Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

TL;DR

This paper classifies outer derivations and central extensions of symmetric modular Lie (super)algebras over algebraically closed fields of positive characteristic, focusing on ranks up to 8 and including specific series like periplectic and exceptional types.

Contribution

It provides a comprehensive listing of derivations and extensions for various symmetric modular Lie (super)algebras, including new results specific to positive characteristic fields.

Findings

Outer derivations and central extensions are classified for symmetric modular Lie (super)algebras.

Results include series like periplectic and exceptional Lie algebras, with some findings counterintuitive to characteristic 0 theories.

New insights into the structure of these algebras in positive characteristic are provided.

Abstract

Over algebraically closed fields of positive characteristic, for simple Lie (super)algebras, and certain Lie (super)algebras close to simple ones, with symmetric root systems (such that for each root, there is minus it of the same multiplicity) and of ranks less than or equal to 8 - most needed in an approach to the classification of simple vectorial Lie superalgebras (i.e., Lie superalgebras realized by means of vector fields on a supermanifold), - we list the outer derivations and nontrivial central extensions. When the conjectural answer is clear for the infinite series, it is given for any rank. We also list the outer derivations and nontrivial central extensions of one series of non-symmetric (except when considered in characteristic 2), namely periplectic, Lie superalgebras - the one that preserves the nondegenerate symmetric odd bilinear form, and of the Lie algebras obtained from them by desuperization. We also list the outer derivations and nontrivial central extensions of an analog of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several results indigenous to positive characteristic are of particular interest being unlike known theorems for characteristic 0, some results are, moreover, counterintuitive.