Remarks on Quantum Symmetric Algebras

TL;DR

This paper investigates properties of quantum symmetric and exterior algebras of finite-dimensional modules, demonstrating their classical-like differences, commutativity, and universality using the coboundary structure in the module category.

Contribution

It clarifies the relationship between quantum symmetric and exterior powers and establishes the commutative and universal nature of quantum symmetric algebras.

Findings

Quantum symmetric and exterior cubes differ as in the classical case.

Quantum symmetric algebras are commutative in a suitable sense.

Quantum symmetric algebras are universal with their commutative property.

Abstract

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.