Explicit generators of the centre of the quantum group

TL;DR

This paper constructs explicit algebraically independent central elements for a generalized quantum group associated with a simple Lie algebra, proving that these generate its entire centre, which is a polynomial algebra.

Contribution

It introduces a set of explicit generators for the centre of a generalized quantum group and proves they form a polynomial algebra, utilizing a quantum Harish-Chandra isomorphism.

Findings

Centre of the generalized quantum group is a polynomial algebra.

Constructed explicit algebraically independent central elements.

Proved these elements generate the entire centre.

Abstract

For a slightly generalised version of the Jimbo quantum group associated with any finite dimensional simple Lie algebra $\mathfrak{g}$, we show that its centre is a polynomial algebra. We construct a set of algebraically independent central elements, each associated with a fundamental weight of $\mathfrak{g}$, and prove that they generate the entire centre of the quantum group. A key ingredient in the proofs of these results is a quantum Harish-Chandra isomorphism for the generalised quantum group, which is proven in some details.