Deformations of Lie algebras of type ${D}_{l}$ and $\overline{{D}_{l}}$ in characteristic 2

TL;DR

This paper investigates the local and some global deformations of Lie algebras of type D_l and their central quotients over fields of characteristic 2, contributing to the classification of simple Lie algebras in small characteristic.

Contribution

It describes the spaces of local deformations and identifies some non-rigid global deformations of these Lie algebras, expanding understanding of their structure in characteristic 2.

Findings

Described local deformation spaces of D_l and its central quotient.

Identified some global deformations that are not rigid.

Contributed to classification efforts of simple Lie algebras in characteristic 2.

Abstract

The study of global deformations of Lie algebras is related to the problem of classification of simple Lie algebras over fields of small characteristic. The classification of finite-dimensional simple Lie algebras is complete over algebraically closed fields of characteristic $p>3$ (\cite{Strade1}-\cite{Strade3}). Over the fields of characteristic 2, a large number of examples of Lie algebras are constructed that do not fit into previously known schemes. Description of the deformation of classical Lie algebras first gives new examples of simple Lie algebras, and second allows you to describe known examples as deformations of classical Lie algebras, as is done, for example, in \cite{CKK}. This paper describes the spaces of local deformations of Lie algebras of the type $D_l$ for $l>3$ and the factor of the algebra at the center $\overline{D}_l$, and describes some global deformations of Lie algebras of these types that are not rigid.