Remarks on Quantum Symmetric Algebras

We examine the quantum symmetric and exterior algebras of finite-dimensional \uqg-modules first systematically studied by Berenstein and Zwicknagl, and resolve some questions that they raised. 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. Furthermore, we show that quantum symmetric algebras are commutative in an appropriate sense, and are universal with this property. We make extensive use of the coboundary structure on the module category.