Explicit generators and relations for the centre of the quantum group

TL;DR

This paper explicitly constructs generators and relations for the center of the quantum group U_q(g), providing a uniform description for certain Lie algebra types and a generating set for others.

Contribution

It offers explicit generators and relations for the center of U_q(g), extending understanding across various Lie algebra types with a uniform approach.

Findings

Center is isomorphic to a quotient of a polynomial algebra for specific types.

Center is generated by algebraically independent elements for other types.

Provides explicit descriptions applicable to multiple Lie algebra classifications.

Abstract

For the standard Drinfeld-Jimbo quantum group ${\rm U}_q(\mathfrak{g})$ associated with a simple Lie algebra $\mathfrak{g}$, we construct explicit generators of the centre $Z({\rm U}_q(\mathfrak{g}))$, and determine the relations satisfied by the generators. For $\mathfrak{g}$ of type $A_n(n\geq 2)$, $D_{2k+1}(k\geq 2)$ or $E_6$, the centre $Z({\rm U}_q(\mathfrak{g}))$ is isomorphic to a quotient of a polynomial algebra in multiple variables, which is described in a uniform manner for all cases. For $\mathfrak{g}$ of any other type, $Z({\rm U}_q(\mathfrak{g}))$ is generated by $n=$rank$(\mathfrak{g})$ algebraically independent elements.