On the quantum Frobenius map for general linear groups (Q1907165)

Let \(G\) be a complex semisimple Lie group and \({\mathcal O}_q (G)\) its quantum coordinate ring. Assume that \(q\) is an odd root of unity. Then the primitive spectrum \(S\) of \({\mathcal O}_q (G)\) consists of maximal ideals of finite codimension and there is a \(q\)-analogue of the Frobenius homomorphism (sometimes called the quantum Frobenius homomorphism), which provides an embedding of the classical coordinate ring \({\mathcal O} (G)\) into the center of \({\mathcal O}_q (G)\) and thus also a finite surjection from \(S\) into the primitive spectrum of \({\mathcal O} (G)\). The purpose of this paper is to describe this surjection more precisely in the special cases of \({\mathcal O}_q (GL_n)\) and \({\mathcal O}_q (M_n)\) by showing that its fibers are orbits of a finite group of automorphisms obtained from row and column multiplication of the standard generators \(X_{ij}\) by powers of \(q\). The techniques of the paper are mostly natural extensions of methods used by \textit{K. Goodearl} and the author in [Prime ideals in skew and \(q\)-skew polynomial rings, Mem. Am. Math. Soc. 521 (1994; Zbl 0814.16026)] and [Proc. Am. Math. Soc. 121, 1017-1025 (1994; Zbl 0812.16039)]; they can also be viewed as extensions of earlier techniques used to study solvable Lie superalgebras.