This paper introduces global and local Weyl modules for queer Lie superalgebras tensorized with a commutative algebra, establishing their universality, finite dimensionality under certain conditions, and a tensor product property.
Contribution
It defines and analyzes Weyl modules for queer Lie superalgebras, proving their universality, finite dimensionality, and tensor product behavior under specific assumptions.
Findings
01
Global Weyl modules are universal highest weight objects.
02
Local Weyl modules are finite dimensional under certain conditions.
03
Tensor product property holds for local Weyl modules.
Abstract
We define global and local Weyl modules for q⊗A, where q is the queer Lie superalgebra and A is an associative commutative unital C−algebra. We prove that global Weyl modules are universal highest weight objects in certain category upto parity reversing functor Π. Then with the assumption that A is finitely generated and with a special technical condition which simple root system of q satisfy it is shown that the local Weyl modules are finite dimensional. Further they are universal highest map-weight objects in certain category upto Π. Finally we prove a tensor product property for local Weyl modules.
\displaystyle[e_{i}^{\prime},f_{j}]=\delta_{ij}(k_{i}^{\prime}-k_{i+1}^{\prime}),[k_{l}^{\prime},e_{i}]=\alpha_{i}(k_{l})e_{i}^{\prime}\mbox{ for $i,j\in I,l\in J$},
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TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Advanced Algebra and Geometry
Full text
Weyl modules for queer Lie superalgebras
Saudamini Nayak
Department of Mathematics, National Institute of Technology Calicut, NIT Campus P.O., Kozhikode-673 601, India
We define global and local Weyl modules for q⊗A, where q is the queer Lie superalgebra and A is an associative commutative unital C−algebra. We prove that global Weyl modules are universal highest weight objects in certain category upto parity reversing functor Π. Then with the assumption that A is finitely generated and with a special technical condition which simple root system of q satisfy it is shown that the local Weyl modules are finite dimensional. Further they are universal highest map-weight objects in certain category upto Π. Finally we prove a tensor product property for local Weyl modules.
The theory of Lie superalgebras and their representations have a wide range of applications in many areas of physics and mathematics such as describing supersymmetry, in string theory, conformal field theory and number theory to name a few. In 1977, Kac classified the simple Lie superalgebras g over C [KAC77b]. These are divided into three groups namely: basic Lie superalgebras (which means the classical and exceptional series), the strange ones (often also called periplectic and queer) and the ones of Cartan type. Kac also classified the simple finite dimensional representations of the basic classical Lie superalgebras in [KAC77a, KAC77b, KAC78]. In recent times there has been much interest in understanding finite dimensional modules for the Lie superalgebra g⊗A where g is simple finite dimensional Lie superalgebra and A is commutative associative algebra with unit over complex numbers C. For example, If we take A=C[X], then the Lie superalgebra g⊗C[X] is called a current superalgebra. If we take A=C[X,X−1], then g⊗C[X,X−1] is called a loop superalgebra. If we take A=C[X1±1,…,Xn±1], then g⊗C[X1±1,…,Xn±1] is called a multiloop superalgebra. The classification of finite dimensional irreducible modules for multiloop superalgebra is obtained in [ER13, ERZ04]. In more general setting, the irreducible finite dimensional modules were classified in [SAV14, CMS16].
The Weyl modules play important role in the representation theory of infinite-dimensional Lie algebras. Chari and Pressley [CP01] introduced Weyl modules (global and local) for the loop algebra g⊗C[X±1], where g is simple Lie algebra over C and proved that these modules are indexed by dominant integral weights of g and are closely related to certain irreducible modules for quantum affine algebras. Feigin and Loktev [FL04] extended the notion of Weyl modules to the higher- dimensional case, i.e., instead of the loop algebra they worked with the Lie algebra g⊗A where A is the coordinate ring of an algebraic variety and obtained analogs of some of the results of [CP01]. Later in [CFK10], Chari et. al., consider a more general functorial approach to Weyl modules associated to the algebra g⊗A where A is commutative associative unital algebra over C. Also twisted versions of Weyl modules have been defined and investigated in [CFS08, FMS13]. The Weyl modules for equivariant map algebras has been studied in [FMS15].
However, in super setting the study of Weyl modules has less developed than the corresponding theory in Lie algebras. At first Zhang in [ZHA14], define and study the Weyl modules in the spirit of Chari-Pressley for a quantum analogue in the loop case for g=sl(m,n). In [CLS19], Calixto, Lemay and Savage study Weyl modules for Lie superalgebras of the form g⊗CA, where A is an associative commutative unital C-algebra and g is a classical Lie superalgebra or sl(n,n),n≥2. Particularly, they define Weyl modules (global and local) for the Lie superalgebras g⊗CA and prove that global Weyl modules are universal highest weight objects in a certain category and local Weyl modules are finite dimensional. Furthermore recently, Bagci, Calixto and Macedo [BCM19] study Weyl modules (global and local) and Weyl functors for the superalgebras g⊗A, where g is either sl(n,n),n≥2, or any finite dimensional simple Lie superalgebra not of type q(n), and A is an associative, commutative algebra with unit.
The goal of this paper is to study global and local Weyl modules for Lie superalgebras g⊗CA, where A is an associative commutative unital C-algebra and g is the queer Lie superalgebra. To prove our results, we follow [CLS19].
2. Preliminaries
Throughout the paper ground field will be the field of complex numbers C. By Z≥0 and Z>0 we denote the nonnegative integers and strictly positive integers, respectively. Also we set Z2=Z/2Z. All supervectorspaces, superalgebras, tensor products etc. are defined over C. In this section, we review some facts about associative commutative algebras and queer Lie superalgebras that we need in the sequel.
2.1. Basic definitions
A vector space V is called a supervectorspace if V is Z2-graded, i.e., V=V0ˉ⊕V1ˉ. The dimension of the vector space V is the tuple (dimV0ˉ∣dimV1ˉ). The parity of a homogeneous element v∈Vi is denoted by ∣v∣=i,i∈Z2. An element in V0ˉ is called even, while an element in V1ˉ is called odd. A subspace of V is a Z2-graded vector space W=W0ˉ⊕W1ˉ⊆V with compatible Z2-gration, i.e., Wi⊆Vi, for i∈Z2. We denote by Cm∣n the supervectorspace Cm⊕Cn, where the first summand is even and the second summand is odd.
Given two supervectorspaces V and W, a linear mapping T:V⟶W is homogeneous of degreed∈Z2 if T(V)i⊂Wi+d for i∈Z2. The map T is called even (respectively, odd) if d=0ˉ (respectively, d=1ˉ). Consider the vector space of all linear transformations from V to W denoted as Hom(V,W) is Z2-graded with
[TABLE]
where d∈Z2 . Define End(V):=Hom(V,V). The supervectorspaces and homogeneous mappings define a category. If we restrict the mappings to homogeneous even mappings we obtain an abelian category say Vec.
We denote by Π the parity change functor, on category Vec which is defined as
[TABLE]
and Πf=f for V∈\mboxVec and f:V⟶W∈\mboxVec. For example consider if V is of dimension (1∣0) then ΠV has dimension (0∣1). We assume the field C is of homogeneous even dimension.
A supervectorspace A=A0ˉ⊕A1ˉ, equipped with a bilinear associative multiplication satisfying AiAj⊆Ai+j, for i,j∈Z2 is called a Z2-graded associative algebra or, associative superalgebra. For instance End(V) is an associative superalgebra. A homomorphism between two superalgebras A and B i.e., f:A⟶B, is a even linear map(f(Ai)⊆Bi for i∈Z2) with f(ab)=f(a)f(b). The tensor product A⊗B is a superalgebra, with underlying vector space is the tensor product of supervectorspaces of A and B, with the induced Z2-grading and multiplication is given by (a1⊗b1)(a2⊗b2)=(−1)∣a2∣∣b1∣a1a2⊗b1b2 for homogeneous elements ai∈A and bi∈B. A moduleM over a superalgebra A is always understood in the Z2-graded sense, that is M=M0ˉ⊕M1ˉ such that AiMj⊆Mi+j, for i,j∈Z2. Subalgebras and ideals of superalgebras are Z2-graded subalgebras and ideals. A superalgebra that has no non-trivial ideal is called simple. A homomorphism between A-modules M and N is an even linear map f:M⟶N( i.e., f(Mi)⊆Ni for i∈Z2), with f(am)=af(m), for all a∈A,m∈M.
2.2. Lie superalgebras
Definition 2.1** (Lie superalgebra).**
A Lie superalgebra is a Z2-graded vector space g=g0ˉ⊕g1ˉ with a bilinear multiplication [⋅,⋅] satisfying the following axioms:
(a)
The multiplication respects the grading: [gi,gj]⊆gi+j for all i,j∈Z2.
2. (b)
Skew-supersymmetry: [a,b]=−(−1)∣a∣∣b∣[b,a], for all homogeneous elements a,b∈g.
3. (c)
Super Jacobi Identity: [a,[b,c]]=[[a,b],c]+(−1)∣a∣∣b∣[b,[a,c]], for all homogeneous elements a,b,c∈g.
Example 2.2**.**
Let A be any associative superalgebra. Then we can make A into a Lie superalgebra by defining [a,b]:=ab−(−1)∣a∣∣b∣ba for all homogeneous elements a,b∈A and extending [.,.] by linearity. We call this is the Lie superalgebra associated with A. A concrete example is the general linear Lie superalgebra gl(V) associated with associative superalgebra End(V) of all linear operators on a Z2-graded vectorspace V.
A homomorphism ρ between Lie superalgebras is a map which preserves the structure in them. Precisely ρ:g⟶g1 is an even linear map with ρ([x,y])=[ρx,ρy] for all x,y∈g.
Definition 2.3**.**
A representaion of Lie superalgebra g is a Lie superalgebra homomorphism ρ:g⟶gl(V),i.e., ρ is an even linear with ρ[x,y]=ρ(x)ρ(y)−(−1)∣x∣∣y∣ρ(y)ρ(x).
Alternatively V is called g-module and V is irreducible if there are no submodule other than [math] and V itself.
Lemma 2.4**.**
[SAV14]**
Suppose g is a Lie superalgebra and V is an irreducible g-module such that Iv=0 for some ideal I of g and non-zero vector v∈V. Then IV=0.
Given a Lie superalgebra g, we will denote by U(g) its universal enveloping superalgebra. The universal enveloping superalgebra U(g) is constructed from the tensor algebra T(g) by factoring out the ideal generated by the elements [u,v]−u⊗v+(−1)∣u∣∣v∣v⊗u, for homgeneous elements u,v in g. Now we state an analogous of PBW Theorem in super setting, which ensures that g↦U(g) is an inclusion by precisely giving a basis for U(g).
Let g=g0ˉ⊕g1ˉ be a Lie superalgebra. If x1,…,xm be a basis of g0ˉ and y1,…,yn be a basis of g1ˉ, then the monomials
[TABLE]
form a basis of U(g). In particular, if g is finite dimensional and g0ˉ=0, then U(g) is finite dimensional.
2.3. The queer Lie superalgebra
Let V=V0ˉ⊕V1ˉ be a supervectorspace with dimV0ˉ=dimV1ˉ. Choose P∈\mboxEnd(V)1ˉ such that P2=−1. The subspace
[TABLE]
is a subalgebra of gl(V) called the, queer Lie superalgebra. If V=Cn∣n, then with a homogeneous basis we identify gl(V) with gl(n∣n).
Now for explicit realization of the queer Lie superalgebra q(n) in terms of matrices, set
[TABLE]
Then, for X∈gl(n∣n), we have X∈q(n) if and only if
XP−(−1)∣X∣PX=0 holds. Hence q(n) consisting of matrices of the form
[TABLE]
where A and B arbitrary n×n matrices with
[TABLE]
From now on we denote q(n)=:q.
A subalgebra of q is called a Cartan subalgebra if it is a self-normalizing nilpotent subalgebra. Every such subalgebra has a non-trivial odd part.
Denote by N−,H,N+ respectively the strictly lower triangular, diagonal and strictly upper triangular matrices in gl(n). Then we define
such that n−,n+,h are graded subalgebra of q with n± nilpotent. The subalgebra h is called the standard Cartan subalgebra of q.
Given any Lie superalgebra a, the map
x⟶x⊗1⊕1⊗x,x∈a extends to an algebra homomorphism U(a)⟶U(a)⊗U(a). By the PBW Theorem (see Lemma 2.5), we know that if b and c are subalgebras of a such that a=b⊕c as vector spaces
[TABLE]
Thus, from Lemma 2.6, we obtain the triangular decomposition of U(q):
[TABLE]
2.4. Root system for q.
We fix h=h0ˉ⊕h1ˉ to be standard Cartan subalgebra of q, is given by
[TABLE]
where
[TABLE]
and Ei,j is the n×n matrix having 1 at the (i,j)-entry and [math] elsewhere. The Cartan subalgebra h has a nontrivial odd part h1ˉ and hence is not abelian, as [h0ˉ,h]=0 and [h1ˉ,h1ˉ]=h0ˉ. Note that all Cartan subalgebra of q are conjugate to h. For 1≤i=j≤n, we set
[TABLE]
The set {ei,j,ei,j′∣1≤i,j≤n} is a homogeneous linear basis for q. The even subalgebra is q0ˉ is spanned by {ei,j∣1≤i,j≤n} and hence is isomorphic to the general linear Lie algebra gl(n) and odd space q1ˉ is is isomorphic to the adjoint module.
Let {ϵ1,…,ϵn} be the basis of h0ˉ∗
dual to {k1,⋯,kn} defined as \epsilon_{i}(\left(\begin{array}[]{@{}c|c@{}}h&0\\
\hline\cr 0&h\end{array}\right))=a_{i}, for any diagonal matrix h with diagonal entries (a1,a2,⋯,an). We denote hi:=ki−ki+1 for 1≤i≤n−1. Given α∈h0ˉ∗, let
[TABLE]
Note that q0=h. We call α=0 a root if qα=0. The set Φ={α∣qα=0} is called the root system of q. A root α is called even root if qα∩q0ˉ=0 and it is called odd if qα∩q1ˉ=0. The root system Φ=Φ0ˉ∪Φ1ˉ of q has identical even and odd parts where Φ0ˉ denote the set of even roots and Φ1ˉ denote the set of odd roots. Namely Φ0ˉ=Φ1ˉ={ϵi−ϵj∣1<i=j<n}. For each root α=ϵi−ϵj,1≤i=j≤n, we have root spaces has dimension (1∣1),
[TABLE]
and
[TABLE]
is the root space decomposition of q.
A root α is called positive (resp. negative) if qα∩n+=0 (resp. qα∩n−=0). We denote by Φ+ (resp. Φ−) the subset of positive (resp. negative) roots. Denote by Δ the set of simple roots. Thus,
[TABLE]
Hence,
[TABLE]
A maximal solvable subalgebra of q is called Borel subalgebra b. Borel subalgebra of q is conjugate to the standard Borel subalgebra b+=h⊕n+ of q.
Set αi:=ϵi−ϵi+1 and the root space qαi is spanned by
[TABLE]
while q−αi is spanned by
[TABLE]
Hence n+ is spanned by ei,ei′
and n− is spanned by fi,fi′ and the standard Borel is spanned by ei,ei′,hi,ki′.
For α=ϵi−ϵj∈Φ0ˉ+, let sα:h0ˉ∗⟶h0ˉ∗ be the corresponding reflection and is defined by
[TABLE]
The Weyl group of q is the Weyl group W of q0ˉ generated by sα where α∈Φ0ˉ+which is the symmetric group Sn in n letters.
Let I:={1,2,…,n−1} and J:={1,2,…,n}.
Proposition 2.7**.**
[GJKM10]**
The Lie superalgebra q generated by the elements ei,ei′,fi,fi′ for i∈I, h0ˉ and kj′ for j∈J with the following defining relations
[TABLE]
The simple roots Δ of q satisfy the following property:
[TABLE]
This is true, as every root of q is even as well as odd.
2.5. Clifford Algebra.
Definition 2.8** (Clifford Algebra).**
Let V be a finite dimensional vector space and f:V×V→C be a symmetric bilinear form. We call the pair (V,f) a quadratic pair. Let I be the ideal of the tensor algebra T(V) generated by the elements
[TABLE]
and set \mboxCliff(V,f)=T(V)/I. The algebra \mboxCliff(V,f) is alled the Clifford algebra of the pair (V,f) over C.
Remark 2.9** ([HUS94], Ch. 12, Def. 4.1 and Theorem 4.2).**
For a quadratic pair (V,f), there exists a linear map θ:V→\mboxCliff(V,f) such that the pair (\mboxCliff(V,f),θ) has the following universal property: For all linear maps η:V→A such that η(v)2=f(v,v)1A for all v∈V, where A is a unital algebra, there exists a unique algebra homomorphism η′:\mboxCliff(V,f)→A such that η′∘θ=η, in other words, we have the following commutative diagram.
Clifford algebra have a natural superalgebra structure. In fact, T(V) possess a Z2-grading such that I is homogeneous, so the grading descends to \mboxCliff(V,f). Thus resulting superalgbera \mboxCliff(V,f) is sometimes called the Clifford superalgebra. When f is known from the context, we shall write \mboxCliff(V) instead of \mboxCliff(V,f).
For λ∈h0ˉ∗, define an even super antisymmetric bilinear form Fλ on h1ˉ, by setting Fλ(u,v)=λ([u,v]) and denote Eλ:=h1ˉ/kerFλ. Let \mboxCliff(λ) be the Clifford superalgebra with respect to quadratic pair (Eλ,Fλ) and \mboxCliff(λ) is endowed with a canonical Z2-grading. By definition we have an isomorphism of superalgebras
[TABLE]
where Iλ denoted the ideal of U(h) generated by kerFλ and a−λ(a) for a∈h0ˉ.
Let h1ˉ′⊆h1ˉ be a maximal isotropic subspace with respect to Fλ and define the Lie superalgebra h′:=h0ˉ⊕h1ˉ′. Let Cvλ, be the one-dimensional h0ˉ-module defined by hvλ=λ(h)vλ for all h∈h0ˉ, extends to an h′-module by setting h1ˉ′vλ=0. Then the induced module
[TABLE]
is an irreducible h-module. If \mboxIndh′hCvλ is a finite dimensional irreducible h-module, then \mboxIndh′hCvλ is a finite dimensional irreducible module over \mboxCliff(λ) via the pullback through (2.9).
We may consider \mboxCliff(λ) as the associative C-algebra generated by the identity 1=1+Iλ and tiˉ:=ki+Iλ satisfying the relations
[TABLE]
Let S=⊕i=1nCtiˉ and λ=(λ1,…,λn)∈Cn and denote by Bλ:S×S→C the symmetric bilinear form defined by Bλ(tiˉ,tjˉ)=δijλi. Let \mboxCliffS(λ) be the unique up to isomorphism Clifford algebra associated to S and Bλ. Now define S(λ):=S/kerBλ and denote by βλ the restriction of Bλ on S(λ). Let Nλ={i∣λi=0},Zλ={j∣λj=0} and ℓ=#Nλ. Set
[TABLE]
One can see that kerBλ=⊕j∈ZλCtj and \mboxCliffS(λN)=⊕i∈NλCti is the Clifford algebra corresponding to (S(λ),βλ). Further,
[TABLE]
Here ⋀U denotes the exterior algebra of the vector space U. Thus by the isomorphisms in (2.11), every \mboxCliffS(λ)-module can be considered as a \mboxCliffS(λN)-module under the embedding
[TABLE]
Then one can easily prove the following lemma.
Lemma 2.10**.**
Let M be an irreducible \mboxCliffS(λ) module. Then M is an irreducible \mboxCliffS(λN)-module and tiv=0 for every i∈ZN.
2.6. Highest weight modules over q.
From now on, for a superalgebra A, an A-module will be understood as an A-supermodule. A q-module M is called a weight module if it admits a weight space decomposition
[TABLE]
An element μ∈h0ˉ∗ such that Mμ=0 is called a weight of M and Mμ is called weight space. The set of all weights of M is denoted by wt(M).
Definition 2.11**.**
A weight module M is called a highest weight module with highest weight λ∈h0ˉ∗ if Mλ is finite dimensional and satisfies the following conditions:
(a)
M is generated by Mλ,
2. (b)
eiv=ei′v=0 for all v∈Mλ,i∈I.
Definition 2.12**.**
Let Λ0ˉ+ and Λ+ be the set of gl(n)-dominant integral weights and the set of q-dominant integral weights respectively, given by
The maximal nilpotent subalgebra n+ of b+ acts on v trivially.
2. (b)
There exists a unique weight λ∈h0ˉ∗ such that v is endowed with a canonical left \mboxCliff(λ)-module structure and λ determines v up to the parity reversing functor Π.
3. (c)
For all h∈h0ˉ,v∈v, we have hv=λ(h)v.
Remark 2.14**.**
From Proposition 2.13, we know that the dimension of the highest weight space of a highest weight q-module with highest weight λ is the same as the dimension of an irreducible \mboxCliff(λ)-module. On the other hand all irreducible \mboxCliff(λ)-modules have the same dimension (see, for example, [ABS64, Table 2]). Thus the dimension of the highest weight space is constant for all highest weight modules with highest weight λ.
Proposition 2.15**.**
Let λ∈Λ+ and V(λ) be the irreducible highest weight q-module generated by an irreducible finite dimensional b+-module v. Then fiλ(hi)+1v=0, for all v∈v and i∈I.
Proof.
Note that one can easily show by induction that for k∈Z≥0,
[TABLE]
Since n+v=0 for all v∈v, then
[TABLE]
If k=λ(hi)+1, one can see that eifiλ(hi)+1v=0. For i=j, as [ej′,fi]=0=[ej,fi], we have ejfiλ(hi)+1v=0=ej′fiλ(hi)+1v.
Now suppose that ei′fiλ(hi)+1v=0. Since [ei,ei′]=0 for ∣i−j∣=1, so
[TABLE]
We get ei(ei′fiλ(hi)+1v)=0 and ei′(eifiλ(hi)+1v)=0. Similarly, as [ei′,ej′]=0 for ∣i−j∣=1, we get
[TABLE]
Also, for i=j, we have
[TABLE]
If λ(hi)≥1, then weight of the weight vector ei′fiλ(hi)+1vis λ−λ(hi)αi<λ. Thus, ei′fiλ(hi)+1v would generate a nontrivial proper submodule of V(λ), which contradicts the irreducibility of V(λ).
If λ(hi)=0, then λi=λi+1=0 and since v∈v, so by Lemma 2.10, we get ki′v=ki+1′v=0. Now
[TABLE]
Therefore, in any case ei′fiλ(hi)+1v=0.
Similarly, if fiλ(hi)+1v=0, it would generate a non-trivial proper submodule of V(λ). Hence, fiλ(hi)+1v=0 for all v∈v.
∎
Definition 2.16**.**
Let v(λ) be a finite dimensional irreducible b+-module determined by λ up to Π. The Weyl moduleW(λ) of q with highest weight λ is defined to be
[TABLE]
Note that in the above definition, the structure of W(λ) is determined by λ up to Π.
For any weight λ, W(λ) has a unique maximal submodule N(λ).
2. (b)
For each finite dimensional simple q-module M, there exists a unique weight λ∈Λ0ˉ+ and a surjective homomorphism W(λ)→M (one of the two possible W(λ)).
3. (c)
The irreducible quotient L(λ):=W(λ)/N(λ) is finite dimensional if and only if λ∈Λ+.
Remark 2.18**.**
For λ∈Λ+, up to isomorphism, there exists two simple finite dimensional modules w.r.t. highest weight λ namely
L(λ) and ΠL(λ), where Π is the parity change functor.
Let P(λ)={μ∈h0ˉ∗∣Mμ=0}. Let Q (resp. Q+) be the integer span (resp. Z>0-span ) of the simple roots. Denote by ≤ the usual partial order on P(λ),
[TABLE]
Since q0ˉ=gl(n) is reductive Lie algebra, for each even simple root αi we can choose elements ei∈qαi,fi∈q−αi, and hi∈h0ˉ, such that the subalgebra generated by these elements is isomorphic to sl(2), with these elements satisfying the relations for the standard Chevalley generators. In this case, we say that the set {ei,fi,hi} is an sl(2)-triple.
Denote the irreducible highest weight q-module with highest weight λ∈h0∗, by L(λ) which is unique upto Π and consider the weight space decomposition L(λ)=⨁μ∈h0ˉ∗L(λ)μ.
Definition 2.19** (The module Lˉ(λ)).**
For λ∈Λ+, we define Lˉ(λ) (up to Π) to be the q-module generated by L(λ)λ with defining relations
[TABLE]
Proposition 2.20**.**
The module Lˉ(λ) is finite dimensional for all λ∈Λ+.
Proof.
Let x1,…,xm and y1,…,ym be a homogeneous basis of q0ˉ and q1ˉ, respectively. Then by Lemma 2.5, the monomials
[TABLE]
form a basis of U(q). Since {y1b1⋯ymbm∣bj=0,1} is a finite set, it is enough to show U(q0ˉ)L(λ)λ is finite dimensional.
Consider irreducible q0ˉ-module V(λ) with highest weight λ∈Λ+. Since q0ˉ=gl(n) is reductive Lie algebra, λ(hi)∈N for each even simple root αi with i∈I. Hence we have V(λ) is finite dimensional. Note the centre Z(q0ˉ) acts as a scalar on V(λ). Hence V(λ) is isomorphic to q0ˉ-module generated by a vector uλ with defining relations
[TABLE]
Now U(q0ˉ)L(λ)λ⊂∑kλ∈L(λ)λU(q0ˉ)kλ⊆Lˉ(λ) be the q0ˉ-submodule of Lˉ(λ). For any kλ∈L(λ)λ, we known that U(q0ˉ)kλ is a highest weight module over q0ˉ with highest weight λ satisfying fαλ(hα)+1kλ=0. Thus, U(q0ˉ)kλ is cyclic and kλ satisfies (2.12) for any kλ∈L(λ)λ. Then there exists a unique surjective homomorphism of q0ˉ-submodules satisfying
[TABLE]
for all x∈U(q0ˉ). Since ψ is surjective and V(λ) is finite dimensional, it follows that U(q0ˉ)kλ finite dimensional for any kλ∈L(λ)λ and hence U(q0ˉ)L(λ)λ is finite dimensional.
∎
Proposition 2.21**.**
For highest weight λ∈Λ+, consider finite dimensional highest weight q-module V. Then there exists a surjective homomorphism of q-modules ψ1:Lˉ(λ)⟶V up to Π. Moreover, there exists a unique upto Π submodule W of Lˉ(λ) such that V≅Lˉ(λ)/W or V≅Π(Lˉ(λ))/Π(W)
as q-modules.
Proof.
Consider the highest weight q-module V, with highest weight λ, is generated by an irreducible b+-module v. For vλ∈v the first two relations in (2.12) hold. Since V is finite dimensional, then by Proposition 2.15, fiλ(hi)+1vλ=0, for all vλ∈v,i∈I. Also, by Remark 2.14, the dimension of the highest weight space Lˉ(λ)λ is equal to the dimension of the highest weight space v. Thus the map ψ1:Lˉ(λ)⟶V induced by Lˉ(λ)λ⟶v, is a surjective homomorphism of q-modules up to Π. Since module homomorphism preserve weight spaces, the kernel of ψ1 is unique upto Π and say ker(ψ1)=W.
∎
Since every simple finite dimensional q-module is a highest weight module with highest weight λ∈Λ+, Proposition 2.21 applies to all simple finite dimensional q-modules.
3. Global Weyl modules
Let A denote a finitely generated commutative associative unital algebra and q(n)=:q with n≥2. Take q⊗A, with Z2-grading is given by (q⊗A)j=qj⊗A,j∈Z2. Then q⊗A with bracket of any two homogeneous elements
[TABLE]
is a Lie superalgebra.
Further, we identify q with a subalgebra of q⊗A via the isomorphism q≅q⊗C and the inclusion q⊗C⊆q⊗A. Let I be the full subcategory of the category of q modules whose objects are those modules that are isomorphic to direct sums of irreducible finite dimensional q0ˉ modules. Note that if V∈I then any element of V lies in a finite dimensional q0ˉ submodule of V. Let Iq⊗A,q0ˉ denote the full subcategory of the category of q⊗A-modules whose objects are the q⊗A-modules whose restriction to q0ˉ lies in I.
Lemma 3.1**.**
Category I is closed under taking submodules, quotients, arbitrary direct sums and finite tensor products.
Regard U(q⊗A) as a right q-module via right multiplication and given a left q-
module V (up to Π), set
[TABLE]
Then PA(V) is left q⊗A-module by left multiplication and we have an isomorphism of vector spaces
[TABLE]
where A+ is a vector space complement to C⊆A. Note that PA(V) is defined up to Π.
Lemma 3.2**.**
Let V be a q-module whose restriction to q0ˉ lies in I. Then PA(V)∈Iq⊗A,q0ˉ.
Proof.
Note that q is finitely semisimple, that is, it is isomorphic to direct sums of irreducible finite dimensional q0ˉ-modules via adjoint representation. So q∈I and q⊗A≅q⊕dim(A) as q0ˉ-modules. By Lemma 3.1, q⊗A is a finitely semisimple q0ˉ-module, that is, it is isomorphic to direct sums of irreducible finite dimensional q0ˉ modules. Again, by Lemma 3.1, as I is closed under finite tensor product and arbitrary direct sum so U(q⊗A) is finitely semisimple q0ˉ-module. Also as V∈I, we have U(q⊗A)⊗CV is a finitely semisimple q0ˉ-module. Consider the map
[TABLE]
For every u∈U(q⊗A),v∈V and every homogenenous element x∈q, we have
[TABLE]
Hence, the map in (3.3) is a surjective homomorphism of q-modules. This shows that PA(V) is a quotient of U(q⊗A)⊗CV. Hence the lemma follows from Lemma 3.1.
∎
Proposition 3.3**.**
If λ∈Λ+, then PA(Lˉ(λ)) (up to Π) is generated as a left U(q⊗A)-module by L(λ)λ satisfying the following relations:
[TABLE]
Proof.
Note that pλ=1⊗kλ∈PA(Lˉ(λ)) when kλ∈Lˉ(λ). Since kλ satisfies the relation in (2.12), 1⊗kλ satisfies relations (3.4). We have to check these are all the relations. To do this, suppose that M is the highest weight q⊗A-module with highest weight λ, generated by an irreducible b+-module m such that
[TABLE]
By Remark 2.14, the dimension of the highest weight space PA(Lˉ(λ)λ) is equal to the dimension of the highest weight space m. Then we have a surjective homomorphism (up to Π) of q⊗A-modules ϕ:M⟶PA(Lˉ(λ)) induced by ϕ(m)=PA(Lˉ(λ)λ).
From (3.4), let m∈m generates q-submodule M′ of M which is isomorphic to Lˉ(λ). Thus, the map ψ:PA(Lˉ(λ))⟶M induced by Lˉ(λ)→M′ is a surjective homomorphism (up to Π). Since ϕ=ψ−1, we have M≅PA(Lˉ(λ)) or M≅Π(PA(Lˉ(λ))).
∎
For ν∈Λ+ and M∈Iq⊗A,q0ˉ, let Mν be the unique maximal q⊗A-module quotient of M satisfying
[TABLE]
or equivalently,
[TABLE]
Let Iq⊗A,q0ˉν be the full subcategory of Iq⊗A,q0ˉ whose objects are the left U(q⊗A)-modules M∈Iq⊗A,q0ˉ such that Mν=M.
Definition 3.4** (Global Weyl module).**
Let λ∈Λ+. We define the global Weyl module (up to Π) associated to λ∈Λ+ to be
[TABLE]
From
[TABLE]
we note that wλ is the image of kλ in WA(λ).
The next result gives a description of global Weyl modules by generators and relations.
Proposition 3.5**.**
For λ∈Λ+, the global Weyl module WA(λ) (up to Π) is generated by WA(λ)λ with defining relations
[TABLE]
and wλ in WA(λ)λ.
Proof.
Let for any kλ∈Lˉ(λ)λ, wλ is the image of kλ in WA(λ). Note that (qα⊗A)Vμ⊆Vμ+α for all α∈Δ,μ∈h0ˉ∗. Since the weights of WA(λ) lie in λ−Q+, it follows that (n+⊗A)wλ=0. The remaining relations are satisfied by wλ since they are satisfied by kλ. To prove that these are the only relations, let W′(λ) be the highest weight module generated by an irreducible b+-module m with relations
[TABLE]
Then we have a surjective homomorphism ϕ:W′(λ)→WA(λ) induced by ϕ(m)=WA(λ)λ. Note that the relations (3.8) implies W′(λ) is a weight module. So m∈m generates q-submodule W′′ of W′(λ) which is isomorphic to Lˉ(λ). Thus, the map ψ:PA(Lˉ(λ))⟶W′ induced by Lˉ(λ)→W′′ is a surjective homomorphism. Further, q-weights of W′(λ) are bounded above by λ, it follows that ψ induces a map WA(λ)→W′(λ) inverse to ϕ.
∎
Theorem 3.6**.**
Any global Weyl module with highest weight λ in the category I(q⊗A,q0ˉ) is isomorphic to WA(λ) or Π(WA(λ)). Furthermore, if any object V(λ)∈I(q⊗A,q0ˉ) is generated by an irreducible b+-module v of weight λ, then there exists a surjective homomorphism form WA(λ) to V(λ) (up to Π).
Proof.
Let V(λ)∈I(q⊗A,q0ˉ) be highest weight q⊗A-module with highest weight λ is generated by an irreducible b+-module v. Then by definition
[TABLE]
Since the q0ˉ-module generated by v is finite dimensional, then by Proposition 2.15, fiλ(hi)+1v=0 for all v∈v and αi∈Δ(q0ˉ),i∈I. Thus, by Proposition 3.5, we have a surjective homomorphism WA(λ)→V(λ) induced by WA(λ)λ→v (up to Π).
Suppose WA′(λ) is another object in I(q⊗A,q0ˉ) that is generated by an irreducible b+-module m with highest weight λ and admits a surjective homomorphism to any object of I(q⊗A,q0ˉ) which is also generated by highest weight vectors of weight λ. In particular, we have a surjective homomorphism ϕ:WA′(λ)→WA(λ). It follows from PBW theorem that WA(λ)λ=U(h⊗A+)⊗Cm. Hence the elements of this weight space that generate WA(λ) are the C-multiples of m for all m∈m. Thus, we have ϕ(m)=WA(λ)λ. Now the relation (3.7) hold for all m∈m. Thus, there exists a homomorphism ψ:WA(λ)→WA′(λ) induced by WA(λ)λ→m (up to Π) and hence WA(λ)≅WA′(λ) or Π(WA(λ))≅WA′(λ).
∎
4. Local Weyl modules
An ideal I of A is said to be of finite co-dimension if dimA/I is finite. Let
[TABLE]
For any ψ∈L(h⊗A), there exists unique, up to Π, simple finite dimensional h⊗A-module H(ψ) such that xv=ψ(x)v, for all x∈h0ˉ⊗A and v∈H(ψ) (see [CMS16, Th. 4.3]). Define an action of b⊗A on H(ψ) by n+⊗A to act by zero. Then consider the induced module
[TABLE]
which is a highest weight module. Notice that Vˉ(ψ) is defined up to the parity reversing functor Π. Further, a submodule of Vˉ(ψ) is proper if and only if its intersection with H(ψ) is zero. Moreover, any q⊗A-submodule of a weight module is also a weight module. Hence, if W⊂Vˉ(ψ) is proper q⊗A-submodule, then
[TABLE]
Therefore, Vˉ(ψ) has a unique maximal proper submodule N(ψ)
[TABLE]
is an irreducible highest weight q⊗A-module. So, every finite dimensional irreducible q⊗A-module is isomorphic to V(ψ) for some ψ∈L(h⊗A)(see [CMS16, Proposition 5.4]). Note that the highest weight space of V(ψ)≅h⊗AH(ψ) or V(ψ)≅h⊗AΠ(H(ψ)).
Definition 4.1**.**
Let ψ∈L(h⊗A) such that λ=ψ∣h0ˉ∈Λ+. We define the local Weyl moduleWA\mboxloc(ψ) associated to ψ upto Π to be the q⊗A-module generated by H(ψ) with defining relations
[TABLE]
A q⊗A-module generated by H(ψ) is called highest map-weight module with highest map-weightψ if
[TABLE]
A vector wψ∈H(ψ) is called highest map-weight vector of highest map-weight ψ.
Recall the even part of queer Lie superalgebra q(=q0ˉ⊕q1ˉ) is isomorphic to gl(n+1). For each α∈Φ0ˉ+ we have an sl(2)-triple xα,yα,hα.
Lemma 4.2**.**
Suppose ψ∈L(h⊗A) such that λ=ψ∣h0ˉ∈Λ+. Consider q⊗A-module generated by an irreducible module H(ψ). If αi∈Φ0ˉ+, then fiλ(hi)+1wψ=0 for all wψ∈H(ψ),i∈I.
Proof.
Let h=h0ˉ⊕h1ˉ be the Cartan subalgebra of q and h⊗A be the Cartan subalgebra of q⊗A. Since h⊂h⊗A is a subalgebra, hwψ=λ(h)wψ for all h∈h0ˉ,wψ∈H(ψ) and h1ˉwψ=0. So λ is the highest weight with highest weight vector wψ. The vector fiλ(hi)+1wψ has weight λ−(λ(hi)+1)αi. On the other hand, by Theorem 3.6, WA\mboxloc(ψ) is a quotient of the global Weyl module WA(λ) up to Π, hence is direct sum of finite-dimensional irreducible q0ˉ-modules. This implies the weights of WA\mboxloc(ψ) are invariant under the action of Weyl group of q0ˉ. Let sαi denotes the reflection associated to the root αi. Then sαi(λ−(λ(hi)+1)=λ+αi, and this implies fiλ(hi)+1wψ=0.
∎
Let u be an indeterminate and for a∈A,α∈Φ0ˉ+, define a power series with coefficients in U(h⊗A) by
[TABLE]
For i∈N, let pa,αi be the coefficient of ui in pa,α(u).
Lemma 4.3**.**
Suppose r∈N, a∈A and α∈Φ0ˉ+ then
[TABLE]
Proof.
When A=C[t±1], the formula in (4.1) is proved in [CP01]. Further since the fact that t is an invertible element in C[t±1] is not used in that proof, the result is still true when A=C[t]. Applying, the Lie algebra homomorphism,
[TABLE]
gives the result.
∎
From now on we assume that A is finitely generated (say a1,…,am be a set of generators of A). Using the first and third relation of the Definition 4.1 and Lemma 4.3, and then applying induction, one can prove the following.
Lemma 4.4**.**
Suppose ψ∈L(h⊗A) such that λ=ψ∣h0ˉ∈Λ+. If α∈Φ0ˉ+, a1,a2,…,am∈A, and s1,s2,⋯,sm∈N,wψ∈H(ψ), then
[TABLE]
In particular, (yα⊗A)Hψ) is finite dimensional.
Lemma 4.5**.**
If ψ∈L(h⊗A), then WA\mboxloc(ψ)=0.
Proof.
Let λ=ψ∣h0ˉ∈Λ+, let α be a positive root of q and let Iα be the kernel of the linear map
[TABLE]
Notice that Iα={a∈A∣(u⊗a)v=0,v∈WA\mboxloc(ψ)λ,u∈q−α}. Since q−α=Cyα, Lemma 4.4 gives that (q−α⊗A)wψ) is finite dimensional. Thus, Iα is a linear subspace of A of finite co-dimension. We claim that Iα is an ideal of A. Since α=0, we can choose h∈h0ˉ such that α(h)=0. Then, for all b∈A,a∈Iα,v∈WA\mboxloc(ψ)λ,u∈q−α, we have
[TABLE]
Since (h⊗b)v∈WA\mboxloc(ψ)λ and a∈Iα, the last term above is zero. Since we have assumed that α(h)=0, this implies that (u⊗ba)v. As this holds for all v∈WA\mboxloc(ψ)λ and u∈q−α, we have ba∈Iα. Hence Iα is an ideal of A.
Let I=⋂α∈Φ0ˉ+Iα. Since q is finite dimensional and hence a finite number of positive roots, this intersection is finite and thus I is an ideal of A of finite co-dimension. Then we have
[TABLE]
Since λ is the highest weight of WA\mboxloc(ψ)λ, we also have (n+⊗A)WA\mboxloc(ψ)λ=0. Further, since h0ˉ⊗I⊆[n+⊗A,n0ˉ−⊗I], we have (h0ˉ⊗I)WA\mboxloc(ψ)λ=0. In particular, (h0ˉ⊗a)wψ=0 for all a∈I. Since ψ∈L(h⊗A), then there exists a∈I such that ψ(h⊗a)=0. So, we must have wψ=0 and hence
[TABLE]
∎
Definition 4.6**.**
Suppose ψ∈L(h⊗A) such that λ=ψ∣h0ˉ∈Λ+. Let Iψ be the sum of all ideals I⊆A such that (h0ˉ⊗I)WA\mboxloc(ψ)λ=0.
Remark 4.7**.**
It follows from the proof of Lemma that, Iψ is a finite co-dimensional ideal in A and that (yα⊗Iψ)wψ=0 for all α∈Φ0ˉ+. Furthermore, since Iψ has finite co-dimension and A is assumed to be finitely generated, we have that Iψn has finite co-dimension, for all n∈N (see [CLS19, Lemma 2.1(a), (b)])
Lemma 4.8**.**
Suppose ψ∈L(h⊗A) such that λ=ψ∣h0ˉ∈Λ+. Then there exists nψ∈N such that
[TABLE]
Proof.
For α=∑i=1naiαi, with ai∈N and where the αi are the simple roots of q, we define the height of α to be
[TABLE]
By induction on the height of α, we will show that (q−α⊗Iψ\mboxht(α))wψ=0 for all α∈Φ+,wψ∈H(ψ). Since q is finite dimensional, the heights of elements of Φ+ are bounded above, and thus the lemma will follow.
For the base case, first we will show that
[TABLE]
since the set {fi∣αi∈∑} of generators of n−. By the above remark, it suffices to consider the case αi∈∑1ˉ. At first fix such an αi. By (2.8), there exists αj∈Φ1ˉ such that αk:=αi+αj∈Φ0ˉ+. Note that dimqα=(1∣1) for any α∈Φ, that is, qα is generated by an even vector and an odd vector. Further, since [h,[ej,fk]]=αi(h)[ej,fk], we can write after rescaling, if necessary
[TABLE]
Then
[TABLE]
Since by Definition 4.1, (ej⊗A)wψ and by the above remark, (fk⊗Iψ)wψ=0, then from above, (4.3) holds.
Now suppose that αj∈Φ+ with \mboxht(αj)>1. Then there exists αk,αℓ∈Φ+ with \mboxht(αk),\mboxht(αℓ)<\mboxht(αi) such that fi∈C[fk,fℓ]. Then by induction hypothesis
[TABLE]
∎
Corollary 4.9**.**
Suppose ψ∈L(h⊗A) such that λ=ψ∣h0ˉ∈Λ+ and let n∈N as in Lemma 4.8. Then
[TABLE]
Proof.
By [CMS16, Lemma 2.12], to prove that (q⊗Iψnψ)H(ψ)=0, it is enough to prove that (q⊗Iψnψ)wψ=0 for any non-zero wψ∈H(ψ). Since the triangular decomposition of q=n−+h+n+, we have to show that
[TABLE]
From the first relation in Definition 4.1 we have (n+⊗Iψ)wψ=0 for all wψ∈H(ψ) and this implies (n+⊗Iψnψ)wψ=0. Also, (h0ˉ⊗I)wψ=0 by the definition of Iψ. Again by [CMS16, Lemma 4.1], (h1ˉ⊗I)wψ=0. From Lemma 4.8, we get (n−⊗Iψnψ)wψ=0.
∎
Now we give a sufficient conditions for local Weyl modules to be finite dimensional.
Theorem 4.10**.**
Assume that A is finitely generated. Then the local Weyl module WA\mboxloc(ψ) is finite dimensional for all ψ∈L(h⊗A) such that λ=ψ∣h0ˉ∈Λ+.
Proof.
By Definition 4.1, we have WA\mboxloc(ψ)=U(n−⊗A)H(ψ). Also, by Lemma 4.8, we have (n−⊗Iψnψ)H(ψ)=0. Thus,
[TABLE]
Since the set of q-weights of WA\mboxloc(ψ) is finite, there exists N∈N such that
[TABLE]
where U(g)=∑n=0∞Un(g) is the usual filtration on the universal enveloping algebra of a Lie superalgebra g induced from the natural grading on the tensor algebra. Since the Lie superalgebra n−⊗A/Iψnψ and H(ψ) are finite dimensional, the local Weyl module WA\mboxloc(ψ) is finite dimensional.
∎
Theorem 4.11**.**
Let ψ∈L(h⊗A) such that λ=ψ∣h0ˉ∈Λ+. Then any finite dimensional local Weyl module with highest map-weight ψ in the category I(q⊗A,q0ˉ) is isomorphic to WA\mboxloc(ψ) or ΠWA\mboxloc(ψ). Furthermore, if any finite dimensional object V(ψ)∈I(q⊗A,q0ˉ) is generated by an irreducible module H(ψ) of map-weight ψ, then there exists a surjective homomorphism form WA\mboxloc(ψ) to V(ψ) (up to Π).
Proof.
Let V(ψ)∈I(q⊗A,q0ˉ) be a finite dimensional highest map-weight q⊗A-module with highest map-weight ψ is generated by an irreducible module H(ψ). Then by Definition 4.1,
[TABLE]
Since the q0ˉ-module generated by H(ψ) is finite dimensional, then fiλ(hi)+1v=0 for all v∈H(ψ) and i∈I. Thus, we have a surjective homomorphism WA\mboxloc(ψ)→V(ψ) induced by WA\mboxloc(ψ)λ→H(ψ) (up to Π).
Suppose WA′(ψ) is another object in I(q⊗A,q0ˉ) that is generated by an irreducible module m(ψ) with highest map-weight ψ and admits a surjective homomorphism to any object of I(q⊗A,q0ˉ) which is also generated by highest map-weight vectors of map-weight ψ. Then WA′(ψ) is a quotient of WA\mboxloc(ψ) (up to Π) and vice-versa. Since both modules are finite dimensional, WA\mboxloc(ψ)≅WA′(ψ) or Π(WA\mboxloc(ψ))≅WA′(ψ).
∎
Corollary 4.12**.**
Let ψ∈L(h⊗A) such that λ=ψ∣h0ˉ∈Λ+. Then the local Weyl module WA\mboxloc(ψ) is the maximal finite dimensional quotient of the global Weyl module WA(λ) (up to Π) that is a highest map-weight module of highest map-weight ψ.
Remark 4.13**.**
By [CMS16, Theorem 5.6], any finite dimensional q⊗A-module is a highest map-weight module for some ψ∈L(h⊗A) such that λ=ψ∣h0ˉ∈Λ+. Then by Theorem 4.11, there exists a surjective homomorphism from the local Weyl module WA\mboxloc(ψ) to such a module (up to Π). Equivalently, all finite dimensional q⊗A-modules are quotients of loal Weyl modules (up to Π).
5. Tensor product of local Weyl modules
If A and B are associative unitary algebras, then all irreducible representations of A⊗B are of the form VA⊗VB. Further, all such modules are irreducible. However, when A and B are allowed to be superalgebras, then VA⊗VB is not necessarily irreducible.
If gi for i=1,2 are two finite dimensional Lie superalgebras, and Vi is an irreducible finite-dimensional gi-module for i=1,2, then g1⊕g2-module V1⊗V2 is irreducible only if Endgi(Vi)1ˉ=0 for some i=1,2. When Endgi(Vi)1ˉ=Cϕi,ϕi2=−1 for i=1 and i=2, we have
[TABLE]
is an irreducible g1⊕g2-submodule of V1⊗V2 such that V1⊗V2≅V⊕V (see [CHE95, p.27]). Now we set
[TABLE]
If Vi is an irreducible finite-dimensional gi-module for i=1,2, then it is proved that every irreducible finite dimensional g1⊕g2-module is isomorphic to a module of the form V1⊗V2 (see [CHE95, Prop. 8.4]).
Given an ideal I of A, we define its support to be the set
[TABLE]
Theorem 5.1**.**
Assume that A is finitely generated. Let ψ1,ψ2∈L(h⊗A) such that λ1∣h0ˉ=ψ1,λ2∣h0ˉ=ψ2 and suppose that λ1,λ2∈Λ+ such that λ1+λ2∈Λ+. If \mboxSupp(Iψ1)∩\mboxSupp(Iψ2)=∅, then we have
[TABLE]
as q⊗A-modules.
Proof.
Let ψ1,ψ2∈L(h⊗A) such that λ1∣h0ˉ=ψ1,λ2∣h0ˉ=ψ2 and suppose that λ1,λ2∈Λ+ such that λ1+λ2∈Λ+. Let the local Weyl module WA\mboxloc(ψi) associated to ψi (upto Π) be the q⊗A-module generated by H(ψi) for i=1,2. Let ρi be the representation corresponding to WA\mboxloc(ψi) for i=1,2. By Corollary 4.9, there exists n1,n1∈N such that (q⊗Iψini)wψi=0 for all wψi∈H(ψi) for i=1,2. Notice that WA\mboxloc(ψ1)⊗WA\mboxloc(ψ2), as (q⊗A/Iψ1n1)⊕(q⊗A/Iψ2n2)-module generated by H(ψ1)⊗H(ψ2) is either irreducible or is isomorphic to V⊕V where V⊊WA\mboxloc(ψ1)⊗WA\mboxloc(ψ2) is an irreducible (q⊗A/Iψ1n1)⊕(q⊗A/Iψ2n2)-module.
Now the representation ρ1⊗ρ2 factors through the composition
[TABLE]
where first map is the diagonal map and the second map is the projection on each summand. By [CLS19, Lemma 2.1], we have that A=Iψ1n1+Iψ2n2 and Iψ1n1∩Iψ2n2=Iψ1n1Iψ2n2, since \mboxSupp(Iψ1)∩\mboxSupp(Iψ2)=∅. Therefore, we have the following commutative diagram:
It follows that the composition (5.1) is surjective. By the surjective of (5.1), it follows that WA\mboxloc(ψ1)⊗WA\mboxloc(ψ2), as (q⊗A)-module generated by H(ψ1)⊗H(ψ2) is either irreducible or is isomorphic to V⊕V where V⊊WA\mboxloc(ψ1)⊗WA\mboxloc(ψ2). Moreover, h0ˉ⊗A acts on wψ1⊗wψ2 as follows:
[TABLE]
and
[TABLE]
Thus, WA\mboxloc(ψ1)⊗WA\mboxloc(ψ2) is a finite dimensional highest map-weight module generated by H(ψ1)⊗H(ψ2) of highest map-weight ψ1+ψ2. Therefore, by Theorem 4.11, WA\mboxloc(ψ1)⊗WA\mboxloc(ψ2) is a quotient of WA\mboxloc(ψ1+ψ2).
By [CMS16, Theorem 4.3], there exists a unique (up to Π) irreducible finite dimensional h⊗A-module H(ψ1+ψ2) such that xv=(ψ1+ψ2)(x)v,x∈h0ˉ⊗A,v∈H(ψ1+ψ2). Let I=Iψ1∩Iψ2=Iψ1Iψ2 and n=nψ1+ψ2. Then I⊆Iψ1+ψ2 and hence (h0ˉ⊗Iψ1+ψ2n)H(ψ1+ψ2)=0. So, the action of b⊗A on H(ψ1+ψ2) descends to an action of b⊗A/In on H(ψ1+ψ2). Now consider the induced module
[TABLE]
It follows that WA\mboxloc(ψ1+ψ2) is a quotient of M(ψ1+ψ2).
On the other hand, since b⊗A module H(ψ1+ψ2) is irreducible, by [CMS16, Prop 6.3], we have H(ψ1+ψ2) and H(ψ1)⊗H(ψ2) are isomorphic. Hence,
[TABLE]
Since WA\mboxloc(ψ1+ψ2) is a quotient of M(ψ1+ψ2), so WA\mboxloc(ψ1+ψ2) is a quotient of M(ψ1)⊗M(ψ2) and hence we can fix a surjection
[TABLE]
Then one can show that the image of M(ψ1)μ1⊗M(ψ2)μ2 under the map η is zero except for a finite number of weights μ1 and μ2 and let Di be the such finite set of weights. Now for i=1,2, let M(ψi)′ be the submodule of M(ψi) generated by the weight subspaces M(ψi)λi with λi∈Di, and let Mˉ(ψi)=M(ψi)/M(ψi)′. Then WA\mboxloc(ψ1+ψ2) is a quotient of Mˉ(ψ1)⊗Mˉ(ψ2). Since Iψi has finite co-dimension and there are only a finite number of weights occurring in the quotient Mˉ(ψi), this module is a finite dimensional highest map-weight module of highest map-weight ψi. Hence, by Theorem 4.11, it is a quotient of WA\mboxloc(ψi). Thus, Mˉ(ψ1)⊗Mˉ(ψ2) is a quotient of WA\mboxloc(ψ1)⊗WA\mboxloc(ψ2) and this implies WA\mboxloc(ψ1+ψ2) is a quotient of WA\mboxloc(ψ1)⊗WA\mboxloc(ψ2). Since the modules WA\mboxloc(ψ1+ψ2) and WA\mboxloc(ψ1)⊗WA\mboxloc(ψ2) are both finite dimensional, and one is quotient of other implies that WA\mboxloc(ψ1+ψ2)≅WA\mboxloc(ψ1)⊗WA\mboxloc(ψ2).
∎
Note that WA\mboxloc(ψ1) and WA\mboxloc(ψ2) satisfy the hypothesis of Theorem 5.1, then
[TABLE]
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