Centers of Universal Enveloping Algebras

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.