebra of type A_1
aa r X i v : . [ m a t h . R T ] F e b IRREDUCIBLE TENSOR PRODUCT MODULES OVER THEAFFINE-VIRASORO ALGEBRA OF TYPE A QIU-FAN CHEN, YU-FENG YAO
Abstract.
In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra of type A by taking tensor products of irreducibles defined in [7] withirreducible highest weight modules. The irreducibility and the isomorphism classes of thesemodules are determined. Moreover, we show that these tensor product modules are differ-ent from the known non-weight modules. Finally, we realize some tensor product modulesas induced modules from modules over certain subalgebras of the affine-Virasoro algebraof type A , and give sufficient and necessary conditions for these induced modules to bereducible. Introduction
Throughout the paper, we denote by C , Z , C ∗ , Z + , N the sets of complex numbers, inte-gers, nonzero complex numbers, nonnegative integers and positive integers, respectively. Allalgebras (modules, vector spaces) are assumed to be over C . For a Lie algebra g , we use U ( g ) to denote the universal enveloping algebra of g . More generally, for a subset X of g ,we use U ( X ) to denote the universal enveloping algebra of the subalgebra of g generated by X .It is well known that representation theory of the Virasoro algebra and affine Kac MoodyLie algebras plays an important role both in physics and in mathematics. The Virasoro alge-bra acts on any (except when the level is negative the dual coxeter number) highest weightmodule of the affine Lie algebra through the use of the famous Sugawara operators. Theaffine Lie algebras admit representations on the Fock space and hence admit representations Mathematics Subject Classification.
Key words and phrases. affine-Virasoro algebra, tensor product, non-weight module, highest weightmodule.This work is supported by National Natural Science Foundation of China (Grant Nos. 11801363, 11771279,12071136 and 11671247). of the Virasoro algebra. This close relationship strongly suggests that they should be con-sidered simultaneously, i.e., as one algebraic structure, and hence has led to the definitionof the so-called affine-Virasoro algebra [14, 15]. Highest weight representations and inte-grable representations of the affine-Virasoro algebras have been extensively studied (cf. [2],[9], [13]-[18], [25]). Quite recently, the authors gave the classification of irreducible quasi-finite modules over the affine-Virasoro algebras in [17]. The affine-Virasoro algebras are verymeaningful in the sense that they are closely connected to the conformal field theory. Forexample, the even part of the N = 3 superconformal algebra [8] is just the affine-Virasoroalgebra of type A . The affine-Virasoro algebra of type A , denoted by L , is defined as theLie algebra with C -basis { e i , f i , h i , d i , C | i ∈ Z } subject to the following Lie brackets:[ e i , f j ] = h i + j + iδ i + j, C, [ h i , e j ] = 2 e i + j , [ h i , f j ] = − f i + j , [ d i , d j ] = ( j − i ) d i + j + δ i + j, i − i C, [ d i , h j ] = jh i + j , [ h i , h j ] = − iδ i + j, C, [ d i , e j ] = je i + j , [ d i , f j ] = jf i + j , [ e i , e j ] = [ f i , f j ] = [ C, L ] = 0 . It is clear that h := C d + C h + C C is the Cartan subalgebra of L . Moreover, L admits atriangular decomposition: L = L − ⊕ h ⊕ L + , where L − = span C { e − i , f − i , h − i , d − i , f | i ∈ N } and L + = span C { e i , f i , h i , d i , e | i ∈ N } . The classification of all irreducible Harish-Chandra modules over L was achieved in [10].In recent years, many authors constructed various irreducible non-Harish-Chandra mod-ules and irreducible non-weight modules (cf. [1], [3, 4], [12], [19]-[24]). In particular, J.Nilsson [21] constructed a class of sl n + -modules that are free of rank one when restricted tothe Cartan subalgebra. Since then, this kind of non-weight modules, which many authorscall U ( h )-free modules, have been extensively studied. Especially, the authors classified the RREDUCIBLE TENSOR PRODUCT MODULES 3 U ( h )-free modules of rank one for L in [7]. Moreover, the irreducibility and isomorphismclasses of these modules were determined therein.It is well known that an important way to construct new modules over an algebra is toconsider the linear tensor product of two known modules over the algebra (cf. [5, 6], [11],[23], [26]). The purpose of the present paper is to construct new irreducible non-weight L -modules by taking tensor product of irreducible modules defined in [7] with irreduciblehighest weight modules.The present paper is organized as follows. In Section 2, we recall some known modulesand results from [7]. Section 3 is devoted to studying the irreducibility of the tensor prod-uct modules M ( λ, α, β, γ ) ⊗ V ( θ, ǫ, η ), where M ( λ, α, β, γ ) = Ω( λ, α, β, γ ) , ∆( λ, α, β, γ ) orΘ( λ, α, β, γ ) (2 β / ∈ Z + ) (see § L , and give sufficient andnecessary conditions for these induced modules to be reducible.2. Preliminaries
Let us first recall the definitions of the L -modules Ω( λ, α, β, γ ) , ∆( λ, α, β, γ ) , Θ( λ, α, β, γ )and V ( η, ǫ, θ ) concerned in this paper, and some basic properties of them. Denote by C [ s, t ] the polynomial algebra in variables s and t with coefficients in C . As vector spaces,Ω( λ, α, β, γ ) , ∆( λ, α, β, γ ) and Θ( λ, α, β, γ ) coincide with C [ s, t ]. Definition 2.1.
For λ, α ∈ C ∗ , β, γ ∈ C , i ∈ Z and g ( s, t ) ∈ C [ s, t ] , define the L -moduleaction on C [ s, t ] as follows: Ω( λ, α, β, γ ) : e i · g ( s, t ) = λ i αg ( s − i, t − ,f i · g ( s, t ) = − λ i α ( t − β )( t β + 1) g ( s − i, t + 2) ,h i · g ( s, t ) = λ i tg ( s − i, t ) , d i · g ( s, t ) = λ i ( s + iγ ) g ( s − i, t ) ,C · g ( s, t ) = 0;∆( λ, α, β, γ ) : e i · g ( s, t ) = − λ i α ( t β )( t − β − g ( s − i, t − ,f i · g ( s, t ) = λ i αg ( s − i, t + 2) , h i · g ( s, t ) = λ i tg ( s − i, t ) , QIU-FAN CHEN, YU-FENG YAO d i · g ( s, t ) = λ i ( s + iγ ) g ( s − i, t ) ,C · g ( s, t ) = 0;Θ( λ, α, β, γ ) : e i · g ( s, t ) = λ i α ( t β ) g ( s − i, t − ,f i · g ( s, t ) = − λ i α ( t − β ) g ( s − i, t + 2) ,h i · g ( s, t ) = λ i tg ( s − i, t ) , d i · g ( s, t ) = λ i ( s + iγ ) g ( s − i, t ) ,C · g ( s, t ) = 0 . It is worthwhile to point out that C [ s, t ] in each case has the same module structure overthe subalgebra span C { h i , d i , C | i ∈ Z } . For later use, we need the following known resulton conditions for irreducibility and a classification of isomorphism classes for the modulesconstructed above. Proposition 2.2. (cf. [7] ) Keep notations as above, then the following statements hold. (1) Ω( λ, α, β, γ ) and ∆( λ, α, β, γ ) are irreducible for any λ, α ∈ C ∗ and β, γ ∈ C ; While Θ( λ, α, β, γ ) is irreducible if and only if β / ∈ Z + . (2) Let λ , λ , α , α ∈ C ∗ , β , β , γ , γ ∈ C . Then Ω( λ , α , β , γ ) , ∆( λ , α , β , γ ) , Θ( λ , α , β , γ ) are pairwise non-isomorphic for all parameter choices. Moreover, Ω( λ , α , β , γ ) ∼ = Ω( λ , α , β , γ ) ⇐⇒ ( λ , α , β , γ ) = ( λ , α , β , γ )or ( λ , α , β , γ ) = ( λ , α , − β − , γ );∆( λ , α , β , γ ) ∼ = ∆( λ , α , β , γ ) ⇐⇒ ( λ , α , β , γ ) = ( λ , α , β , γ )or ( λ , α , β , γ ) = ( λ , α , − β − , γ );Θ( λ , α , β , γ ) ∼ = Θ( λ , α , β , γ ) ⇐⇒ ( λ , α , β , γ ) = ( λ , α , β , γ ) . For any η, ǫ, θ ∈ C , let I ( η, ǫ, θ ) be the left ideal of U ( L ) generated by the followingelements { e , e i , f i , h i , d i | i ∈ N } ∪ { d − η, h − ǫ, C − θ } . The Verma L -module with highest weight ( η, ǫ, θ ) is defined as the quotient module V ( η, ǫ, θ ) = U ( L ) /I ( η, ǫ, θ ) . RREDUCIBLE TENSOR PRODUCT MODULES 5
By the PBW theorem, V ( η, ǫ, θ ) has a basis consisting of all vectors of the form f F − q − q · · · f F e E − p − p · · · e E − − h H − m − m · · · h H − − d D − n − n · · · d D − − · v h , where v h is the coset of 1 in V ( η, ǫ, θ ), and D − , · · · , D − n , H − , · · · , H − m , E − , · · · , E − p , F , · · · , F − q ∈ Z + . Then we have the irreducible highest weight module V ( η, ǫ, θ ) = V ( η, ǫ, θ ) /J , where J is theunique maximal proper submodule of V ( η, ǫ, θ ). Readers can refer to [2, 14] for the structureof V ( η, ǫ, θ ).In the rest of this paper, we will always assume λ, α ∈ C ∗ , β, γ, η, ǫ, θ ∈ C , M ( λ, α, β, γ ) =Ω( λ, α, β, γ ), ∆( λ, α, β, γ ) or Θ( λ, α, β, γ ) constructed in Definition 2.1, and V ( η, ǫ, θ ) isan irreducible highest weight L -module. Take a tensor product of the non-weight module M ( λ, α, β, γ ) with irreducible highest weight module V ( η, ǫ, θ ). Clearly, the tensor product L -module M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) is not a weight module.3. Irreducibility of the tensor product modules
In this section, we show the irreducibility of the tensor product L -module M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ), where 2 β / ∈ Z + when M = Θ. Theorem 3.1.
The tensor product module M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) is irreducible providedthat β / ∈ Z + when M = Θ .Proof. Let W be a nonzero submodule of M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ). We need to show that W = M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ). It is clear that, for any v ∈ V ( η, ǫ, θ ), there exists a positiveinteger K ( v ) such that d m · v = e m · v = f m · v = h m · v = 0 for all m ≥ K ( v ). Take anynonzero element w = P ri =0 a i ( t ) s i ⊗ v i ∈ W with a i ( t ) ∈ C [ t ] , v i ∈ V ( η, ǫ, θ ) , a r ( t ) = 0 , v r = 0and r ∈ Z + is minimal. Claim 1. r = 0 . Let K = max { K ( v i ) | i = 0 , , · · · , r } . Then we have(3.1) λ − m d m · w = r X i =0 ( s + mγ ) a i ( t )( s − m ) i ⊗ v i ∈ W, ∀ m ≥ K. Case (i): γ = 0. QIU-FAN CHEN, YU-FENG YAO
In this case, we write the right hand side of (3.1) as r +1 X i =0 m i w i ∈ W, ∀ m ≥ K, where all w i ∈ M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) are independent of m . Taking m = K, K +1 , · · · , K + r + 1, we see that the coefficient matrix of w i is a Vandermonde matrix. So each w i ∈ W .Especially, w r +1 = ( − r γa r ( t ) ⊗ v r ∈ W . Thus, r must be zero by its minimality.Case (ii): γ = 0.Similar arguments as in case (i) yield that w r = sa r ( t ) ⊗ v r ∈ W . Moreover, when M ( λ, α, β, γ ) = Ω( λ, α, β, γ ), we have α − ( λ − K e K − λ − K − e K +1 ) · w r = ( − r a r ( t − ⊗ v r ∈ W. When M ( λ, α, β, γ ) = ∆( λ, α, β, γ ) or Θ( λ, α, β, γ ), it follows from similar computation that a r ( t + 2) ⊗ v r ∈ W , ( t + β ) a r ( t − ⊗ v r ∈ W , respectively. In conclusion, r must be zeroin each situation by its minimality. Claim 2. W = M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) . By Claim 1, a ( t ) ⊗ v ∈ W . Fix this v and let P = { a ( s, t ) ∈ C [ s, t ] | a ( s, t ) ⊗ v ∈ W } . For any a ( t ) ∈ C [ t ] , k ∈ N and m ≥ K ( v ), by induction on k , we have the following twoformulae. λ − mk h km · (cid:0) a ( t ) ⊗ v (cid:1) = t k a ( t ) ⊗ v , (3.2) λ − mk d km · (cid:0) a ( t ) ⊗ v (cid:1) = k − Y i =0 ( s + mγ − mi ) a ( t ) ⊗ v . (3.3)We write W as W Ω , W ∆ and W Θ to emphasis that W is an L -submodule of Ω( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) , ∆( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) and Θ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ), respectively. By Definition2.1, one can inductively show that for m ≥ K ( v ) λ − mk α − k e km · (cid:0) a ( t ) ⊗ v (cid:1) = a ( t − k ) ⊗ v ∈ W Ω ,λ − mk α − k f km · (cid:0) a ( t ) ⊗ v (cid:1) = a ( t + 2 k ) ⊗ v ∈ W ∆ ,λ − mk α − k e km · (cid:0) a ( t ) ⊗ v (cid:1) = k − Y n =0 ( t β − n ) a ( t − k ) ⊗ v ∈ W Θ , RREDUCIBLE TENSOR PRODUCT MODULES 7 λ − mk ( − α ) − k f km · (cid:0) a ( t ) ⊗ v (cid:1) = k − Y n =0 ( t − β + n ) a ( t + 2 k ) ⊗ v ∈ W Θ . Note that we can choose k large enough so that (cid:0) a ( t ) , a ( t − k ) (cid:1) = 1 , (cid:0) a ( t ) , a ( t + 2 k ) (cid:1) = 1and (cid:0) k − Y n =0 ( t β − n ) a ( t − k ) , k − Y n =0 ( t − β + n ) a ( t + 2 k ) (cid:1) = 1 , if 2 β / ∈ Z + . It follows from these and (3.2) that1 ⊗ v ∈ W Ω , ⊗ v ∈ W ∆ and 1 ⊗ v ∈ W Θ , which in turn force C [ t ] ⊆ P in each case. Since (3.3) implies that P is stable underthe multiplication by s , it follows that P = C [ s, t ] = M ( λ, α, β, γ ). Hence, M ( λ, α, β, γ ) ⊗ U ( L ) v ⊆ W . Consequently, W = M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) by the irreducibility of V ( η, ǫ, θ ).We complete the proof. (cid:3) Isomorphism classes of the tensor product modules
In this section, we give the classification of the isomorphism classes of the tensor product L -modules M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ), where 2 β / ∈ Z + when M = Θ. Theorem 4.1.
Let λ i , α i ∈ C ∗ , β i , γ i , θ i , ǫ i , η i ∈ C , where i = 1 , , and V ( η , ǫ , θ ) , V ( η , ǫ , θ ) are irreducible highest weight modules. Let M ( λ, α, β, γ ) = Ω( λ, α, β, γ ) , ∆( λ, α, β, γ ) or Θ( λ, α, β, γ ) (2 β / ∈ Z + ) constructed in Definition 2.1. Then M ( λ , α , β , γ ) ⊗ V ( η , ǫ , θ ) and M ( λ , α , β , γ ) ⊗ V ( η , ǫ , θ ) are isomorphic as L -modules if and only if M ( λ , α , β , γ ) ∼ = M ( λ , α , β , γ ) and V ( η , ǫ , θ ) ∼ = V ( η , ǫ , θ ) . Proof.
We only tackle the case M ( λ, α, β, γ ) = Ω( λ, α, β, γ ), since the other two cases can betreated similarly. The sufficiency is obvious and it suffices to show the necessity. Let φ be an L -module isomorphism from Ω( λ , α , β , γ ) ⊗ V ( η , ǫ , θ ) to Ω( λ , α , β , γ ) ⊗ V ( η , ǫ , θ ).Take a nonzero element v ∈ V ( η , ǫ , θ ). Suppose φ (1 ⊗ v ) = n X i =0 a i ( t ) s i ⊗ w i , where a i ( t ) ∈ C [ t ] , w i ∈ V ( η , ǫ , θ ) with a n ( t ) = 0 , w n = 0. There exists a positive integer K = max { K ( v ) , K ( w i ) | i = 0 , · · · , n } such that d m · v = d m · w i = e m · v = e m · w i = f m · v = QIU-FAN CHEN, YU-FENG YAO f m · w i = h m · v = h m · w i = 0 for all m ≥ K and 0 ≤ i ≤ n . For any two different integers m , m ≥ K , we have( λ − m d m − λ − m d m ) · (1 ⊗ v ) = ( m − m ) γ (1 ⊗ v ) . Then by applying φ on both sides, we obtain( m − m ) γ n X i =0 a i ( t ) s i ⊗ w i = ( λ − m d m − λ − m d m ) · n X i =0 a i ( t ) s i ⊗ w i = n X i =0 ( λ λ ) m ( s + m γ )( s − m ) i a i ( t ) ⊗ w i − n X i =0 ( λ λ ) m ( s + m γ )( s − m ) i a i ( t ) ⊗ w i . (4.1)Comparing the coefficients of s n +1 in the above formula, we deduce that (cid:0) ( λ λ ) m − ( λ λ ) m (cid:1) ( a n ( t ) ⊗ w n ) = 0 , forcing λ = λ . Then (4.1) can be simplified as( m − m ) γ n X i =0 a i ( t ) s i ⊗ w i = n X i =0 (cid:0) ( s + m γ )( s − m ) i − ( s + m γ )( s − m ) i (cid:1) a i ( t ) ⊗ w i . (4.2)Regard it as a polynomial in m , m with coefficients in Ω( λ , α , β , γ ) ⊗ V ( η , ǫ , θ ). If γ = 0, it follows from (4.2) that n = 0, since otherwise the coefficient ( − n γ a n ( t ) ⊗ w n of m n +11 would be zero, yielding a contradiction. Then it follows from (4.2) that γ = γ = 0.If γ = 0, (4.2) becomes( m − m ) γ n X i =0 a i ( t ) s i ⊗ w i = s n X i =0 (cid:0) ( s − m ) i − ( s − m ) i (cid:1) a i ( t ) ⊗ w i , Also, by regarding it as a polynomial in m , m with coefficients in Ω( λ , α , β , γ ) ⊗ V ( η , ǫ , θ ), we see that n ≤
1. If n = 1, then the above formula becomes( m − m ) γ (cid:0) a ( t ) ⊗ w + a ( t ) s ⊗ w (cid:1) = ( m − m ) a ( t ) s ⊗ w , RREDUCIBLE TENSOR PRODUCT MODULES 9 yielding γ = − a ( t ) ⊗ w = 0. So, φ (1 ⊗ v ) = a ( t ) s ⊗ w . Since(4.3) λ − m α − e m · (1 ⊗ v ) = 1 ⊗ v for any m ≥ K, by applying φ on both sides of (4.3), it follows that a ( t ) s ⊗ w = λ − m α − e m · (cid:0) a ( t ) s ⊗ w (cid:1) = α − α a ( t − s − m ) ⊗ w , forcing α − α a ( t − ⊗ w = 0, a contradiction with the assumption that a ( t ) s ⊗ w = 0.Therefore, n = 0 and γ = 0. The preceding discussion shows that φ (1 ⊗ v ) = a ( t ) ⊗ w and γ = γ . Now applying φ to (4.3) gives α − α a ( t − ⊗ w = a ( t ) ⊗ w , which yields a ( t ) ∈ C ∗ and α = α . Without loss of generality, we assume that a ( t ) = 1.Denote λ = λ = λ , α = α = α and γ = γ = γ in what follows. Thus there exists alinear injection τ : V ( η , ǫ , θ ) → V ( η , ǫ , θ ) such that(4.4) φ (1 ⊗ v ) = 1 ⊗ τ ( v ) , ∀ v ∈ V ( η , ǫ , θ ) . For any m ≥ K , the equations φ ( d m · (1 ⊗ v )) = d m · φ (1 ⊗ v ) ,φ ( h m · (1 ⊗ v )) = h m · φ (1 ⊗ v ) ,φ ( f m · (1 ⊗ v )) = f m · φ (1 ⊗ v )are respectively equivalent to λ m φ ( s ⊗ v ) + λ m mγ (1 ⊗ τ ( v )) = λ m ( s ⊗ τ ( v )) + λ m mγ (1 ⊗ τ ( v )) ,λ m φ ( t ⊗ v ) = λ m ( t ⊗ τ ( v )) , − λ m α φ (cid:0) ( t + t − β ( β + 1)) ⊗ v (cid:1) = − λ m α ( t + t − β ( β + 1)) ⊗ τ ( v ) , we see that φ ( s ⊗ v ) = s ⊗ τ ( v ) , (4.5) φ ( t ⊗ v ) = t ⊗ τ ( v ) , (4.6) φ (cid:0) ( t + t − β ( β + 1)) ⊗ v (cid:1) = ( t + t − β ( β + 1)) ⊗ τ ( v ) . (4.7) From (4.6) and φ ( h m · ( t ⊗ v )) = h m · φ ( t ⊗ v ) , where m ≥ K, we deduce that(4.8) φ ( t ⊗ v ) = t ⊗ τ ( v ) . This along with (4.6)-(4.7) gives β = β or β = − β −
1. Hence, Ω( λ , α , β , γ ) ∼ =Ω( λ , α , β , γ ) by Proposition 2.2. Combining (4.4)-(4.6) with (4.8), we obtain φ (( X m · ⊗ v ) = ( X m · ⊗ τ ( v ) , where X m ∈ { d m , h m , e m , f m | ∀ m ∈ Z } . This together with φ ( X m · (1 ⊗ v )) = X m · φ (1 ⊗ v ) , where X m ∈ { d m , h m , e m , f m | ∀ m ∈ Z } gives φ (1 ⊗ ( X m · v )) = 1 ⊗ ( X m · τ ( v )) , where X m ∈ { d m , h m , e m , f m | ∀ m ∈ Z } . Therefore, τ ( X m · v ) = X m · τ ( v ) , where X m ∈ { d m , h m , e m , f m | ∀ m ∈ Z } , v ∈ V ( θ , ǫ , η ) . Since φ ( C · (1 ⊗ v )) = C · φ (1 ⊗ v ) , ∀ v ∈ V ( θ , ǫ , η ) , we see that τ ( C · v ) = C · τ ( v ). Thus, τ is a nonzero L -module homomorphism. Note that V ( η , ǫ , θ )) and V ( η , ǫ , θ ) are simple L -modules, τ is an L -module isomorphism. Wecomplete the proof. (cid:3) Comparison of tensor product modules with known non-weight modules
In this section, we compare the tensor product modules constructed in the present pa-per with all other known non-weight L -modules, i.e., U ( h )-free modules of rank one andWhittaker modules (cf. [7]).Let µ = ( µ , · · · , µ ) ∈ C . Assume that J µ is the left ideal of U ( L + ) generated by { d − µ , d − µ , e − µ , f − µ , C − µ , d j , e k , f l , h m | j ≥ , k ≥ , l ≥ , m ≥ } . Denote N µ := U ( L + ) /J . Then Ind( N µ ) := U ( L ) ⊗ U ( L + ) N µ is a universal Whittaker module, andany Whittaker module is a quotient of some universal Whittaker module. RREDUCIBLE TENSOR PRODUCT MODULES 11
For any r ∈ Z + , l, m ∈ Z , as in [16], we denote ω ( r ) l,m = r X i =0 (cid:18) ri (cid:19) ( − r − i d l − m − i d m + i ∈ U ( L ) . Lemma 5.1.
Let M ( λ, α, β, γ ) and V ( η, ǫ, θ ) be the L -modules defined as in §
2, and η, ǫ, θ can not be identically zero. Then the following statements hold. (1) d i acts injectively on M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) for any i ∈ Z . (2) For any g ( s, t ) ∈ M ( λ, α, β, γ ) , we have ω ( r ) l,m ( g ( s, t )) = 0 , ∀ l, m, r ∈ Z , r > . (3) For any r > and = g ( s, t ) ∈ M ( λ, α, β, γ ) , there exists v ∈ V ( η, ǫ, θ ) and l, m ∈ Z such that ω ( r ) l,m ( g ( s, t ) ⊗ v ) = 0 . Proof. (1) For any 0 = v = m X j =0 n j X k =0 s j t k ⊗ v jk ∈ M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ )with v jk ∈ V ( η, ǫ, θ ) for any j, k , and v mn m = 0, it follows from the L -module structure on M ( λ, α, β, γ ) in Definition 2.1 that d i · v = n m X k =0 λ i s m +1 t k ⊗ v mk + m X j =0 X k s j t k ⊗ w jk = 0 , where w jk ∈ V ( η, ǫ, θ ) for any j, k . Hence, (1) follows.(2) For any h ( t ) ∈ C [ t ] and j ∈ Z + , we have ω ( r ) l,m ( s j h ( t )) = r X i =0 (cid:18) ri (cid:19) ( − r − i d l − m − i d m + i ( s j h ( t ))= r X i =0 (cid:18) ri (cid:19) ( − r − i d l − m − i (cid:0) λ m + i ( s + ( m + i ) γ )( s − m − i ) j h ( t ) (cid:1) = r X i =0 (cid:18) ri (cid:19) ( − r − i λ l (cid:0) s + ( l − m − i ) γ (cid:1)(cid:0) s + ( m + i ) γ − l + m + i (cid:1) ( s − l ) j h ( t ) . This together with the following identity r X i =0 (cid:18) ri (cid:19) ( − r − i i j = 0 , ∀ j, r ∈ Z + with j < r forces ω ( r ) l,m ( g ( s, t )) = 0 provided r >
2, proving (2).(3) Fix any r >
2. Take v to be the highest weight vector of V ( η, ǫ, θ ). It is important toobserve that the vectors d − v, d − v, · · · , d − r − v are linearly independent in V ( η, ǫ, θ ). Take l = r + 1 and m = − r −
2. As ω ( r ) l,m ( g ( s, t )) = 0 by (ii), we have ω ( r ) l,m ( g ( s, t ) ⊗ v ) = r X i =0 (cid:18) ri (cid:19) ( − r − i d l − m − i d m + i ( g ( s, t ) ⊗ v )= r X i =0 (cid:18) ri (cid:19) ( − r − i d l − m − i ( g ( s, t )) ⊗ d m + i ( v )= r X i =0 (cid:18) ri (cid:19) ( − r − i λ l − m − i (cid:0) s + ( l − m − i ) γ (cid:1) g ( s − l + m + i, t ) ⊗ d m + i ( v ) = 0 . Hence (3) follows. (cid:3)
As a consequence of Lemma 5.1, we have the following result which asserts that the tensorproduct modules constructed in the present paper are different from the other known non-weight modules.
Proposition 5.2.
Let λ, α ∈ C ∗ , β, γ, η, ǫ, θ ∈ C , and η, ǫ, θ can not be identically zero. Thenthe tensor product module M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) is a new non-weight L -module.Proof. We need to show that the tensor product module M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) is neitherisomorphic to a Whittaker module nor isomorphic to a U ( h )-free modules of rank one in [7].For that, let W be a Whittaker module, then W is isomorphic to a quotient of N µ for some µ = ( µ , · · · , µ ) ∈ C . It follows from Lemma 5.1 (i) that M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) ≇ W ,since for any nonzero element v ∈ Ind( N µ ) (resp. w ∈ W ), there exists a positive integer i such that d i acts on v (resp. w ) trivially. Moreover, thanks to Lemma 5.1 (ii) and (iii), M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) ≇ M ( λ ′ , α ′ , β ′ , γ ′ ) for any λ ′ , α ′ ∈ C ∗ , β ′ , γ ′ ∈ C . We complete theproof. (cid:3) RREDUCIBLE TENSOR PRODUCT MODULES 13 Realization of tensor product modules as induced modules
In this section, we realize the tensor product modules M ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) as certaininduced modules. For that, fix λ ∈ C ∗ , let b λ = span C { d m − λ m d , f m , h n , e n , C | m ∈ N , n ∈ Z + } be the subalgebra of L . Definition 6.1.
Let C [ t ] be the polynomial algebra in the variable t with coefficients in C .For λ, α ∈ C ∗ , β, γ, η, ǫ, θ ∈ C and g ( t ) ∈ C [ t ], define the action of b λ on C [ t ] as follows: C [ t ] Ω λ,α,β,γ,η,ǫ,θ : ( d m − λ m d ) ◦ g ( t ) = λ m ( mγ − η ) g ( t ) ,f m ◦ g ( t ) = − λ m α ( t − β )( t β + 1) g ( t + 2) ,h n ◦ g ( t ) = λ n ( t + δ n, ǫ ) g ( t ) , e n ◦ g ( t ) = λ n αg ( t − ,C ◦ g ( t ) = θg ( t ); C [ t ] ∆ λ,α,β,γ,η,ǫ,θ : ( d m − λ m d ) ◦ g ( t ) = λ m ( mγ − η ) g ( t ) ,f m ◦ g ( t ) = λ m αg ( t + 2) ,h n ◦ g ( t ) = λ n ( t + δ n, ǫ ) g ( t ) ,e n ◦ g ( t ) = − λ n α ( t β )( t − β − g ( t − ,C ◦ g ( t ) = θg ( t ); C [ t ] Θ λ,α,β,γ,η,ǫ,θ : ( d m − λ m d ) ◦ g ( t ) = λ m ( mγ − η ) g ( t ) ,f m ◦ g ( t ) = − λ m α ( t − β ) g ( t + 2) ,h n ◦ g ( t ) = λ n ( t + δ n, ǫ ) g ( t ) , e n ◦ g ( t ) = λ n α ( t β ) γg ( t − ,C ◦ g ( t ) = θg ( t ) . where m ∈ N , n ∈ Z + . Proposition 6.2.
Keep notations as above, then C [ t ] Ω λ,α,β,γ,η,ǫ,θ , C [ t ] ∆ λ,α,β,γ,η,ǫ,θ and C [ t ] Θ λ,α,β,γ,η,ǫ,θ are b λ -modules under the actions given in Definition 6.1.Proof. The assertion can be verified straightforwardly, we omit the details. (cid:3)
Remark 6.3.
Let λ, α ∈ C ∗ , β, γ, η, ǫ, θ ∈ C . Then by a similar argument as that in theproof in [7, Proposition 2.5 ] , the b λ -modules C [ t ] Ω λ,α,β,γ,η,ǫ,θ and C [ t ] ∆ λ,α,β,γ,η,ǫ,θ are alwaysirreducible, while the b λ -module C [ t ] Θ λ,α,β,γ,η,ǫ,θ is irreducible if and only if β / ∈ Z + . We always assume that λ, α ∈ C ∗ , β, γ, η, ǫ, θ ∈ C in the following. Now one can form theinduced L -modules as follows:Ind( C [ t ] Ω λ,α,β,γ,η,ǫ,θ ) := U ( L ) ⊗ U ( b λ ) C [ t ] Ω λ,α,β,γ,η,ǫ,θ ;Ind( C [ t ] ∆ λ,α,β,γ,η,ǫ,θ ) := U ( L ) ⊗ U ( b λ ) C [ t ] ∆ λ,α,β,γ,η,ǫ,θ ;Ind( C [ t ] Θ λ,α,β,γ,η,ǫ,θ ) := U ( L ) ⊗ U ( b λ ) C [ t ] Θ λ,α,β,γ,η,ǫ,θ . Theorem 6.4.
Keep notations as above. Then as L -modules Ω( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) ∼ = Ind ( C [ t ] Ω λ,α,β,γ,η,ǫ,θ );∆( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) ∼ = Ind ( C [ t ] ∆ λ,α,β,γ,η,ǫ,θ );Θ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) ∼ = Ind ( C [ t ] Θ λ,α,β,γ,η,ǫ,θ ) . Proof.
In the following we only prove the first case, since a similar argument can be appliedto the other two cases.According to the PBW Theorem, we see that Ind( C [ t ] Ω λ,α,β,γ,η,ǫ,θ ) has a basis B = { f F − q − q · · · f F e E − p − p · · · e E − − h H − m − m · · · h H − − d D − n − n · · · d D ⊗ t i | D , · · · , D − n ,H − , · · · , H − m , E − , · · · , E − p , F , · · · , F − q , i ∈ Z + } and Ω( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) has a basis B = { t i s D ⊗ ( f F − q − q · · · f F e E − p − p · · · e E − − h H − m − m · · · h H − − d D − n − n · · · d D − − · v h ) | D , · · · , D − n ,H − , · · · , H − m , E − , · · · , E − p , F , · · · , F − q , i ∈ Z + } . Now we define the following linear map φ : Ind( C [ t ] Ω λ,α,β,γ,η,ǫ,θ ) → Ω( λ, α, β, γ ) ⊗ V ( η, ǫ, θ )by φ ( f F − q − q · · · f F e E − p − p · · · e E − − h H − m − m · · · h H − − d D − n − n · · · d D ⊗ t i )= f F − q − q · · · f F e E − p − p · · · e E − − h H − m − m · · · h H − − d D − n − n · · · d D ( t i ⊗ v h ) RREDUCIBLE TENSOR PRODUCT MODULES 15
We claim that φ is an L -module homomorphism. By Definition 6.1, one can observe that( d m − λ m d )( t i ⊗ v h ) = (( d m − λ m d ) ◦ t i ) ⊗ v h , ∀ m ∈ N . Combining this with Definition 6.1, for any i ∈ Z + , m ∈ N , n ∈ Z + and x = f F − q − q · · · f F e E − p − p · · · e E − − h H − m − m · · · h H − − d D − n − n · · · d D , we have φ ( xd m ⊗ t i ) = φ ( λ m xd ⊗ t i + x ( d m − λ m d ) ⊗ t i )= φ ( λ m xd ⊗ t i + x ⊗ ( d m − λ m d ) ◦ t i )= λ m xd ( t i ⊗ v h ) + x (( d m − λ m d ) ◦ t i ⊗ v h ) , = λ m xd ( t i ⊗ v h ) + x ( d m − λ m d )( t i ⊗ v h ) , = xd m ( t i ⊗ v h ) ,φ ( xf m ⊗ t i ) = φ ( x ⊗ f m ◦ t i ) = x ( f m ◦ t i ⊗ v h ) = xf m ( t i ⊗ v h ) ,φ ( xY n ⊗ t i ) = φ ( x ⊗ Y n ◦ t i )= x ( Y n ◦ t i ⊗ v h ) = xY n ( t i ⊗ v h ) , where Y n = e n or h n and φ ( xC ⊗ t i ) = θφ ( x ⊗ t i ) = θx ( t i ⊗ v h ) = xC ( t i ⊗ v h ) . Then for any j ∈ Z , by the PBW Theorem, we can write z j x = X j a X j a x j a + X j b Y j b y j b d j b + X j c Z j c z j c h j c + X j p U j p u j p e j p + X j q V j q v j q f j q + X j l W j l w j l C, where z j ∈ { d j , h j , e j , f j | j ∈ Z } , X j a , Y j b , Z j c , U j p , V j q , W j l ∈ C , x j a ⊗ t i , y j b ⊗ t i , z j c ⊗ t i , u j p ⊗ t i , v j q ⊗ t i , w j l ⊗ t i ∈ B and j b , j q ∈ N , j c , j p ∈ Z + . Now we deduce that for any j ∈ Z , φ ( z j x ⊗ t i ) = φ (cid:16) X j a X j a x j a ⊗ t i + X j b Y j b y j b d j b ⊗ t i + X j c Z j c z j c h j c ⊗ t i + X j p U j p u j p e j p ⊗ t i + X j q V j q v j q f j q ⊗ t i + X j l W j l w j l C ⊗ t i (cid:17) = X j a X j a x j a ( t i ⊗ v h ) + X j b Y j b y j b d j b ( t i ⊗ v h )+ X j c Z j c z j c h j c ( t i ⊗ v h ) + X j p U j p u j p e j p ( t i ⊗ v h )+ X j q V j q v j q f j q ( t i ⊗ v h ) + X j l W j l w j l C ( t i ⊗ v h )= z j x ( t i ⊗ v h ) = z j φ ( x ⊗ t i )and φ ( Cx ⊗ t i ) = θφ ( x ⊗ t i ) = θx ( t i ⊗ v h ) = xC ( t i ⊗ v h ) = Cx ( t i ⊗ v h ) = Cφ ( x ⊗ t i ) . This implies that φ is an L -module homomorphism.Next we shall show that φ is an L -module isomorphism. Clearly, t i ⊗ v h ∈ Im( φ ) forall i ∈ Z + . Moreover, d j ( t i ⊗ v h ) ∈ Im( φ ), this implies that t i s j ⊗ v h ∈ Im( φ ) for all j ∈ Z + , i.e., Ω( λ, α, β, γ ) ⊗ v h ⊂ Im( φ ). By applying the action of the elements of the form f F − q − q · · · f F e E − p − p · · · e E − − h H − m − m · · · h H − − d D − n − n · · · d D on Ω( λ, α, β, γ ) ⊗ v h , we further deduce thatΩ( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ) ⊂ Im( φ ) and φ is surjective. In order to obtain the injectivity of φ ,we first define a total order “ ≺ ” on B t i s D ⊗ ( f F − q − q · · · f F e E − p − p · · · e E − − h H − m − m · · · h H − − d D − n − n · · · d D − − · v h ) ≺ t i ′ s D ′ ⊗ ( f F ′− q − q · · · f F ′ e E ′− p − p · · · e E ′− − h H ′− m − m · · · h H ′− − d D ′− n − n · · · d D ′− − · v h )if and only if( D − , . . . , D − n , n z }| { , . . . , , H − , . . . , H − m , m z }| { , . . . , , E − , . . . , E − p , p z }| { , . . . , ,F , . . . , F − q , q z }| { , . . . , , D , i ) < ( D ′− , . . . , D ′− n , n z }| { , . . . , , H ′− , . . . , H ′− m , m z }| { , . . . , , E ′− , . . . , E ′− p , p z }| { , . . . , ,F ′ , . . . , F ′− q , q z }| { , . . . , , D ′ , i ′ ) , where( a , . . . , a l ) < ( b , . . . , b l ) ⇐⇒ ∃ k > a i = b i for all i < k and a k < b k . RREDUCIBLE TENSOR PRODUCT MODULES 17
An explicit calculation gives f F − q − q · · · f F e E − p − p · · · e E − − h H − m − m · · · h H − − d D − n − n · · · d D ( t i ⊗ v h )= t i s D ⊗ ( f F − q − q · · · f F e E − p − p · · · e E − − h H − m − m · · · h H − − d D − n − n · · · d D − − · v h ) + lower terms.Since B is a basis of Ω( λ, α, β, γ ) ⊗ V ( η, ǫ, θ ), so is the set { f F − q − q · · · f F e E − p − p · · · e E − − h H − m − m · · · h H − − d D − n − n · · · d D ( t i ⊗ v h ) | D , · · · , D − n ,H − , · · · , H − m , E − , · · · , E − p , F , · · · , F − q , i ∈ Z + } . Thus, φ is injective, completing the proof. (cid:3) As a consequence of Theorem 3.1, Theorem 6.4, Proposition 2.2 and [14, Corollary 1], wehave the following sufficient and necessary conditions for reducibility of the induced modules
Corollary 6.5.
Let λ, α ∈ C ∗ , β, γ, η, ǫ, θ ∈ C . Then the following statements hold. (1) The induced module
Ind( C [ t ] Mλ,α,β,γ,η,ǫ,θ ) for M ∈ { Ω , ∆ } is reducible if and only ifone of the following conditions holds. (i) ǫ + m ( θ + 2) − n = 0 for some m, n ∈ Z + . (ii) − ǫ + m ( θ + 2) − n − for some m, n ∈ N . (iii) θ = − and there exists some m, n ∈ N such that η + ǫ + 2 ǫ θ + 2) + 148 (cid:16) − θ + 14 θ + 26 θ + 2 ( m + n )+ s ( θ − θ − θ − θ − θ + 2) ( m − n ) − mn + 2 θ − θ − θ + 2 (cid:17) = 0 . (2) The induced module
Ind( C [ t ] Θ λ,α,β,γ,η,ǫ,θ ) is reducible if and only if β ∈ Z + or oneof the conditions (i), (ii), (iii) in (1) holds.Proof. It follows from Theorem 3.1 and Theorem 6.4 that Ind( C [ t ] Mλ,α,β,γ,η,ǫ,θ ) is irreducible for M ∈ { Ω , ∆ , Θ } if and only if both M ( λ, α, β, γ ) and V ( η, ǫ, θ ) are irreducible. Consequently,the desired assertion follows directly from Proposition 2.2 and [14, Corollary 1]. (cid:3) Acknowledgements.
Yu-Feng Yao is grateful to professor Yufeng Pei for providing thereferences [14] and [17].
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Department of Mathematics, Shanghai Maritime University, Shanghai, 201306, China.
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