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Simple Vectorial Lie Algebras in Characteristic 2 and their Superizations

Sofiane Bouarroudj, Pavel Grozman, Alexei Lebedev, Dimitry Leites, Irina Shchepochkina

Abstract

We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectorial Lie superalgebras. We consider odd parameters of deformations. For all 15 Weisfeiler gradings of the 5 exceptional families, and one Weisfeiler grading for each of 2 serial simple complex Lie superalgebras (with 2 exceptional subseries), we describe their characteristic-2 analogs - new simple Lie algebras. Descriptions of several of these analogs, and of their desuperizations, are far from obvious. One of the exceptional simple vectorial Lie algebras is a previously unknown deform (the result of a deformation) of the characteristic-2 version of the Lie algebra of divergence-free vector fields; this is a new simple Lie algebra with no analogs in characteristics distinct from 2. In characteristic 2, every simple Lie superalgebra can be obtained from a simple Lie algebra by one of the two methods described in arXiv:1407.1695. Most of the simple Lie superalgebras thus obtained from simple Lie algebras we describe here are new.

Simple Vectorial Lie Algebras in Characteristic 2 and their Superizations

Abstract

We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectorial Lie superalgebras. We consider odd parameters of deformations. For all 15 Weisfeiler gradings of the 5 exceptional families, and one Weisfeiler grading for each of 2 serial simple complex Lie superalgebras (with 2 exceptional subseries), we describe their characteristic-2 analogs - new simple Lie algebras. Descriptions of several of these analogs, and of their desuperizations, are far from obvious. One of the exceptional simple vectorial Lie algebras is a previously unknown deform (the result of a deformation) of the characteristic-2 version of the Lie algebra of divergence-free vector fields; this is a new simple Lie algebra with no analogs in characteristics distinct from 2. In characteristic 2, every simple Lie superalgebra can be obtained from a simple Lie algebra by one of the two methods described in arXiv:1407.1695. Most of the simple Lie superalgebras thus obtained from simple Lie algebras we describe here are new.

Paper Structure

This paper contains 53 sections, 3 theorems, 127 equations.

Table of Contents

  1. Notation and background
  2. Main points
  3. Generalities
  4. Conventions and notation often used
  5. Definition of Lie superalgebras for $\boldsymbol{p\neq 2, 3}$
  6. Definition of Lie superalgebras for $\boldsymbol{p=2}$
  7. Definition of Lie superalgebras for $\boldsymbol{p=3}$
  8. Basics on the functor of points
  9. PBW-theorem for Lie superalgebras
  10. Deformations of the brackets
  11. Linear (matrix) Lie superalgebras
  12. The queer version of $\boldsymbol{{\mathfrak{gl}}(n)}$
  13. The supermatrix of the dual operator (after Lsos)
  14. Lie superalgebras preserving bilinear forms
  15. Notational convention
  16. ...and 38 more sections

Key Result

Theorem 2.1

Brackets and squares of contact vector fields, and the corresponding contact brackets of generating functions, are given by formulas K_fbr. Both the contact brackets $\{-, -\}_{\rm k.b.}$ and $\{-, -\}_{\rm m.b.}$ are of the shape Then, for any $f, f_1, g, g_1\in \widehat{{\mathcal{F}}}$, we deduce

Theorems & Definitions (8)

  • Conjecture 1.1
  • Theorem 2.1: on explicit squaring and contact brackets for $p=2$
  • Lemma 2.2: A helpful lemma
  • Claim 2.3: Grading operators in ${\mathfrak{k}}(2n_{\bar{0}}+1;\underline{N}|m)$ and ${\mathfrak{m}}(n;\underline{N}|n+1)$
  • Claim 2.4: brackets in ${\mathfrak{k}}(2k_{\bar{0}}+1;\underline{N}|2k_{\bar{1}}+1)$
  • Claim 2.5: the second divergence on ${\mathfrak{k}}(1|2)$
  • Remark 2.7
  • Proposition 2.8: two exceptional deforms of the Poisson algebra