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

Abstract

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