Remarks on quantum symmetric algebras

We examine the quantum symmetric and exterior algebras of finite-dimensional Uq(g)Uq(g)-modules first systematically studied by Berenstein and Zwicknagl, and we resolve some of their questions. We show that the difference (in the Grothendieck group) between the quantum symmetric and exterior cubes of a finite-dimensional module is the same as it is classically. This implies a certain “numerical Koszul duality” between the algebras. Furthermore, we show that quantum symmetric algebras are “commutative” in an appropriate sense, and moreover that they possess a universal mapping property with respect to such commutative algebras. We make extensive use of the coboundary structure on the module category.