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

2024/10/09 by Bin Shu, Shu, Bin, Lisun Zheng +3
Mathematics · #14E08 #14M20 #17B45 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Primary 17B50 #Representation Theory (math.RT) #Secondary 17B35

paper · pdf · doi:10.48550/arxiv.2410.07292

openalex publication_date 2024/10/09 · openalex created_date 2024/10/13 · openalex updated_date 2026/07/28

Abstract

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

Related

• Zassenhaus varieties of general linear Lie algebras
• Centers of universal enveloping algebras of Lie superalgebras in prime characteristic
• Centralizers of nilpotent elements in basic classical Lie superalgebras in good characteristic
• Lie superalgebras in characteristic 2 and mixed characteristic
• Typical blocks of Lie superalgebras in prime characteristic