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Almost-simple algebraic supergroups

S. Bouarroudj, A. N. Zubkov

TL;DR

The paper studies almost-simple algebraic supergroups over an algebraically closed field of characteristic zero or odd characteristic, grounding the theory in Harish-Chandra pairs and distinguishing SAS from WAS via structural conditions on the even part and odd component. It proves that in characteristic zero SAS coincides with Lie superalgebras that are simple, while in positive characteristic SAS remains strictly smaller than WAS and may admit non-simple Lie superalgebras; it develops a normality criterion ensuring SAS behavior and demonstrates the constraints on possible Lie superalgebras. By leveraging Harish-Chandra pairs, the authors lift the simple Lie superalgebras in Kac’s list (excluding Cartan series) to algebraic SAS or WAS supergroups and analyze numerous families, including PSL, SpO, P(n), Q(n), and several exceptional cases such as AG(2), AB(3), and OSp_alpha(4|2). Finally, they construct a new 10|12-dimensional SAS-supergroup BRJ(2;5), proving its algebraicity in characteristic 5 and showing a contrasting non-algebraic example in characteristic 3, thereby expanding the landscape beyond Kac’s list and illuminating the diversity of SAS structures in positive characteristic.

Abstract

We describe certain almost-simple algebraic supergroups over an algebraically closed field of odd or zero characteristic. In addition to supergroups with simple Lie superalgebras from Kac's theorem, we construct new supergroups whose Lie superalgebra is either non-simple or simple but is not part of Kac's list.

Almost-simple algebraic supergroups

TL;DR

The paper studies almost-simple algebraic supergroups over an algebraically closed field of characteristic zero or odd characteristic, grounding the theory in Harish-Chandra pairs and distinguishing SAS from WAS via structural conditions on the even part and odd component. It proves that in characteristic zero SAS coincides with Lie superalgebras that are simple, while in positive characteristic SAS remains strictly smaller than WAS and may admit non-simple Lie superalgebras; it develops a normality criterion ensuring SAS behavior and demonstrates the constraints on possible Lie superalgebras. By leveraging Harish-Chandra pairs, the authors lift the simple Lie superalgebras in Kac’s list (excluding Cartan series) to algebraic SAS or WAS supergroups and analyze numerous families, including PSL, SpO, P(n), Q(n), and several exceptional cases such as AG(2), AB(3), and OSp_alpha(4|2). Finally, they construct a new 10|12-dimensional SAS-supergroup BRJ(2;5), proving its algebraicity in characteristic 5 and showing a contrasting non-algebraic example in characteristic 3, thereby expanding the landscape beyond Kac’s list and illuminating the diversity of SAS structures in positive characteristic.

Abstract

We describe certain almost-simple algebraic supergroups over an algebraically closed field of odd or zero characteristic. In addition to supergroups with simple Lie superalgebras from Kac's theorem, we construct new supergroups whose Lie superalgebra is either non-simple or simple but is not part of Kac's list.

Paper Structure

This paper contains 14 sections, 31 theorems, 171 equations.

Table of Contents

  1. Basic definitions and results
  2. Auxiliary results on WAPS and SAPS-supergroups
  3. Projective special linear supergroups
  4. More properties of SAS-supergroups
  5. SAS-supergroup with not simple Lie superalgebra
  6. About other alternatives from Proposition \ref{['Lie algebra of SAS']}
  7. Almost-simple supergroups from Kac's classification list
  8. Ortho-symplectic supergroups
  9. Periplectic supergroups
  10. Queer supergroups
  11. The supergroups $\mathrm{OSp}_{\alpha}(4|2)$
  12. The exceptional supergroup $\mathrm{AG}(2)$
  13. The exceptional supergroup $\mathrm{AB}(3)$
  14. The new $10|12$-dimensional SAS-supergroup

Key Result

Lemma 1.1

If $\mathbb{H}$ and $\mathbb{K}$ are normal supersubgroups of $\mathbb{G}$, then $\mathbb{H}\mathbb{K}$ is a normal supersubgroup again, and its Harish-Chandra pair is equal to $(H K, \mathfrak{H}_{\bar{1}}+\mathfrak{K}_{\bar{1}})$.

Theorems & Definitions (70)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Lemma 2.1
  • proof
  • ...and 60 more