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.
Categories: math.RT math.RA