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

We consider the BGG category O of a quantized universal enveloping algebra Uq(g). We call a module M ∈ O tensor-closed if M ⊗ N ∈ O for any N ∈ O. In this paper we prove that M ∈ 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.