Mathematical Research Letters

Volume 22 (2015)

Number 2

Lie algebra deformations in characteristic $2$

Pages: 353 – 402

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n2.a3

Authors

Sofiane Bouarroudj (Division of Science and Mathematics, New York University Abu Dhabi, United Arab Emirates)

Alexei Lebedev (Equa Simulation AB, Stockholm, Sweden)

Dimitry Leites (Department of Mathematics, Stockholm University, Stockholm, Sweden)

Irina Shchepochkina (Independent University of Moscow, Russia)

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.

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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