2017/08/17 by Edward S. Letzter, Linhong Wang, Letzter, Edward S. +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1708.05432
openalex publication_date 2017/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We continue the first and second authors' study of q-commutative power series rings R=kq[[x1,…,xn]] and Laurent series rings L=kq[[x± 11,…,x± 1n]], specializing to the case in which the commutation parameters qij 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 qij. Our main result establishes an exact criterion (dependent on the qij) 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).