Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic

TL;DR

This paper proves that the Zassenhaus varieties of basic classical Lie superalgebras and their purely-even reductive subalgebras are birationally equivalent in odd characteristic, revealing their rationality and structural similarities.

Contribution

It establishes the birational equivalence of Zassenhaus varieties for Lie superalgebras and their even parts in odd characteristic, extending understanding of their algebraic geometry.

Findings

Fraction fields of centers are isomorphic.

Zassenhaus varieties are birationally equivalent.

Centers are rational under standard conditions.

Abstract

Let $\mathfrak{g}=\mathfrak{g}_{\bar 0}\oplus\mathfrak{g}_{\bar 1}$ be a basic classical Lie superalgebra over an algebraically closed field $\textbf{k}$ of characteristic $p>2$. Denote by $\mathcal{Z}$ the center of the universal enveloping algebra $U(\mathfrak{g})$. Then $\mathcal{Z}$ turns out to be finitely-generated purely-even commutative algebra without nonzero divisors. In this paper, we demonstrate that the fraction $\text{Frac}(\mathcal{Z})$ is isomorphic to $\text{Frac}(\mathfrak{Z})$ for the center $\mathfrak{Z}$ of $U(\mathfrak{g}_{\bar 0})$. Consequently, both Zassenhaus varieties for $\mathfrak{g}$ and $\mathfrak{g}_{\bar 0}$ are birationally equivalent via a subalgebra $\widetilde{mathcal{Z}}\subset\mathcal{Z}$, and $\text{Spec}(\mathcal{Z})$ is rational under the standard hypotheses.