Centers of universal enveloping algebras of Lie superalgebras in prime characteristic

TL;DR

This paper investigates the structure of the center of the universal enveloping algebra of basic classical Lie superalgebras over fields of prime characteristic, revealing its relation to invariants and the p-center.

Contribution

It proves that the quotient field of the center coincides with that generated by the G_ev-invariants and the p-center, clarifying the algebraic structure in prime characteristic.

Findings

Center is a domain.

Quotient field matches that of invariants and p-center.

Provides structural insight into Lie superalgebra enveloping algebras.

Abstract

Let $\ggg=\ggg_\bz+\ggg_\bo$ be a basic classical Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$, and $G$ be an algebraic supergroup satisfying $\Lie(G)=\ggg$, with the purely even subgroup $G_\ev$ which is a reductive group. The center $\cz:=\cz(\ggg)$ of the universal enveloping algebra of $\ggg$ easily turns out to be a domain. In this paper, we prove that the quotient field of $\cz$ coincides with that of the subalgebra generated by the $G_{\ev}$-invariant ring $\cz^{G_\ev}$ of $\cz$ and the $p$-center $\cz_0$ of $U(\ggg_\bz)$.