On $q$-commutative power and Laurent series rings at roots of unity

TL;DR

This paper investigates $q$-commutative power and Laurent series rings at roots of unity, establishing criteria for their centers to be commutative and demonstrating their unique factorization properties as noncommutative PI domains.

Contribution

It provides an exact criterion for the centers of these rings at roots of unity and proves their status as unique factorization rings, offering new examples of noncommutative UFDs.

Findings

Centers are commutative Laurent/ power series rings under specific conditions.

L is an Azumaya algebra with degree determined by $q_{ij}$.

Both rings are proven to be unique factorization rings.

Abstract

We continue the first and second authors' study of $q$-commutative power series rings $R=k_q[[x_1,\ldots,x_n]]$ and Laurent series rings $L=k_q[[x^{\pm 1}_1,\ldots,x^{\pm 1}_n]]$, specializing to the case in which the commutation parameters $q_{ij}$ are all roots of unity. In this setting, $R$ is a PI algebra, and we can apply results of De Concini, Kac, and Procesi to show that $L$ is an Azumaya algebra whose degree can be inferred from the $q_{ij}$. Our main result establishes an exact criterion (dependent on the $q_{ij}$) for determining when the centers of $L$ and $R$ are commutative Laurent series and commutative power series rings, respectively. In the event this criterion is satisfied, it follows that $L$ is a unique factorization ring in the sense of Chatters and Jordan, and it further follows, by results of Dumas, Launois, Lenagan, and Rigal, that $R$ is a unique factorization ring. We thus produce new examples of complete, local, noetherian, noncommutative, unique factorization rings (that are PI domains).