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

TL;DR

This paper characterizes tensor-closed modules in the BGG category of a quantized semisimple Lie algebra, showing they are precisely the finite-dimensional modules, with methods applicable to the classical case.

Contribution

It provides a complete characterization of tensor-closed modules in the BGG category for quantized Lie algebras, extending to the classical unquantized case.

Findings

Tensor-closed modules are exactly the finite-dimensional modules.

The characterization applies to both quantized and unquantized cases.

The method used is versatile and broadly applicable.

Abstract

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. The method used in this paper applies to the unquantized case as well.