Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 089, 101 pages      arXiv:1510.07255      https://doi.org/10.3842/SIGMA.2020.089
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Simple Vectorial Lie Algebras in Characteristic 2 and their Superizations

Sofiane Bouarroudj a, Pavel Grozman b, Alexei Lebedev b, Dimitry Leites ac and Irina Shchepochkina d
a) New York University Abu Dhabi, Division of Science and Mathematics, P.O. Box 129188, United Arab Emirates
b) Equa Simulation AB, Raasundavägen 100, Solna, Sweden
c) Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
d) Independent University of Moscow, Bolshoj Vlasievsky per. 11, 119002 Moscow, Russia

Received September 25, 2019, in final form August 25, 2020; Published online September 24, 2020

Abstract
We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectorial Lie superalgebras. We consider odd parameters of deformations. For all 15 Weisfeiler gradings of the 5 exceptional families, and one Weisfeiler grading for each of 2 serial simple complex Lie superalgebras (with 2 exceptional subseries), we describe their characteristic-2 analogs - new simple Lie algebras. Descriptions of several of these analogs, and of their desuperizations, are far from obvious. One of the exceptional simple vectorial Lie algebras is a previously unknown deform (the result of a deformation) of the characteristic-2 version of the Lie algebra of divergence-free vector fields; this is a new simple Lie algebra with no analogs in characteristics distinct from 2. In characteristic 2, every simple Lie superalgebra can be obtained from a simple Lie algebra by one of the two methods described in arXiv:1407.1695. Most of the simple Lie superalgebras thus obtained from simple Lie algebras we describe here are new.

Key words: modular vectorial Lie algebra, modular vectorial Lie superalgebra.

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