Cohomology of a restricted Lie algebra with a restricted derivation in characteristic 2

TL;DR

This paper investigates the cohomology and deformation theory of restricted Lie algebras with restricted derivations in characteristic 2, establishing conditions for rigidity, extensibility, and classification of central extensions.

Contribution

It introduces the cohomology complex for ResLieDer pairs, linking second cohomology to rigidity, extensibility, and classification of extensions in characteristic 2.

Findings

ResLieDer pairs are rigid if second cohomology is trivial

Deformations are extensible when obstruction classes vanish

Central extensions correspond to second cohomology groups

Abstract

This paper mainly studies the ResLieDer pair in characteristic 2, that is, a restricted Lie algebra with a restricted derivation. We define the restricted representation of a ResLieDer pair and the corresponding cohomology complex. We show that a ResLieDer pair is rigid if the second cohomology group is trivial and a deformation of order $n$ is extensible if and only if its obstruction class is trivial. Moreover, we prove that the central extensions of a ResLieDer pair are classified by the second cohomology group. Finally, we show that a pair of restricted derivations is extensible if and only if its obstruction class is trivial.