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.