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 ¯ {g=g₀g₁} be a basic classical Lie superalgebra over an algebraically closed field 𝐤 {k} of characteristic p > 2 p>2. Denote by 𝒵 Z the center of the universal enveloping algebra U ⁢ (𝔤) U ({g) }. Then 𝒵 Z turns out to be finitely-generated purely-even commutative algebra without nonzero divisors. In this paper, we demonstrate that the fraction Frac ⁡ (𝒵) Frac (Z) is isomorphic to Frac ⁡ (ℨ) Frac (Z) for the center ℨ Z of U ⁢ (𝔤 0 ¯) U ({g₀) }. Consequently, both Zassenhaus varieties for 𝔤 {g} and 𝔤 0 ¯ {g₀} are birationally equivalent via a subalgebra 𝒵 ~ ⊂ 𝒵 {Z}, and Spec ⁡ (𝒵) Spec (Z) is rational under the standard hypotheses.