Lie algebra deformations in characteristic $2$

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.

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.

Keywords

Lie algebra, characteristic $2$, Kostrikin-Shafarevich conjecture, Jurman algebra, Kaplansky algebra, deformation

2010 Mathematics Subject Classification

Primary 17B20, 17B50. Secondary 17B25, 17B55, 17B56.

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Published 16 April 2015