# Deformations of symmetric simple modular Lie (super)algebras

Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites

Arxiv ID: 0807.3054•Last updated: 12/1/2022

We say that a Lie (super)algebra is "symmetric" if with every root (with
respect to the maximal torus) it has its opposite of the same multiplicity.
Over algebraically closed fields of positive characteristics we describe the
deforms (results of deformations) of all known simple finite-dimensional
symmetric Lie (super)algebras of rank $<9$, except for superizations of the Lie
algebras with ADE root systems.
The moduli of deformations of any Lie superalgebra constitute a supervariety.
Any infinitesimal deformation given by any odd cocycle is integrable with an
odd parameter running over a supervariety. All deforms corresponding to odd
cocycles are new. Among new results are classification of the deforms of the
29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras
and orthogonal Lie algebras without Cartan matrix in characteristic 2.
For the Lie (super)algebras considered, all cocycles are integrable, the
deforms corresponding to the weight cocycles are usually linear in the
parameter. Problems: describe isomorphic deforms and deforms of superized
algebras of ADE series in characteristic 2.
Appendix: For several modular analogs of complex simple Lie algebras, and
simple Lie algebras indigenous to characteristics 3 and 2, we describe the
space of cohomology with trivial coefficients. We show that the natural
multiplication in this space is very complicated.

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