Tensor-closed objects in the BGG category of a quantized semisimple Lie algebra

We consider the BGG category $\mathcal{O}$ of a quantized universal enveloping algebra $U_q(\mathfrak{g})$. We call a module $M\in \mathcal{O}$ tensor-closed if $M\otimes N\in\mathcal{O}$ for any $N\in \mathcal{O}$. In this paper we prove that $M\in \mathcal{O}$ is tensor-closed if and only if $M$ is finite dimensional.


Introduction
BGG category O plays a central role in representation theory, see [1]. For a complex semisimple Lie algebra g we can consider its quantized universal enveloping algebra U q (g) and the category O of U q (g) as in [4].
The large category U q (g)-Mod has tensor product but category O is not closed under the tensor product. We call a module M ∈ O tensor-closed if M ⊗ N ∈ O for any N ∈ O. It is easy to show that finite dimensional modules are tensor-closed. Actually in [3] used tensor products of finite dimensional U q (g)-modules to construct the coordinate ring of the deformed flag variety of g.
For (unquantized) complex semisimple Lie algebra g it is a folklore theorem that any tensor-closed module in O must be finite dimensional, see [2] for an outline of the proof.
The main result of this paper is Theorem 4.2, which claims that M ∈ O of U q (g) is tensor-closed if and only if M is finite dimensional. This result gives a categorical characterization of finite dimensional modules in category O. The proof is based on the idea in [2] together with a careful study of rational expressions of formal characters of modules in O.

A review of the BGG category O of a quantized universal enveloping algebra 2.1 A review of quantized universal enveloping algebras
We follow the notations in [4]. Let g a semisimple Lie algebra over C of rank N . We fix a Cartan subalgebra h ⊂ g. Let ∆ be the set of roots and we fix Σ = {α 1 , . . . , α N } ⊂ ∆ the set of simple roots. We write ( , ) for the bilinear form on h * obtained by rescaling the Killing form such that the shortest root α of g satisfies (α, α) = 2. For a root β ∈ ∆ we set d β = (β, β) 2 and let β ∨ = β d β be the corresponding coroot. In particular let for the weight, root and coroot lattices of g, respectively. It is well-known that β ∨ ∈ Q ∨ for each β ∈ ∆.
The set P + of dominant integral weights is the set of all non-negative integer combinations of the fundamental weights. We also write Q + for the non-negative integer combinations of the simple roots. Let ∆ + = Q + ∩ ∆ be the set of positive roots.
The Cartan matrix for g is the matrix (a ij ) 1≤i,j≤N with coefficients . We write W for the Weyl group, that is, the finite group of automorphisms of P generated by the reflections s i given by s i (µ) = µ − (α ∨ i , µ)α i for α i ∈ Σ and all µ ∈ P. Definition 2.1. [[4, Definition 2.13]] Let q = e h ∈ R × be an invertible element for h ∈ R × . It is clear q is not a root of 1. The algebra U q (g) over C has generators K λ for λ ∈ P, and E i , F i for i = 1, . . . , N , and the defining relations for U q (g) are for all λ, µ ∈ P and all i, j, together with the quantum Serre relations In the above formulas we abbreviate K i = K α i for all simple roots, and we use the notation q i = q d i .
Let U q (n + ) be the subalgebra of U q (g) generated by the elements E 1 , . . . , E N , and let U q (n − ) be the subalgebra generated by F 1 , . . . , F N . Moreover we let U q (h) be the subalgebra generated by the elements K λ for λ ∈ P. [4,Proposition 2.14]] Multiplication in U q (g) induces a linear isomorphism We write U q (b + ) for the subalgebra of U q (g) generated by E 1 , . . . , E N and all K λ for λ ∈ P, and similarly we write U q (b − ) for the subalgebra generated by the elements F 1 , . . . , F N , K λ for λ ∈ P. These algebras are Hopf subalgebras.

A review of the BGG category O
Recall that 1 ≠ q = e h for an h ∈ R × . We shall also use the notation c) The action of U q (n + ) on M is locally nilpotent, that is, for each v ∈ M , the subspace U q (n + ) ⋅ v of M is finite dimensional.
Morphisms in category O are all U q (g)-linear maps.
We list some basic properties of category O.

All finite dimensional weight modules of
Examples of objects in category O include Verma modules, which we define as follows. First for a λ ∈ h * q we have the character χ λ of U q (b + ) determined by where K λ denotes the one-dimensional U q (b + )-module K with the action induced from the character χ λ . The vector v λ = 1 ⊗ 1 ∈ U q (g) ⊗ Uq(b+) K λ is called the highest weight vector of M (λ). It is clear that every highest weight module of highest weight λ is isomorphic to a quotient of M (λ) and every simple highest weight module of highest weight λ is isomorphic to V (λ) .
Moreover, the number of subquotients isomorphic to V (λ) for λ ∈ h * q is independent of the decomposition series and will be denoted by We want to know when [M (µ) ∶ V (λ)] ≠ 0 for a Verma module M (µ). For this purpose we need the following concepts.
It is clear that W acts on Y q and we have the following definition.
Definition 2.6. The extended Weyl groupŴ is defined as the semidirect product with respect to the action of W on Y q .Ŵ is a finite group.
Definition 2.8. We define a partial order ≥ on h * q by saying that λ ≥ µ if λ − µ ∈ Q + . Here we are identifying Q + with its image in h * q . SinceŴ is a finite group, for each µ ∈ h * q there exists only finitely many Remark 1. The reason that we allow the 1 2 i ̵ h −1 Q ∨ -translation is that the image of the Harish-Chandra map Z(U q (g)) → U q (h) W consists of elements generated by K 2µ , µ ∈ P. Therefore q (α ∨ ,−) , α ∨ ∈ We write X q for the set of weights ω ∈ h * q satisfying q (ω,α) = ±1 for all α ∈ Q. We define the extended integral weight lattice by We also put P + q = P + + X q . Note that P ∩ X q = {0} by the assumption that q is not a root of unity.

Proposition 2.6. [[4, Proposition 2.34]]
For every ω ∈ X q , there is a one-dimensional representation χ ω ∶ U q (g) → K defined on generators by and one-dimensional representation of U q (g) is of this form.
The following result characterizes finite dimensional weight modules of U q (g) here the expression on the right hand side is interpreted as a formal sum.
We have the following more general definition Definition 3.2. Let X be the formal sums of the form ∑ λ∈h * q f (λ)e λ where f ∶ h * q → Z is any integer valued function whose support lies in a finite union of sets of the form ν − Q + with ν ∈ h * q . The product in X is the convolution product given by It is clear that the right hand side is still in X .
We have the following formula for ch(M (λ)).

Definition 3.3.
We introduce an element p ∈ X as where P (ν) is the Kostant partition function. Here e µ p is the convolution product of e µ and p.
The following result on ch(V (µ)) is also well-known.
where m µ i ,λ are integers such that m µ i ,µ i = 1 and m µ i ,λ = 0 unless λ ≤ µ i and λ isŴ -linked to µ i . In particular we have finite sum on the right hand side.

Reduced rational expressions of formal characters of modules in O
Notice that we can write the formal character p = ∏ β∈∆ + ∑ ∞ m=0 e −mβ as so by Corollary 3.3 for each M ∈ O, we can write its formal character as We want to simplify ch(M ) to obtain a reduced fraction, which needs some work because the ring X is not a UFD.
Let S be the ring of Z-coefficient polynomials generated by e −α i , i = 1, . . . , N , where {α 1 , . . . , α N } is the set of simple roots. It is clear that We have the following definition.
Definition 3.4. Let X be as in Definition 3.2. We say that a ∈ X can be written in reduced rational form if there exists a subset T a ⊂ ∆ + and a finite collection {µ 1 , . . . , µ m } ⊂ h * q such that where 1. µ i − µ j is not in the root lattice Q for each i ≠ j; 2. f i is a polynomial in S with nonzero constant term for each i; 3. n β is a positive integer for each β ∈ T a ; 4. The numerator and denominator of (2) are coprime. More precisely, for each β ∈ T a , there exists an f i in the numerator such that 1 − e −β is not a factor of f i .
We call the set T a the denominator roots of a.
and Property   Proof. It is implied by the uniqueness of reduced rational form.
Example 2. For a Verma module M (µ), the reduced rational form of its formal character is hence T M (µ) = ∆ + . On the other hand, for each simple highest weight module V (µ) we have where f V (µ) ∈ S. In particular if µ ∈ P + q then T V (µ) = ∅ and ch(V (µ)) is given explicitly by the Weyl character formula. Actually [4,Proposition 4.4] only covers the µ ∈ P + case but the general case can be easily obtained by [4,Lemma 2.41].

Tensor closed objects in category O
The category U q (g)-Mod has a tensor product since U q (g) is a Hopf algebra. Moreover U q (g)-Mod is a braided category since U q (g) is quasitriangular in the sense of [4, Theorem 2.108]. In particular for any left U q (g)-modules V and W we have a U q (g)- However category O is not closed under tensor product. The following result is well-known.
Proof. The proof is the same as that of [1, Theorem 1.1 (d)].
In this section we prove the following result.

Theorem 4.2. A module V ∈ O is tensor-closed if and only if it is finite dimensional.
To give the proof more rigorously we introduce the following auxiliary category. b) There exists finitely many weights ν 1 , . . . , ν l ∈ h * q such that suppM Morphisms in categoryÕ are all U q (g)-linear maps.
It is clear that O is a full subcategory ofÕ.Õ is closed under tensor product and modules inÕ have formal characters in the ring X in Definition 3.  Proof. For each simple highest weight module V (µ) we have the reduced rational form where f V (µ) is in the polynomial ring S such that 1 − e −β is not a factor of f V (µ) for each β ∈ T V (µ) . Therefore ch(V (µ) ⊗ V (λ)) = ch(V (µ))ch(V (λ)) = .