Harish-Chandra Theorem for the Multi-Parameter Quantum Groups of Okado-Yamane Type

TL;DR

This paper proves the Harish-Chandra theorem for multi-parameter quantum groups of Okado-Yamane type, establishing an isomorphism of the Harish-Chandra homomorphism and describing the structure of the center.

Contribution

It extends the Harish-Chandra theorem to multi-parameter quantum groups of Okado-Yamane type and characterizes their centers as polynomial or quotient algebras.

Findings

Harish-Chandra homomorphism is an isomorphism for these quantum groups

Center of the quantum group is isomorphic to a polynomial or quotient algebra

Results also apply to two-parameter quantum groups

Abstract

This paper is devoted to studying the centre of the multi-parameter quantum group $U_{q,G}(\mathfrak{g})$ introduced by Okado and Yamane, where $\mathfrak{g}$ is a complex simple Lie algebra, and all parameters lie in general position. We mainly establish the Harish-Chandra theorem, proving that the Harish-Chandra homomorphism is an isomorphism; in particular, we determine the centre $Z(U_{q,G})\cong (U^0_\flat)^W$ is isomorphic to a polynomial algebra or a quotient algebra of a polynomial algebra. The same result holds for the $(U^0_\flat)^W$ of the two-parameter quantum group $U_{r,s}(\mathfrak{g})$.