Explicit generators and relations for the centre of the quantum group

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.