Lie algebra deformations in characteristic $2$

Abstract

Of four types of Kaplansky algebras, type-2 and type-4 algebras have previously unobserved Z/2-gradings: nonlinear in roots. A method assigning a simple Lie superalgebra to every 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. 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.