Centers of Universal Enveloping Algebras

TL;DR

This paper investigates the structure of the center of universal enveloping algebras for current and loop (super)algebras over algebraically closed fields, highlighting differences between zero and prime characteristic cases.

Contribution

It characterizes the generators of the center of $U(rak{g})$ in both zero and prime characteristic, including the role of $p$-centers in prime characteristic.

Findings

In zero characteristic, the center is generated by the centers of $rak{g}$.

In prime characteristic, the center includes the centers of $rak{g}$ and the $p$-centers.

Structure of the center in semisimple Lie (super)algebras is also analyzed.

Abstract

The universal enveloping algebra $U(\mathfrak{g} )$ of a current (super)algebra or loop (super)algebra $\mathfrak{g} $ is considered over an algebraically closed field $\mathbb{K} $ with characteristic $p\ge 0$. This paper focuses on the structure of the center $Z(\mathfrak{g} )$ of $U(\mathfrak{g} )$. In the case of zero characteristic, $Z(\mathfrak{g} )$ is generated by the centers of $\mathfrak{g} $. In the case of prime characteristic, $Z(\mathfrak{g} )$ is generated by the centers of $\mathfrak{g} $ and the $p$-centers of $U(\mathfrak{g} )$. We also study the structure of $Z(\mathfrak{g} )$ in the semisimple Lie (super)algebra.