A new class of irreducible Virasoro modules from tensor product

Journal of Algebra 541 (2020) 324–344

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Journal of Algebra www.elsevier.com/locate/jalgebra

A new class of irreducible Virasoro modules from tensor product Xuewen Liu, Xiangqian Guo ∗ , Jing Wang a r t i c l e

i n f o

Article history: Received 27 April 2019 Available online 2 October 2019 Communicated by Volodymyr Mazorchuk MSC: 17B10 17B20 17B65 17B66 17B68

a b s t r a c t In this paper, we obtain a class of Virasoro modules by taking tensor products of the irreducible Virasoro modules Ω(λ, α, h) and Ω(μ, β) with irreducible highest weight modules V (θ, h) or with irreducible Virasoro modules Indθ (N ) defined in [20]. We obtain the necessary and sufficient conditions for such tensor product modules to be irreducible, and determine the necessary and sufficient conditions for two of them to be isomorphic. We also compare these modules with other known non-weight Virasoro modules, showing that most of these simple modules are new. © 2019 Published by Elsevier Inc.

Keywords: Virasoro algebra Tensor product modules Irreducible modules

1. Introduction Let C, Z, Z+ and N be the sets of all complexes, all integers, all non-negative integers and all positive integers respectively. The Virasoro algebra Vir is an infinite dimensional Lie algebra over the complex numbers C, with basis {di , c | i ∈ Z} and defining relations * Corresponding author. E-mail address: [email protected] (X. Guo). https://doi.org/10.1016/j.jalgebra.2019.08.035 0021-8693/© 2019 Published by Elsevier Inc.

X. Liu et al. / Journal of Algebra 541 (2020) 324–344

[di , dj ] = (j − i)di+j + δi+j,0 [c, di ] = 0,

i3 − i c, 12

325

i, j ∈ Z,

i ∈ Z.

The algebra Vir is one of the most important infinite-dimensional Lie algebras both in mathematics and in mathematical physics, see for example [11,10] and references therein. The representation theory of the Virasoro algebra has been widely used in many physics areas and other mathematical branches, for example, quantum physics [9], conformal field theory [6], vertex algebras [13], and so on. The theory of Harish-Chandra modules (that is, weight modules with finitedimensional weight spaces) for the Virasoro algebra is fairly well developed (see [11] and references therein). In particular, a classification of irreducible Harish-Chandra modules was given by Mathieu [17], and a classification of weight Virasoro modules with at least one finite-dimensional nonzero weight space was given in [19]. Recently, many authors constructed and studied many classes of simple non-Harish-Chandra modules, including simple weight modules with infinite-dimensional weight spaces (see [3,4,14,16]) and simple non-weight modules (see [1,12,14,18,19,21,22]). The purpose of the present paper is to construct new irreducible non-weight Virasoro modules by taking tensor product of several known irreducible Virasoro modules defined in [2,15] and [20]. The paper is organized as follows. In section 2, we recall the definitions of various modules involved in this paper and some basic results of them. In section 3, we obtain a class of modules over Vir by taking tensor products of several modules of type Ω(λ, α, h), several modules of type Ω(μ, β) and the irreducible highest module V (θ, h) or irreducible module Indθ (N ). We also determine the irreducibility of these modules and study their submodule structure when they are reducible. In section 4, we determine the necessary and sufficient conditions for two such irreducible tensor modules to be isomorphic. In section 5, we compare the irreducible tensor product modules with other known non-weight irreducible modules, and conclude that these irreducible modules provide a large family of new irreducible Virasoro modules. 2. Preliminaries We first recall the definitions of some Virasoro modules considered in this paper, i.e., the Omega modules Ω(λ, α, h) and Ω(μ, β), the irreducible highest weight modules V (θ, h) and the irreducible modules Indθ (N ) introduced in [20] and some basic properties of them. For any Lie algebra L, denote by U (L) the universal enveloping algebra of L. It is clear that Cd0 ⊕ Cc is the Cartan subalgebra of Vir. However, since the central element c acts as a scalar for most modules in this paper, h = Cd0 actually plays the role of a Cartan subalgebra. A Vir-module V is called a U (h)-free module of rank r if V is a free U (h)-module of rank r when restricting the action to U (h). Here r may be a positive integer or infinity. We first recall the definitions of some well known U (h)-free Vir-modules.

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Definition 2.1. For any λ ∈ C ∗ = C \ {0}, α ∈ C, let Ω(λ, α) = C[∂] as a vector space and we define the Vir-module action on Ω(λ, α) as follows: dk · ∂ i = λk (∂ − αk)(∂ − k)i ,

c · ∂ i = 0, ∀ k ∈ Z, i ∈ Z+ .

It is well known that Ω(λ, α) is simple if and only if α = 0. These modules were first constructed in [8,15]. It was shown that Ω(λ, α) are exactly all U (h)-free Vir-modules of rank 1 in [23]. Now we can generalize this construction to another class of Vir-modules. Definition 2.2. Fix any λ ∈ C ∗ = C \ {0}, α ∈ C and h(t) ∈ C[t]. Let Ω(λ, α, h) = C[t, ∂] as a vector space and we define the Vir-module action as follows: dk · f (t)∂ i = λk (∂ − k)i



∂ + kh(t) − k(k − 1)α

 h(t) − h(α)  f (t) − k(t − kα)f  (t) , t−α

c · f (t)∂ i = 0, ∀ k ∈ Z, i ∈ Z+ , where f ∈ C[t] and f  (t) is the usual derivative of f with respect to t. Sometimes, an alternative formula of the module action may be convenient. Define the following operators F (f ) =

h(t) − h(α) f (t) − f  (t), t−α

G(f ) = h(α)f + tF (f ), ∀ f ∈ C[t],

(2.1)

then the module action on Ω(λ, α, h) can be rewritten as   dk · f (t)∂ i = λk (∂ − k)i ∂f + kG(f ) − k2 αF (f ) ,

∀ k ∈ Z, i ∈ Z+ , f ∈ C[t].

(2.2)

This kind of modules are U (h)-free Vir-modules of infinite rank and were constructed and fully studied in [2]. In particularly, their irreducibility was determined. Theorem 2.3. [[2]] Ω(λ, α, h) is simple if and only if deg(h) = 1 and α = 0. Now we define another class of Vir-modules, on which the positive part of the Virasoro algebra acts locally finitely. For any θ, h ∈ C, let I(θ, h) be the left ideal of U (Vir) generated by the set {di | i > 0}

 {d0 − h · 1, c − θ · 1}.

The Verma module with highest weight (θ, h) for Vir is defined as the quotient module V (θ, h) = U (Vir)/I(θ, h). Then we have the irreducible highest weight module V (θ, h) = V (θ, h)/J, where J is the unique maximal proper submodule of V (θ, h). For the detailed structure of V (θ, h), refer to [5].

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Denote by Vir+ the Lie subalgebra of Vir spanned by all di with i ≥ 0. For k ∈ (k) Z+ , denote by Vir+ the Lie subalgebra of Vir generated by all di for i > k. For any Vir+ -module N and θ ∈ C, consider the induced module Ind(N ) = U (Vir) ⊗U (Vir+ ) N , and denote by Indθ (N ) the module Ind(N )/(c − θ)Ind(N ). These modules are used to give a characterization of the irreducible Vir-modules such that the action of dk are locally finite for sufficiently large k. Theorem 2.4 ([20]). Assume that N is an irreducible Vir+ -module such that there exists k ∈ N satisfying the following two conditions: (a) dk acts injectively on N ; (b) di N = 0 for all i > k. Then for any θ ∈ C the Vir module Indθ (N ) is simple. Theorem 2.5 ([20]). Let V be an irreducible Vir module. Then the following conditions are equivalent: (k)

(1) There exists k ∈ N such that V is a locally finite Vir+ -module; (k) (2) There exists k ∈ N such that V is a locally nilpotent Vir+ -module; (3) Either V is a highest weight module or V ∼ = Indθ (N ) for some θ ∈ C, k ∈ N and an irreducible Vir+ -module N satisfying the conditions (a) and (b) in Theorem 2.4. The purpose of this paper is to consider the irreducibility and structures of tensor products of several modules of type Ω(λ, α), several modules of type Ω(λ, α, h) and an irreducible module satisfying conditions in Theorem 2.5. 3. Irreducibility and structures of tensor product modules In this section we will investigate the structure of the tensor products of the Virasoro modules defined in the previous section, that is, the modules Ω(λ, α), Ω(λ, α, h) and simple modules as in Theorem 2.5. In particular, we will determine their irreducibility. Let ∂1 , . . . , ∂m , ∂m+1 , . . . , ∂m+n , t1 , . . . , tm be commuting variables, where m, n ∈ Z+ . Fix any complex numbers λi ∈ C ∗ , αi ∈ C, 1 ≤ i ≤ m +n and polynomials hi ∈ C[ti ], 1 ≤ i ≤ m, we have the modules Ω(λi , αi , hi ), 1 ≤ i ≤ m and Ω(λi , αi ), m + 1 ≤ i ≤ m + n defined as in Definition 2.1 and 2.2. Then we have that Ω(λi , αi , hi ) = C[∂i , ti ] for 1 ≤ i ≤ m and Ω(λi , αi ) = C[∂i ] for m < i ≤ m + n as vector spaces. Choosing an irreducible Vir-module V satisfying the conditions in Theorem 2.5 (possibly being a 1-dimensional trivial module), we can form the tensor product T (λ λ, α, h; V ) =

m  i=1

Ω(λi , αi , hi )

m+n  i=m+1

 Ω(λi , αi ) ⊗ V,

(3.1)

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where λ = (λ1 , . . . , λm , λm+1 , . . . , λm+n ), α = (α1 , . . . , αm , . . . , αm+n ) and h = (h1 , . . . , hm ). Denote even simply T = T (λ λ, α, h; V ). Note that we can define different operators Fi , Gi on C[ti ] using different polynomials hi (ti ) as in (2.1). To be simple, we will denote them by F, G uniformly, that is F (fi (ti )) =

hi (ti ) − hi (αi ) fi (ti ) − fi (ti ), ti − αi

G(fi ) = hi (αi )fi + ti F (fi )

for any fi (ti ) ∈ C[ti ], 1 ≤ i ≤ m. Here, one only needs to be careful when applying F or G to a constant polynomial, where suitable ti should be chosen according to which component the constant is taken from. Fix these notations in this section. We will study the structure and determine the irreducibility of the module T . Any element f ∈ T can be written as a finite sum f (∂1 , · · · , ∂m+n , t1 , · · · , tm ) =



r

r

m+1 m+n rm pm ∂1r1 tp11 ⊗ · · · ⊗ ∂m tm ⊗ ∂m+1 · · · ⊗ ∂m+n ⊗ vr,p ,

(r,p)∈S

where r = (r1 , . . . , rm , rm+1 , . . . , rm+n ) ∈ (Z+ )m+n , p = (p1 , . . . , pm ) ∈ (Z+ )m , vr,p ∈ V and S is a finite subset of (Z+ )2m+n . For convenience, we may write r

r

m+1 m+n rm pm ∂ r tp = ∂1r1 tp11 ⊗ · · · ⊗ ∂m tm ⊗ ∂m+1 · · · ⊗ ∂m+n

and the element f ∈ T can be rewritten as 

f (∂ ∂ , t) =

∂ r tp ⊗ vr,p

(3.2)

(r,p)∈S 







In what follows, we will also write (∂ ∂ r tp )(∂ ∂ r tp ) = ∂ r+r tp+p for short. Denote by ≤ the lexicographical order on Zl for any positive integer l, that is, (a1 , . . . , al ) ≤ (b1 , . . . , bl ) if and only if there exists 1 ≤ k ≤ l such that ak ≤ bk and ai = bi for all 1 ≤ i < k. Then we define the degree deg(f ) of f as the maximal (r, p) ∈ S with vr,p = 0. For convenience, we set deg(1 ⊗· · ·⊗1 ⊗v) = 0 = (0, 0, . . . , 0) ∈ (Z+ )2m+n for 0 = v ∈ V . Then we present two technical results. The first one is taken from [22] and the second one resembles Proposition 3.2 of [2]. Lemma 3.1 ([22]). Let μ1 , μ2 , . . . , μl ∈ C, s1 , s2 , . . . , sl ∈ N with s1 + s2 + · · · + sl = s. Define a sequence of functions on Z as follows: f1 (k) = μk1 , f2 (k) = kμk1 , . . . , fs1 (k) = ks1 −1 μk1 , fs1 +1 (k) = μk2 , . . . , fs1 +s2 (k) = ks2 −1 μk2 , . . . , fs (k) = ksl −1 μkl for any k ∈ Z. Let Y = (ypq ) be the s × s matrix with ypq = fq (p − 1), q = 1, . . . , s, p = p0 + 1, . . . , p0 + s for some fixed p0 ∈ Z+ . Then det(Y ) =

l j=1

s (sj +2p0 −1)/2

(sj − 1)!!μj j

1≤i
(μj − μi )si sj ,

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where sj !! = sj ! × (sj − 1)! × · · · × 2! × 1! and 0!! = 1. Proposition 3.2. Suppose that λ1 , . . . , λm , λm+1 , . . . , λm+n are distinct. Let W ⊆ T be a subspace which is stable under the actions of all dk for k sufficiently large. Take any

element f (∂ ∂ , t) = (r,p)∈S ∂ r tp ⊗vr,p ∈ W as in (3.2) and denote si = max{ri | (r, p) ∈ S} for all 1 ≤ i ≤ m + n. Then we have j

aij := (−1)



∂1r1 tp11

(r,p)∈S

−



ri ri −j+1 pi ri ∂i ∂ ri −j+1 G(tpi i ) ⊗ ··· ⊗ ti − j j−1 i 

ri ∂ ri −j+2 αi F (tpi i ) j−2 i

r

r

m+1 m+n rm pm ⊗ · · · ⊗ ∂m tm ⊗ ∂m+1 ⊗ · · · ⊗ ∂m+n ⊗ vr,p ∈ W,

(3.3) for 1 ≤ j ≤ si + 2, 1 ≤ i ≤ m; and aij := (−1)j



r

m+1 rm pm ∂1r1 tp11 ⊗ · · · ⊗ ∂m tm ⊗ ∂m+1 ⊗

(r,p)∈S



 ri ri rm+n + αi ∂iri −j+1 ⊗ · · · ⊗ ∂m+n ⊗ vr,p ∈ W, ··· ⊗ j j−1

(3.4)

for 1 ≤ j ≤ si + 1, m + 1 ≤ i ≤ m + n, where the operators F and G are defined as in    (2.1) and we make the convention that 00 = 1 and rji = 0 whenever j > ri or j < 0. Taking special values of i, j, we have some simpler expressions. (1). For any 1 ≤ i ≤ m + n, taking j = 0 in (3.3) and (3.4)), we have  ai,0 = ∂i f (∂ ∂ , t) = ∂ r+i tp ⊗ vr,p ∈ W, ∀ 1 ≤ i ≤ m + n,

(3.5)

(r,p)∈S

where i = (δi1 , . . . , δi,m+n ) ∈ (Z+ )m+n . (2). For any 1 ≤ i ≤ m + n, taking j = 1 in (3.3) and (3.4), we have  ai,1 = − (ri tpi i − G(tpi i ))∂ ∂ r tp−pi i ⊗ vr,p ∈ W, 1 ≤ i ≤ m,

(3.6)

(r,p)∈S

and ai,1 = −



(ri + αi )∂ ∂ r ⊗ vr,p ∈ W, ∀ m + 1 ≤ i ≤ m + n.

(3.7)

(3). For any 1 ≤ i ≤ m, taking j = si + 2 in (3.3), we have  ai,si +2 = (−1)si +1 αi ∂ r−si i F (tpi i )tp−pi i ⊗ vr,p ∈ W.

(3.8)

(r,p)∈S

(r,p)∈S,ri =si

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(4). For any m + 1 ≤ i ≤ m + n, taking j = si + 1 in (3.4), we have 

ai,si +1 = (−1)si +1

αi ∂ r−si i tp ⊗ vr,p ∈ W.

(3.9)

(r,p)∈S,ri =si

Proof. By the assumption on the module V , there exists a positive integer k0 such that dk vr,p = 0 for all (r, p) ∈ S and k ≥ k0 . Then applying dk , k sufficiently large, on f , we have 

dk · f (∂ ∂ , t) =

(dk · ∂ r tp ) ⊗ vr,p

(r,p)∈S

=

m  

r

r

m+1 m+n rm pm ∂1r1 tp11 ⊗ · · · ⊗ (dk · (∂iri tpi i )) ⊗ · · · ⊗ ∂m tm ⊗ ∂m+1 ⊗ · · · ⊗ ∂m+n ⊗ vr,p

(r,p)∈S i=1

+

m+n 



r

r

m+1 m+n rm pm ∂1r1 tp11 ⊗ · · · ⊗ ∂m tm ⊗ ∂m+1 ⊗ · · · ⊗ (dk · ∂iri ) ⊗ · · · ⊗ ∂m+n ⊗ vr,p

(r,p)∈S i=m+1

=

m  

  ∂1r1 tp11 ⊗ · · · ⊗ λki (∂i − k)ri ∂i tpi i + kG(tpi i ) − k2 αi F (tpi i ) ⊗

(r,p)∈S i=1 r

r

m+1 m+n rm pm tm ⊗ ∂m+1 ⊗ · · · ⊗ ∂m+n ⊗ vr,p · · · ⊗ ∂m

+

m+n 



r

m+1 rm pm ∂1r1 tp11 ⊗ · · · ⊗ ∂m tm ⊗ ∂m+1 ⊗

(r,p)∈S i=m+1 r

m+n ⊗ vr,p · · · ⊗ λki (∂i − k)ri (∂i − kαi ) ⊗ · · · ⊗ ∂m+n

 m r i +2   ri ri −j+1 pi ri ∂i ∂iri −j+1 G(tpi i ) = (−1)j λki kj ∂1r1 tp11 ⊗ · · · ⊗ ti − j j − 1 i=1 j=0

(r,p)∈S

+

 ri rm+1 rm+n rm pm ∂iri −j+2 αi F (tpi i ) ⊗ · · · ⊗ ∂m tm ⊗ ∂m+1 ⊗ · · · ⊗ ∂m+n ⊗ vr,p j−2



− 

m+n i +1  r

r

m+1 rm pm (−1)j λki kj ∂1r1 tp11 ⊗ · · · ⊗ ∂m tm ⊗ ∂m+1 ⊗

(r,p)∈S i=m+1 j=0

··· ⊗

  ri ri rm+n + αi ∂iri −j+1 ⊗ · · · ⊗ ∂m+n ⊗ vr,p ∈ W. j j−1

(3.10)

Regarding the above element as an exp-polynomial in k with coefficients in T and noticing that the determinant in Lemma 3.1 is nonzero (taking μi = λi and replacing si with si + 3 for i = 1, . . . , l = m + n in the lemma), we see that the coefficient of each λki kj lies in W , for any 0 ≤ j ≤ si + 2, 1 ≤ i ≤ m and 0 ≤ j ≤ si + 1, m + 1 ≤ i ≤ m + n. These coefficients give rise to the elements in (3.3) and (3.4). 2

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Theorem 3.3. Suppose that λ1 , . . . , λm , . . . , λm+n are distinct, deg hi = 1 for i = 1, . . . , m and V is irreducible. Then the Vir-module T is generated by 1 ⊗ v for any 0 = v ∈ V . Proof. Assume hi (ti ) = ξi ti + ηi for some ξi , ηi ∈ C with ξi = 0. Then F (tki ) = ξi tki − ktk−1 and G(tki ) = (ξi αi + ηi − k)tki + ξi tk+1 for any 1 ≤ i ≤ m and k ∈ Z+ . i i Fix any nonzero v ∈ V . Let W be the submodule of T generated by 1 ⊗ v. We first claim that tp ⊗ v ∈ W implies tp+i ⊗ v ∈ W for any p = (p1 , . . . , pm ) ∈ (Z+ )m and 1 ≤ i ≤ m. Indeed, if tp ⊗ v ∈ W , then equation (3.6) gives G(tpi i )tp−pi i ⊗ v = ((ξi αi + ηi − pi )tp + ξi tp+i ) ⊗ v ∈ W, which implies tp+i ⊗ v ∈ W . We can deduce tp ⊗ v ∈ W for all p ∈ (Z+ )m by induction on p. Then from equation (3.5), we can further deduce that ∂ r tp ⊗ v ∈ W for all r ∈ (Z+ )m+n and p ∈ (Z+ )m by induction on r. That is C[∂ ∂ , p] ⊗ v ⊆ W .  Now denote V = {w ∈ V | C[∂ ∂ , p] ⊗ w ⊆ W }. The previous argument shows V  = 0.  Then for any w ∈ V , the module action of Vir on W x(f (∂ ∂ , t) ⊗ w) = (xf (∂ ∂ , t)) ⊗ w + f (∂ ∂ , t) ⊗ (xw) ∈ W, ∀ f (∂ ∂ , t) ∈ C[∂ ∂ , t], x ∈ Vir implies that xw ∈ V  , that is, V  is stable under the action of Vir. Since V is an irreducible Vir-module, we have V  = V and hence W = T , as desired. 2 Theorem 3.4. Suppose that λ1 , . . . , λm+n are distinct, α1 , . . . , αm+n are nonzero, deg hi = 1 for i = 1, · · · , m and V is irreducible. Then the tensor product T defined in (3.1) is an irreducible Vir-module. Proof. Let W be a nonzero submodule of T and take any nonzero element f (∂ ∂ , t) ∈ W .

Write f (∂ ∂ , t) in the form of (3.2), i.e., f (∂ ∂ , t) = (r,p)∈S ∂ r tp ⊗vr,p . We assume vr,p = 0 for all (r, p) ∈ S. Set si = max{ri | (r, p) ∈ S} for any 1 ≤ i ≤ m + n. If si = 0 for some 1 ≤ i ≤ m, take j = si + 2 in (3.8), and noticing αi = 0, we get 0 = f1 (∂ ∂ , t) :=



∂ r−ri i F (tpi i )tp−pi i ⊗ vr,p ∈ W.

(r,p)∈S,ri =si

Replacing f with f1 , we may assume that si = 0 for this particular i. Repeat this procedure at most m times, we may suppose si = 0 for all 1 ≤ i ≤ m. If si = 0 for some m + 1 ≤ i ≤ m + n, take j = si + 1 in (3.9) and we have 0 = f2 (∂ ∂ , t) :=



∂ r−ri i tp ⊗ vr,p ∈ W.

(r,p)∈S,ri =si

Replacing f with f2 , we may assume that si = 0 for this particular i. Repeat this procedure at most n times, we may suppose si = 0 for all m + 1 ≤ i ≤ m + n.

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Now si = 0 for all 1 ≤ i ≤ m +n and hence ri = 0 for all 1 ≤ i ≤ m +n and (r, p) ∈ S. If pi = 0 for some 1 ≤ i ≤ m, (r, p) ∈ S, take j = si + 2 = 2 in (3.8), we have

f3 (∂ ∂ , t) =:





F (tpi i )tp−pi i ⊗ vr,p =

(r,p)∈S

(ξi tp − pi tp−i ) ⊗ vr,p ∈ W,

(r,p)∈S

which implies

0 = ξi f (∂ ∂ , t) − f3 (∂ ∂ , t) =



pi tp−i ⊗ vr,p ∈ W.

(r,p)∈S

Noticing that ξi f − f3 has lower degree in ti than f . Repeat this procedure a finite number of times, we may reach a nonzero element in W ∩ (1 ⊗ V ). Then by Theorem 3.3, we have W = T , which means that T is irreducible. 2 Now we consider the case when λ1 , · · · , λm+n are not distinct. In fact, T is reducible in this case. To show this result, it suffices to show the reducibility of the tensor    products Ω(λ, α1 ) Ω(λ, α2 ), Ω(λ, α1 , h1 ) Ω(λ, α2 ) and Ω(λ, α1 , h1 ) Ω(λ, α2 , h2 ) for λ, α1 , α2 ∈ C with λ = 0 and suitable polynomials h1 , h2 . The structure of  Ω(λ, α1 ) Ω(λ, α2 ) has already been determined in [22] and in particular,  Ω(λ, α1 ) Ω(λ, α2 ) is reducible. So we only consider the remaining cases in the following.  For the case Ω(λ, α1 , h1 ) Ω(λ, α2 ), we still use the notation as before, only taking m = n = 1 and V as the 1-dimensional trivial module. We simply write t = t1 and  h = h1 . For convenience, identify Ω(λ, α1 , h) Ω(λ, α2 ) = C[∂1 , t, ∂2 ]. For any l ∈ Z+ , denote

Ml = span{∂1r (∂1 + ∂2 )s C[t] | r, s ∈ Z+ , r ≤ l} and

Ml = span{∂2r (∂1 + ∂2 )s C[t] | r, s ∈ Z+ , r ≤ l}.  Then both Ml and Ml are proper Vir-submodules of Ω(λ, α1 , h) Ω(λ, α2 ). Indeed, fix any l ∈ Z+ , then for any ∂1r (∂1 + ∂2 )s f ∈ Ml , where r, s ∈ Z+ , r ≤ l, f ∈ C[t], and dk ∈ Vir, k ∈ Z, we can compute

X. Liu et al. / Journal of Algebra 541 (2020) 324–344

λ

=

−k

dk (∂1r (∂1

−k

s

+ ∂2 ) f ) = λ

j=0

+

s  s j=0

s 

j

dk (∂1r+j ∂2s−j )f

s (∂1 − k)r+j (∂1 f + kG(f ) − k 2 α1 F (f ))∂2s−j j

s  s

∂1r+j (∂2 − k)s−j (∂2 − α2 k)f

j

j=0

s   s (∂1 − k)j ∂2s−j =(∂1 − k)r ∂1 f + kG(f ) − k 2 α1 F (f ) j j=0

+ 

333

∂1r (∂2

− α2 k)f

(3.11)

s  s j=0

j

∂1j (∂2 − k)s−j

   = (∂1 − k) ∂1 f + kG(f ) − k 2 α1 F (f ) + ∂1r (∂2 − α2 k)f (∂1 + ∂2 − k)s  = ((∂1 − k)r − ∂1r )∂1 f  + (∂1 − k)r (kG(f ) − k 2 α1 F (f )) + ∂1r (∂1 + ∂2 − α2 k)f (∂1 + ∂2 − k)s , r

which again lies in Ml . So Ml is a submodule of Ω(λ, α1 , h) ⊗ Ω(λ, α2 ). Similarly, for any ∂2r (∂1 + ∂2 )s f ∈ Ml , where r, s ∈ Z+ , r ≤ l, f ∈ C[t], and dk ∈ Vir, k ∈ Z, we can deduce −k

−k

s  s

dk (∂1s−j ∂2r+j )f j j=0   r 2 =∂2 (∂1 + ∂2 )f + kG(f ) − k α1 F (f ) (∂1 + ∂2 − k)s   (∂2 − k)r − ∂2r )∂2 f − (∂2 − k)r α2 kf (∂1 + ∂2 − k)s λ

dk (∂2r (∂1

s

+ ∂2 ) f ) = λ

(3.12)

lies in Ml , showing that Ml is a submodule of Ω(λ, α1 , h) ⊗ Ω(λ, α2 ).  It is clear that Ml and Ml are submodules of Ml+1 and Ml+1 respectively for any l ∈ Z+ . We can determine the quotient modules of them. Proposition 3.5. Let the notation be as before. For any l ∈ Z+ , Ml is a proper submodule of Ω(λ, α1 , h) ⊗ Ω(λ, α2 ) and  ∼ Ml /Ml−1 ∼ = Ml /Ml−1 = Ω(λ, α1 , h − α2 − l)  as Vir-modules, where we make the convention M−1 = M−1 = 0.

Proof. Identify Ω(λ, α1 , h − α2 − l) = C[∂, t] with module action given similarly as in (2.2). Denote M l = Ml /Ml−1 for short. For any element w ∈ Ml , let w be its image in M l . It is clear that ∂1l (∂1 + ∂2 )s ti , s, i ∈ Z+ , form a basis of M l .

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Take any ∂1l (∂1 + ∂2 )s f ∈ Ml , where s ∈ Z+ , f ∈ C[t], and dk ∈ Vir, k ∈ Z, then by formula (3.11), we have λ−k dk (∂1l (∂1 + ∂2 )s f )   = ((∂1 − k)l − ∂1l )∂1 f + (∂1 − k)l (kG(f ) − k 2 α1 F (f )) + ∂1l (∂1 + ∂2 − α2 k)f (∂1 + ∂2 − k)s = (−lk∂1l f + ∂1l (kG(f ) − k 2 α1 F (f )) + ∂1l (∂1 + ∂2 − α2 k)f )(∂1 + ∂2 − k)s   = ∂1l (∂1 + ∂2 )f + k(G(f ) − lf − α2 f ) − k 2 α1 F (f ) (∂1 + ∂2 − k)s .

We can check that the linear map ϕ from M l to Ω(λ, α1 , h − l − α2 ) given by ϕ(∂1l (∂1 + ∂2 )s f ) = ∂ s f is a Vir-module isomorphism.  Similarly, denote M  l = Ml /Ml−1 for short and let w be the image in M  l of an element w ∈ Ml . Take any ∂2l (∂1 + ∂2 )s f ∈ Ml , where s ∈ Z+ , f ∈ C[t], and dk ∈ Vir, k ∈ Z, then by formula (3.12), we have λ−k dk ∂2l (∂1 + ∂2 )s f   = ∂2l (∂1 + ∂2 )f + k(G(f ) − α2 f − lf ) − k 2 α1 F (f ) (∂1 + ∂2 − k)s , which implies M  l ∼ = Ω(λ, α1 , h − l − α2 ). This completes the proof. 2  Now we consider the case Ω(λ, α1 , h1 ) Ω(λ, α2 , h2 ), we also use the notation as in the beginning of this section, only taking m = 2, n = 0 and V as a 1-dimensional trivial module.  We identify Ω(λ, α1 , h1 ) Ω(λ, α2 , h2 ) = C[∂1 , t1 , ∂2 , t2 ]. For any l ∈ Z+ , denote Nl = span{∂1r (∂1 + ∂2 )s C[t1 , t2 ] | r, s ∈ Z+ , r ≤ l}.  Then each Nl is a proper Vir-submodule of Ω(λ, α1 , h1 ) Ω(λ, α2 , h2 ). To see this, we fix any l ∈ Z+ . Then for any ∂1r (∂1 + ∂2 )s f1 f2 ∈ Nl , where r, s ∈ Z+ , r ≤ l, f1 ∈ C[t1 ], f2 ∈ C[t2 ] and dk ∈ Vir, k ∈ Z, we can deduce λ−k dk (∂1r (∂1 + ∂2 )s f1 f2 )      = (∂1 − k)r − ∂1r ∂1 f1 f2 + (∂1 − k)r kG(f1 ) − k2 α1 F (f1 ) f2 (∂1 + ∂2 − k)s (3.13)   + ∂1r (∂1 + ∂2 )f2 + kG(f2 ) − k2 α2 F (f2 ) f1 (∂1 + ∂2 − k)s , lies in Nl . The computation is straightforward and similar as for (3.11) and (3.12). So Nl is a proper submodule of Ω(λ, α1 , h1 ) ⊗ Ω(λ, α2 , h2 ). It is clear that Nl is a submodule of Nl+1 for any l ∈ Z+ , and the structure of the corresponding quotient modules can be given as follows. Proposition 3.6. Let the notation be as before. For any l ∈ Z+ , Nl is a proper submodule of Ω(λ, α1 , h1 ) ⊗ Ω(λ, α2 , h2 ) and the quotient module N l := Nl /Nl−1 (set N−1 = 0) has a filtration

X. Liu et al. / Journal of Algebra 541 (2020) 324–344

(0)

Nl = Nl (l )

(l +1)

such that N l /N l

(1)

⊇ Nl

335

(2)

⊇ Nl . . .

∼ = Ω(λ, 1, h), where h(t) = h1 (α1 t) + h2 (α2 t) − l − l .

Proof. Fix an l ∈ Z+ . Let w be the image of w ∈ Nl in N l . Take any ∂1l (∂1 + ∂2 )s f1 f2 ∈ Nl , where s ∈ Z+ , fi ∈ C[ti ], i = 1, 2, and dk ∈ Vir, k ∈ Z, then from (3.13), we deduce   λ−k dk ∂1l (∂1 + ∂2 )s f1 f2 = ∂1l (∂1 + ∂2 )f1 f2 + kG (f1 f2 ) − k2 F  (f1 f2 ) (∂1 + ∂2 − k)s , where G (f1 f2 ) = G(f1 )f2 + G(f2 )f1 − lf1 f2 and F  (f1 f2 ) = α1 F (f1 )f2 + α2 F (f2 )f1 . By (2.1), we have explicitly that F  (f1 f2 ) =



∂ ∂  α1 (h1 (t1 ) − h1 (α1 )) α2 (h2 (t2 ) − h2 (α2 ))  + − α1 + α2 f1 f2 t1 − α1 t2 − α2 ∂t1 ∂t2

and



G (f1 f2 ) =

 ∂ t1 h1 (t1 ) − α1 h1 (α1 ) t2 h2 (t2 ) − α2 h2 (α2 )) ∂  + − l − t1 + t2 f1 f2 . t1 − α1 t2 − α2 ∂t1 ∂t2

Now fix an l ∈ Z+ and denote (l )

Nl

= {∂1l (∂1 + ∂2 )s tp1 (α1 t2 − α2 t1 )q | s, p, q ∈ Z+ , q ≥ l}.

For any p, q ∈ Z+ , q ≥ l , we can compute that F  (tp1 (α1 t2 − α2 t1 )q ) =



α1 (h1 (t1 ) − h1 (α1 )) + g1 (t2 ) − α1 pt−1 tp1 (α1 t2 − α2 t1 )q , t1 − α1

and G (tp1 (α1 t2 − α2 t1 )q ) =



t1 h1 (t1 ) − α1 h1 (α1 ) + g2 (t2 ) − (l + p + q) tp1 (α1 t2 − α2 t1 )q , t1 − α1

)−h2 (α2 )) 2 h2 (α2 )) where g1 (t2 ) = α2 (h2 (tt22−α and g2 (t2 ) = t2 h2 (t2t)−α . Expand g1 (t2 ), g2 (t2 ) as 2 2 −α2  p polynomials in t1 and α1 t2 −α2 t1 , we see that both F (t1 (α1 t2 −α2 t1 )q ) and G (tp1 (α1 t2 −   α2 t1 )q ) are linear combinations of elements of the form tp1 (α1 t2 −α2 t1 )q , p , q  ∈ Z+ , q  ≥ (l )

(l )

q. This indicates dk ∂1l (∂1 + ∂2 )s tp1 (α1 t2 − α2 t1 )q ∈ N l and in particular, N l is a submodule of N l . (l ) Note that ∂1l (∂1 + ∂2 )s tp1 (α1 t2 − α2 t1 )l , s, p ∈ Z+ forms a basis of N l . For any s, p ∈ Z+ , and dk ∈ Vir, k ∈ Z, we can deduce that

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336

λ−k dk ∂1l (∂1 + ∂2 )s tp1 (α1 t2 − α2 t1 )l ≡∂1l (α1 t2 − α2 t1 )l (∂1 + ∂2 − k)s ×   t h (t ) − α h (α )  1 1 1 1 1 1 (∂1 + ∂2 )tp1 + k + g2 (α1−1 α2 t1 ) − (l + p + l ) tp1 t1 − α1    (l +1) −1 −1 p 2 α1 (h1 (t1 ) − h1 (α1 )) −k + g1 (α1 α2 t1 ) − α1 pt1 t1 . mod N l t1 − α1 Set g(t2 ) = h2 (α1−1 α2 t2 ), then we have g1 (α1−1 α2 t1 ) =

α2 (h2 (α1−1 α2 t1 ) − h2 (α2 )) α1 (g(t1 ) − g(α1 )) = t1 − α1 α1−1 α2 t1 − α2

and g2 (α1−1 α2 t1 ) =

α1−1 α2 t1 h2 (α1−1 α2 t1 ) − α2 h2 (α2 )) t1 g(t1 ) − α1 g(α1 )) = . t1 − α1 α1−1 α2 t1 − α2

Now we obtain that λ−k dk ∂1l (∂1 + ∂2 )s tp1 (α1 t2 − α2 t1 )l ≡∂1l (α1 t2 − α2 t1 )l (∂1 + ∂2 − k)s ×   t (h + g)(t ) − α (h + g)(α )  1 1 1 1 1 1 (∂1 + ∂2 )tp1 + k − (l + l + p) tp1 t1 − α1    (l +1) −1 p 2 α1 ((h1 + g)(t1 ) − (h1 + g)(α1 )) −k − α1 pt1 t1 mod N l . t1 − α1 Set h(t) = (h1 + g)(α1 t) − l − l = h1 (α1 t) + h2 (α2 t) − l − l ∈ C[t]. Define a linear map (l )

(l +1)

φ : N l /N l

→ Ω(λ, 1, h) = C[∂, t]

  by φ ∂1l (α1 t2 − α2 t1 )l (∂1 + ∂2 )s (α1−1 t1 )p = ∂ s tp , it is straightforward to check that φ is a Vir-module isomorphism. This completes the proof. 2 Combining the previous results, more precisely, the facts that Ω(λ, α) is irreducible if and only if α = 0, that Ω(λ, α, h) is irreducible if and only if α = 0 and deg(h) = 1 (Theorem 2.3), and that all Ω(λ, α1 ) ⊗Ω(λ, α2 ), Ω(λ, α1 , h1 ) ⊗Ω(λ, α2 ) and Ω(λ, α1 , h1 ) ⊗ Ω(λ, α2 , h2 ) are reducible (Proposition 3.5 and Proposition 3.6) and Theorem 3.4, we conclude that Theorem 3.7. The Vir-module T (λ λ, α, h; V ) is irreducible if and only if λ1 , . . . , λm+n are distinct, α1 , . . . , αm+n are nonzero and deg hi = 1 for i = 1, · · · , m.

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337

4. Isomorphism In this section we will determine the necessary and sufficient conditions for two simple tensor product modules studied in Section 3 to be isomorphic. We first provide some properties of the tensor product module T = T (λ λ, α, h; V ), as defined in (3.1). By Theorem 3.4, to ensure that T is irreducible, we also suppose that λ1 , · · · , λm+n are distinct, αi is nonzero for 1 ≤ i ≤ m + n and deg(hi ) = 1 for all 1 ≤ i ≤ m throughout this section. For any f ∈ T , we define Rf = lim rank{f, dk (f ) | k ≥ l} and RT = inf{Rf | 0 = f ∈ T }. l→∞

If we write f = f (∂ ∂ , t) in the form of (3.2), there exists a minimal positive integer k(f ) such that dk vr,p = 0 for all k ≥ k(f ) and (r, p) ∈ S. Then we have Lemma 4.1. Suppose m + n ≥ 1. Take any nonzero f = f (∂ ∂ , t) ∈ T . (1) Rf = rank{f, dk (f ) | k ≥ l} for all l ≥ k(f ). (2) Rf ≥ 2m + n + 1 and the equality holds if and only if f = 1 ⊗ v for some 0 = v ∈ V . (3) RT = 2m + n + 1. Proof. (1) Write f in the form of (3.2). Denote Rf,l = rank{f, dk (f ) | k ≥ l} for any l ∈ N, then it suffices to show Rf,l = Rf,k(f ) for all l ≥ k(f ). For any k ≥ l ≥ k(f ), by equation (3.10), we have dk · f (∂ ∂ , t) = dk ·



∂ r tp ⊗ vr,p

(r,p)∈S

=



m r i +2 

(−1)j λki kj ∂1r1 tp11 ⊗ · · · ⊗

(r,p)∈S i=1 j=0

 ri ri −j+1 pi ri ∂i ∂ ri −j+1 G(tpi i ) ti − j j−1 i

 ri rm+1 rm+n rm pm ∂iri −j+2 αi F (tpi i ) ⊗ · · · ⊗ ∂m − tm ⊗ ∂m+1 ⊗ · · · ⊗ ∂m+n ⊗ vr,p j−2

+



m+n i +1  r

r

m+1 rm pm (−1)j λki kj ∂1r1 tp11 ⊗ · · · ⊗ ∂m tm ⊗ ∂m+1 ⊗

(r,p)∈S i=m+1 j=0

  ri ri rm+n + αi ∂iri −j+1 ⊗ · · · ⊗ ∂m+n ··· ⊗ ⊗ vr,p . j j−1

Regard the above element as an exp-polynomial in k with coefficients in T . Let aij ∈ T be the coefficient of λki kj for 0 ≤ j ≤ si + 2, 1 ≤ i ≤ m and 0 ≤ j ≤ si + 1, m + 1 ≤ i ≤ m + n in the above formula, where si = max{ri | (r, p) ∈ S}. Then aij are just the elements in (3.3) and (3.4).

X. Liu et al. / Journal of Algebra 541 (2020) 324–344

338

It is easy to see that span{dk (f ) | k ≥ l} is stable under dk , k ≥ l and hence contains all aij ∈ T , forcing span{f, dk (f ) | k ≥ l} = span{f, ai,j | 0 ≤ j ≤ si + 2, 1 ≤ i ≤ m + n}, which is independent of l. As a result, Rf,l = Rf,k(f ) for all l ≥ k(f ). (2) If m = 0, the result follows from Lemma 5 of [22]. So we assume m ≥ 1 from now on. For convenience, we also suppose hi (ti ) = ξi ti + ηi for some ξi , ηi ∈ C, ξi = 0, 1 ≤ i ≤ m, then we have explicitly that F (tpi i ) = ξi tpi i − pi tipi −1 , G(tpi i ) = (hi (αi ) − pi )tpi i + ξi tpi i +1 , ∀ 1 ≤ i ≤ m. Now we continue with the arguments in (1). By Lemma 3.2 (3.5)-(3.9), we see that

ai,1

ai,0 = ∂i f (∂ ∂ , t), ∀ 1 ≤ i ≤ m + n, m   =− (ri + pi − hi (αi ) − ξi ti )∂ ∂ r tp ⊗ vr,p , ∀ 1 ≤ i ≤ m, (r,p)∈S i=1

ai,si +2 = (−)si +1 αi



p ∂ r−si i (ξi − pi t−1 i )t ⊗ vr,p , ∀ 1 ≤ i ≤ m,

(r,p)∈S,ri =si

and ai,si +1 = (−1)si +1



αi ∂ r−si i ⊗ vr,p , ∀ m + 1 ≤ i ≤ m + n.

(r,p)∈S,ri =si

If we set deg(f ) = (s, q), then it is easy to see that deg(ai,0 ) = (s +i , p) for 1 ≤ i ≤ m +n and deg(ai,1 ) = (s, p + i ) for 1 ≤ i ≤ m. If there exists 1 ≤ i ≤ m such that ri ≥ 1, let i1 be the minimal i with ri ≥ 1. Then we have deg(ai1 ,si1 +2 ) < (s, p) and the space spanned by f, ai,0 , aj,1 , ai1 ,si1 +2 , 1 ≤ i ≤ m + n, 1 ≤ j ≤ m has dimension 2m + n + 2, or in other words, Rf ≥ 2m + n + 2. If ri = 0 for all 1 ≤ i ≤ m and there exists m + 1 ≤ i ≤ m + n such that ri ≥ 1, let i2 be the minimal one. Again we have deg(ai2 ,si2 +1 ) < (s, p), hence the space spanned by f, ai,0 , aj,1 , ai2 ,si2 +1 , 1 ≤ i ≤ m + n, 1 ≤ j ≤ m has dimension 2m + n + 2 and Rf ≥ 2m + n + 2. If ri = 0 for all 1 ≤ i ≤ m + n and there exists 1 ≤ i ≤ m such that pi ≥ 1, let i3 be the smallest one. Similarly, we have deg(ai3 ,si3 +2 ) < (s, p) and Rf ≥ 2m + n + 2. Otherwise, we must have f ∈ 1 ⊗ V and we may assume f = 1 ⊗ v for some nonzero v ∈ V , then si = 0 for all 1 ≤ i ≤ m + n and we have ai,0 = ∂i ⊗ v, aj,1 = (ξj tj + ξj αj + ηj ) ⊗ v, aj,2 = −αj ξj ⊗ v, ∀ 1 ≤ i ≤ m + n, 1 ≤ j ≤ m. Hence Rf = rank{f, ai,0 , aj,1 , aj,2 | 1 ≤ i ≤ m + n, 1 ≤ j ≤ m} = 2m + n + 1, as desired. This proves (2), and (3) follows directly from (2). 2

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339

Now we are ready to determine the isomorphism between two irreducible tensor product modules. Let T be the tensor module as given at the beginning of this section. We choose another tensor module, say, T  = T (λ λ , α  , h  ; V  ) =

m 

Ω(λi , αi , hi )

 m +n

 Ω(λi , αi ) ⊗ V  ,

j=m +1

i=1

  where λ = (λ1 , . . . , λm , λm +1 , . . . , λm +n ), α = (α1 , . . . , αm =  , . . . , αm +n ), h (h1 , . . . , hm ) and V  is an irreducible Vir-module satisfying the conditions in Theorem 2.5. We identify Ω(λi , αi , hi ) = C[∂i , ti ] for 1 ≤ i ≤ m and Ω(λi , αi ) = C[∂i ] for 1 ≤ i ≤ m + n . To ensure that T and T  are irreducible, we suppose that λ1 , · · · , λm+n are distinct, λ1 , · · · , λm +n are distinct, αi and αj are nonzero for 1 ≤ i ≤ m + n and 1 ≤ j ≤ m + n , and hi = ξi ti + ηi , hj = ξj tj + ηj , where ξi , ηi , ξj , ηj ∈ C with ξi , ξj = 0, for 1 ≤ i ≤ m and 1 ≤ j ≤ m .

Theorem 4.2. T ∼ = T  as Vir-modules if and only if m = m , n = n , V ∼ = V  and (λi , αi ξi ) = (λi , αi ξi ) for 1 ≤ i ≤ m and (λi , αi ) = (λi , αi ) for m + 1 ≤ i ≤ m + n by reordering the indices 1 ≤ i ≤ m + n if necessary. ∼ Ω(λi , αi ξi , ti ) and Proof. By Corollary 3.5 of [7], we know that Ω(λi , αi , hi ) =    ∼      Ω(λi , αi , hi ) = Ω(λj , αj ξj , tj ) for 1 ≤ i ≤ m and 1 ≤ j ≤ m . So by replacing Ω(λi , αi , hi ) with Ω(λi , αi ξi , ti ) and Ω(λj , αj , hj ) with Ω(λj , αj ξj , tj ), we may assume that ξi = ξj = 1 and ηi = ηj = 0 for all 1 ≤ i ≤ m and 1 ≤ j ≤ m . The sufficiency of the theorem is clear, we need only to prove the necessity. Let φ : T → T  be a Vir-module isomorphism. It is clear that RT = RT  , then by Lemma 4.1 (3), we see 2m + n + 1 = 2m + n + 1. Take f = 1 ⊗ v for some nonzero v ∈ V . Using Lemma 4.1 (2), we have Rφ(f ) = Rf = 2m + n + 1, forcing φ(f ) = 1 ⊗ v  for some nonzero v  ∈ V  . Since v  is uniquely determined by v, we may denote τ (v) = v  . It is obvious that τ is a linear bijection from V to V  . Now for any k ∈ Z with k ≥ k(f ) and k ≥ k(φ(f )), we have 0 =φ(dk (1 ⊗ v)) − dk φ(1 ⊗ v) =

m 

m+n    λki φ (∂i + k(αi + ti ) − k2 αi ) ⊗ v + λki φ((∂i − kαi ) ⊗ v)

i=1

i=m+1 

−



(4.1)



m m +n  (λi )k (∂i + k(αi + ti ) − k2 αi ) ⊗ τ (v) − (λi )k (∂i − kαi ) ⊗ τ (v). i=1

i=m +1

Regarding the right-hand side expression in (4.1) as an exp-polynomial in k ∈ Z with coefficients in T  , then this exp-polynomial equal to 0 indicates that each coefficient of distinct

340

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λk , kλk , k2 λk , λ ∈ {λ1 , · · · , λm+n , λ1 , . . . , λm +n } is 0, thanks to Lemma 3.1. For any 1 ≤ i ≤ m, if λi = λj for any 1 ≤ j ≤ m , then the coefficient of k2 λki in (4.1) would be −φ(αi ⊗ v) = 0, contradiction. So we must have λi ∈ {λ1 , . . . , λm } for all 1 ≤ i ≤ m. By symmetry, we obtain m = m and {λ1 , . . . , λm } = {λ1 , . . . , λm }. Similarly, for any m + 1 ≤ i ≤ m + n, if λi = λj for any m + 1 ≤ j ≤ m + n , then the coefficient of kλki in (4.1) would be −φ(αi ⊗ v) = 0, again contradiction. So we have λi ∈ {λm+1 , . . . , λm+n } for all m + 1 ≤ i ≤ m + n. By symmetry, we have n = n and {λm+1 , . . . , λm+n } = {λm+1 , . . . , λm+n }. By reordering the indices 1 ≤ i ≤ m + n, we may assume that λi = λi for all 1 ≤ i ≤ m + n. Now the coefficients of k2 λki , 1 ≤ i ≤ m and kλki , m + 1 ≤ i ≤ m + n in (4.1) become  (αi − αi ) ⊗ τ (v), 1 ≤ i ≤ m + n respectively, forcing αi = αi for all 1 ≤ i ≤ m + n. Furthermore, considering the coefficients of λki , 1 ≤ i ≤ m + n, and kλki , 1 ≤ i ≤ m, we obtain that φ(∂i ⊗ v) = ∂i ⊗ τ (v) for 1 ≤ i ≤ m + n and φ(ti ⊗ v) = ti ⊗ τ (v) for 1 ≤ i ≤ m. At last, for any k ∈ Z, the equation φ(dk (1 ⊗ v)) = dk (1 ⊗ τ (v)) gives φ(1 ⊗ dk (v)) = 1 ⊗ dk (τ (v)), yielding that τ (dk v) = dk τ (v) for any v ∈ V . We see that τ is a Vir-module homomorphism and hence an isomorphism of Vir-modules. This completes the proof. 2 5. New irreducible Virasoro module In this section we will compare the irreducible tensor product Vir-modules with other known non-weight irreducible modules constructed and studied in [15,14,20,18, 21,22]. We fix an irreducible tensor module T = T (λ λ, α, h; V ) as defined in (3.1), where λ1 , · · · , λm+n are distinct, αi is nonzero for 1 ≤ i ≤ m + n and deg(hi ) = 1 for all 1 ≤ i ≤ m. Note that the irreducible modules in [18,21] are special cases of modules in [22], while an irreducible module studied in [22] is a special module T with m = 0. The relationship between the modules in [22] with the modules studied in the present pare are clear from Theorem 4.2. We then compare T with other modules. For any s ∈ Z+ , l, k ∈ Z, as in [14], we denote (s)

ωl,k =

s  s i=0

i

(−1)s−i dl−k−i dk+i ∈ U (Vir).

There is an important identity related to this operator, that is, s  s i=0

i

(−1)s−i ij = 0, ∀ j, s ∈ N, j < s.

Lemma 5.1. Let the notation be as before. Then we have

(5.1)

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341

(1) For sufficiently large k ∈ N, the action of dk on T is not locally finite. (2) If V is not the 1-dimensional trivial module, then for any integer s > 4, there exist (s) l, k ∈ Z and v ∈ V such that ωl,k (1 ⊗ v) = 0. (3) Assume that m + n ≥ 2 and V is the 1-dimensional trivial module. Then for any (s) s > 4, there exists k, l ∈ Z such that ωl,k (1) = 0. Proof. (1). For any nonzero f ∈ T , it is easy to see that deg(dk (f )) > deg(f ) for any k ∈ Z, so we have f, dk (f ), d2k (f ), . . . , are linearly independent. The assertion follows easily. (2). Fix any s > 4. Take v to be a highest weight vector in V if V is a highest weight module, otherwise v can be any nonzero vector in V . From [5] and [20] (see Theorem 2.4, 2.5), we know that the vectors v, d−2 v, d−3 v, · · · , d−s−2 v are linearly independent in V . Then we take k = s + 2 and choose l ∈ N such that dl d−i v = dl v = 0 for 0 ≤ i ≤ s + 2 and l ≥ l. Hence (s) ωl,−k (1

⊗ v) =

s  s i=0

=

i

s  s i=0

i

(−1)s−i dl+k−i d−k+i (1 ⊗ v)   (−1)s−i dl+k−i (1) ⊗ d−k+i (v) + dl+k−i d−k+i (1) ⊗ v

which is nonzero. ∂ , t] as a vector space. For any s, k, l ∈ Z, s > 4, applying (3). In this case, T = C[∂ (s) ωl,k to 1 = 1 ⊗ · · · ⊗ 1 ∈ C[∂ ∂ , t], we have (s)

f = ωl,k (1) =

s  s i=0

=

(−1)s−i dl−k−i dk+i (1 ⊗ · · · ⊗ 1)

i

s  s i=0

i

s−i

(−1)

m+n 

(5.2) fl−k−i,k+i,j,j  (i, ∂ , t),

j,j  =1

where fl−k−i,k+i,j,j  (i, ∂ , t) = fl−k−i,j (i, ∂ , t)fk+i,j  (i, ∂ , t), ∀ j = j  , with

fr,j (i, ∂ , t) =

and

⎧   r 2 ⎪ ⎨ λj ∂j + rG(1) − r αj F (1) ∈ Ω(λj , αj , hj ),

1≤j≤m

⎪ ⎩ λr (∂ − rα ) ∈ Ω(λ , α ), j j j j j

m+1≤j ≤m+n

X. Liu et al. / Journal of Algebra 541 (2020) 324–344

342

fl−k−i,k+i,j,j (i, ∂ , t)    =λlj (∂j − l + k + i) ∂j + (l − k − i)G(1) − (l − k − i)2 αj F (1)   + (k + i) ∂j G(1) + (l − k − i)G2 (1) − (l − k − i)2 αj F G(1)   − (k + i)2 αj ∂j F (1) + (l − k − i)GF (1) − (l − k − i)2 αj F 2 (1) ∈ Ω(λj , αj , hj ) for 1 ≤ j ≤ m, and   fl−k−i,k+i,j,j = λlj (∂j − (k + i)αj − (l − k − i)) ∂j − (l − k − i)αj ∈ Ω(λj , αj ), for m + 1 ≤ j ≤ m + n. Noticing that fl−k−i,k+i,j,j , 1 ≤ j ≤ m +n, can be viewed as polynomials in i of degree no larger than 4 with coefficients in C[∂ ∂ , t], we see from (5.1) that s  s i=0

i

(−1)s−i

m+n 

fl−k−i,k+i,j,j (i, ∂ , t) = 0.

j

Since m +n ≥ 2, we see that the monomial ∂1 ∂2 can only occur in the terms fl−k−i,k+i,1,2 and fl−k−i,k+i,2,1 , so the coefficient of ∂1 ∂2 in f is s  s i=0

i

  + λ2l−k−i λk+i = (λ1l−k−s λk2 + (−1)s λ2l−k−s λk1 )(λ2 − λ1 )s , (−1)s−i λ1l−k−i λk+i 2 1

which is nonzero provided (λ1 /λ2 )l−2k−s + (−1)s = 0. The assertion of this part follows. 2 Theorem 5.2. The Vir-module T is not isomorphic to any non-weight irreducible module defined in [20,15,14]. Proof. The result has been proved for the case m = 0, n = 1 in [21] Theorem 3 and for the case m = 1, n = 0 in [7] Theorem 4.2. So we assume m + n ≥ 2 in the following. Given an irreducible modules S in [20], the action of dk on S is locally finite for sufficiently large k ∈ N by Theorem 2.5. So T  S by Lemma 5.1 (1). Then we consider the irreducible non-weight Vir-module Ab defined in [15]. From the proof of Theorem 9 in [14] or the argument in the proof of Corollary 4 in [21], we have (r)

ωl,k (Ab ) = 0, ∀ l, k ∈ Z, r ≥ 3.

(5.3)

Combining this with Lemma 5.1 (2) and (3), we see easily that T  Ab . Now we recall the irreducible non-weight Virasoro modules defined in [14]. Recall (j) that Vir+ = span{di | i ∈ Z+ } and Vir+ = {di | i > j} for j ∈ N. Let M be an (j) irreducible module over the Lie algebra aj := Vir+ /Vir+ such that the action of d¯j on

X. Liu et al. / Journal of Algebra 541 (2020) 324–344

343

M is injective, where d¯j is the image of dj in aj . For any β ∈ C[t±1 ] \ C, the Vir-module structure on N (M, β) := M ⊗ C[t±1 ] is defined by j 

dp · (v ⊗ tq ) = q + i=0

pi+1 ¯ (i+1)! di

 v ⊗ tp+q + v ⊗ (βtp+q ),

c · (v ⊗ tq ) = 0, ∀ v ∈ M, p, q ∈ Z. From the computation in (6.7) of [14] we see that (r)

ωl,k (N (M, β)) = 0, ∀ l, k ∈ Z, r > 2j + 2.

(5.4)

We conclude N (M, β)  T , combining (5.4) with Lemma 5.1 (2) and (3). 2 Acknowledgments This work is partially supported by NSF of China (Grant 11471294, 11971440). X.G. is also supported by the Outstanding Young Talent Research Fund of Zhengzhou University (Grant 1421315071). X. L. is partially supported by the CSC of China (Grant 201606180088). All the authors would like to thank the referees for good suggestions. References [1] P. Batra, V. Mazorchuk, Blocks and modules for Whittaker pairs, J. Pure Appl. Algebra 215 (7) (2011) 1552–1568. [2] H. Chen, X. Guo, A new family of modules over the Virasoro algebra, J. Algebra 457 (2016) 73–105. [3] H. Chen, X. Guo, K. Zhao, Tensor product weight modules over the Virasoro algebra, J. Lond. Math. Soc. (2) 83 (3) (2013) 829–844. [4] C. Conley, C. Martin, A family of irreducible representations of the Witt Lie algebra with infinitedimensional weight spaces, Compos. Math. 128 (2) (2001) 153–175. [5] B. Feigin, D. Fuchs, Verma modules over the Virasoro algebra, Funktsional. Anal. i Prilozhen. 17 (3) (1983) 91–92. [6] P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory, Springer, New York, 1997. [7] X. Guo, X. Liu, J. Wang, New irreducible tensor product modules for the Virasoro algebra, Asian J. Math. (2019), in press. [8] X. Guo, R. Lu, K. Zhao, Fraction representations and highest-weight-like representations of the Virasoro algebra, J. Algebra 387 (2013) 68–86. [9] P. Goddard, D. Olive, Kac-Moody and Viraroso algebra relation to quantum physics, Internat. J. Modern Phys. A (1986) 303–414. [10] K. Iohara, Y. Koga, Representation Theory of Virasoro Algebra, Springer Monographs in Mathematics, Spinger-Verlag London Lthd, London, 2010. [11] V. Kac, A. Raina, Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, World Sci., Singapore, 1987. [12] R. Lü, X. Guo, K. Zhao, Irreducible modules over the Virasoro algebra, Doc. Math. 16 (2011) 709–721. [13] J. Lepowsky, H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Mathematics, vol. 227, Birkhauser Boston Inc., Boston, 2004. [14] G. Liu, R. Lü, K. Zhao, A class of simple weight Virasoro modules, J. Algebra 424 (2015) 506–521. [15] R. Lü, K. Zhao, Irreducible Virasoro modules from irreducible Weyl modules, J. Algebra 414 (2014) 271–287.

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