Deformations of Symmetric Simple Modular Lie (Super)Algebras

TL;DR

This paper computes the cohomology and classifies infinitesimal deformations of symmetric simple modular Lie (super)algebras over fields of positive characteristic, revealing new deformations and complex cohomology structures.

Contribution

It provides the first classification of deformations for a broad class of symmetric simple modular Lie (super)algebras, including new results for specific algebras in small characteristics.

Findings

All odd cocycle deformations are integrable and new.

Classified cocycles for Brown algebra, Weisfeiler-Kac, and orthogonal Lie algebras.

Described cohomology spaces and their complex multiplication structures.

Abstract

We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal 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, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycleis integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. 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.