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

Abstract

Let $\mathfrak{g}={\mathfrak{g}}_{\bar{0}}\oplus {\mathfrak{g}}_{\bar{1}}$ be a basic classical Lie superalgebra over an algebraically closed field $\text{k}$ 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 $\widetilde{mathcalZ}\subset \mathcal{Z}$, and $\text{Spec}(\mathcal{Z})$ is rational under the standard hypotheses.

22 page. Published in Forum Math. arXiv admin note: substantial text overlap with arXiv:1210.8032