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

Bin Shu, Lisun Zheng, Ye Ren

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.

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

Abstract

Let be a basic classical Lie superalgebra over an algebraically closed field of characteristic . Denote by the center of the universal enveloping algebra . Then turns out to be finitely-generated purely-even commutative algebra without nonzero divisors. In this paper, we demonstrate that the fraction is isomorphic to for the center of . Consequently, both Zassenhaus varieties for and are birationally equivalent via a subalgebra , and is rational under the standard hypotheses.

Paper Structure

This paper contains 14 sections, 25 theorems, 32 equations.

Table of Contents

  1. Preliminaries: modular basic classical Lie superalgebras
  2. Center and anti-center of $U({\mathfrak g})$
  3. Restricted Lie superalgebras
  4. Reduced enveloping algebras of restricted Lie superalgebras
  5. Modular basic classical Lie superalgebras and the corresponding algebraic supergroups
  6. Root space decomposition
  7. Basic properties of root spaces
  8. All central elements in $U({\mathfrak g})$ are even
  9. Harish-Chandra projections
  10. Baby Verma modules
  11. Centers of universal enveloping algebras of modular basic classical Lie superalgebras
  12. Restriction homomorphisms
  13. The structure of center: proof of Theorem \ref{['secondtheorem']}
  14. Birational equivalence of the Zassenhaus varieties and their rationality

Key Result

Lemma 1.1

Let ${\mathfrak g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$ be a basic classical Lie superlagebra. There exists a nonzero element $v_\emptyset\in U({\mathfrak g})$ such that the map $\Phi : u\rightarrow (ad' v_\emptyset)u$ provides a linear isomorphism from the ${\mathfrak g}_{\bar{0}}

Theorems & Definitions (46)

  • Lemma 1.1
  • Proposition 1.2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Corollary 1.5
  • proof
  • Proposition 1.6
  • proof
  • ...and 36 more