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

Abstract

We continue the first and second authors’ study of q-commutative power series rings $R=k_q[[x_1,\dots ,x_n]]$ and Laurent series rings $L=k_q[[x_1^{\pm 1},\dots ,x_n^{\pm 1}]]$, 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 results establish 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.

Keywords:

• Skew power series
• skew Laurent series
• q-commutative

2010 Mathematics Subject Classification:

• Primary: 16W60
• Secondary: 16L30
• 16S34

Acknowledgement

We are grateful to the referee for the insightful comments and useful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.