Simple Vectorial Lie Algebras in Characteristic 2 and their Superizations

TL;DR

This paper classifies and constructs new simple Lie algebras and superalgebras in characteristic 2, revealing unique structures and deformations not present in other characteristics, expanding the understanding of modular Lie algebra theory.

Contribution

It provides a comprehensive classification of simple vectorial Lie algebras and superalgebras in characteristic 2, including new deformations and analogs, with explicit descriptions of their structures.

Findings

Identification of new simple Lie algebras in characteristic 2.

Description of deformations of known Lie algebras in characteristic 2.

Construction of new simple Lie superalgebras from simple Lie algebras.

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.