Generalized Joseph's decompositions

TL;DR

This paper generalizes Joseph's decomposition of quantum groups and links it to Drinfeld's work on central elements, providing an explicit basis for the center of quantum groups that resembles symmetric functions.

Contribution

It introduces a generalized decomposition framework for $U_q(rak g)$ and constructs a natural basis in its center, connecting it to symmetric functions.

Findings

Constructed a basis in the center of $U_q(rak g)$ with Schur polynomial behavior

Linked Joseph's decomposition to Drinfeld's central element computation

Explicitly identified the center with the ring of symmetric functions

Abstract

We generalize the decomposition of $U_q(\mathfrak g)$ introduced by A. Joseph and relate it, for $\mathfrak g$ semisimple, to the celebrated computation of central elements due to V. Drinfeld. In that case we construct a natural basis in the center of $U_q(\mathfrak g)$ whose elements behave as Schur polynomials and thus explicitly identify the center with the ring of symmetric functions.