Classification of simple Lie superalgebras in characteristic $2$

TL;DR

This paper classifies all simple finite-dimensional Lie superalgebras over fields of characteristic 2, introducing procedures to generate new superalgebras and discussing deformation and restrictedness concepts.

Contribution

It provides a complete classification of simple finite-dimensional Lie superalgebras in characteristic 2, including new procedures and structures analogous to classical restrictedness.

Findings

All simple finite-dimensional Lie superalgebras are obtained via two procedures.

Most of the resulting superalgebras are new discoveries.

Introduces new structures like (2,4)- and (2|4)-structures for Lie algebras and superalgebras.

Abstract

All results concern characteristic 2. Two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of the latter new, are offered. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures, so we classified all simple finite-dimensional Lie superalgebras modulo non-existing at the moment classification of simple finite-dimensional Lie algebras. This result concerns Lie superalgebras considered naively, as vector spaces. To obtain classification of simple Lie superalgebras in the category of supervarieties, one should list the non-isomorphic deforms (results of deformations) with odd parameter. This problem is open bar several examples described in arXiv~0807.3054. For Lie algebras, in addition to the known ---"classical" --- restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and of Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: $(2|4)$- and $(2|2)$-structures, one more analog --- a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras.