Lie algebra deformations in characteristic 2

TL;DR

This paper explores deformations of Kaplansky Lie algebras in characteristic 2, revealing new gradings, classifying their deformations, and connecting them to Hamiltonian algebras, thereby advancing understanding of simple modular Lie algebras.

Contribution

It introduces new $Z/2$-gradings for Kaplansky algebras, classifies their deformations, and links them to Hamiltonian algebras, supporting broader conjectures in modular Lie algebra theory.

Findings

Type-2 and type-4 Kaplansky algebras have new $Z/2$-gradings.

Type-2 and one type-4 algebra are deformations of Hamiltonian algebras.

Explicit description of Jurman algebra as a semitrivial deformation.

Abstract

Of four types of Kaplansky algebras, type-2 and type-4 algebras have previously unobserved $\mathbb{Z}/2$-gradings: nonlinear in roots. A method assigning a simple Lie superalgebra to every $\mathbb{Z}/2$-graded simple Lie algebra in characteristic 2 is illustrated by seven new series. Type-2 algebras and one of the two type-4 algebras are demystified as nontrivial deforms (the results of deformations) of the alternate Hamiltonian algebras. The type-1 Kaplansky algebra is recognized as the derived of the nonalternate version of the Hamiltonian Lie algebra, the one that preserves a tensorial 2-form, not an exterior one. Deforms corresponding to nontrivial cohomology classes can be isomorphic to the initial algebra, e.g., we confirm Grishkov's implicit claim and explicitly describe the Jurman algebra as such a "semitrivial" deform of the derived of the alternate Hamiltonian Lie algebra. This paper helps to sharpen the formulation of a conjecture describing all simple finite-dimensional Lie algebras over any algebraically closed field of nonzero characteristic and supports a conjecture of Dzhumadildaev and Kostrikin stating that all simple finite-dimensional modular Lie algebras are either of "standard" type or deforms thereof. In characteristic 2, we give sufficient conditions for the known deformations to be semitrivial.