Prime factor algebras of the coordinate ring of quantum matrices

Abstract

It is proved that every prime factor algebra of the coordinate ring O q ( M n ( k ) ) {\mathcal {O}_q}({M_n}(k)) of quantum n × n n \times n matrices over a field k is an integral domain (albeit not necessarily commutative) when q is not a root of unity. The same conclusion follows for the quantum groups O q ( SL n ( k ) ) {\mathcal {O}_q}({\text {SL}_n}(k)) and O q ( GL n ( k ) ) {\mathcal {O}_q}({\text {GL}_n}(k)) . The proof uses a q-analog of Sigurdsson’s theorem bounding the Goldie ranks of prime factors of differential operator rings; this q-analog in turn is based on results from the authors’ recent work on q-skew polynomial rings.