Prime Factor Algebras of the Coordinate Ring of Quantum Matrices

Abstract

It is proved that every prime factor algebra of the coordinate ring ${\mathcal {O}_q}({M_n}(k))$ of quantum $n \times n$ 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 ${\mathcal {O}_q}({\text {SL}_n}(k))$ and ${\mathcal {O}_q}({\text {GL}_n}(k))$. 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.