TENSOR-CLOSED OBJECTS IN THE BGG CATEGORY OF A QUANTIZED SEMISIMPLE LIE ALGEBRA

Abstract

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