Symplectic Double Extensions for Restricted Quasi-Frobenius Lie (Super)Algebras

TL;DR

This paper introduces a method for constructing symplectic double extensions of restricted quasi-Frobenius Lie superalgebras, analyzing cohomological obstructions and providing explicit examples.

Contribution

It develops a new approach for symplectic double extensions in the context of restricted Lie superalgebras, including conditions and obstructions.

Findings

Cocycles in restricted cohomology obstruct symplectic double extensions.

A necessary condition for a restricted Lie superalgebra to be a symplectic double extension.

Explicit examples illustrating the construction and obstructions.

Abstract

In this paper, we present a method of symplectic double extensions for restricted quasi-Frobenius Lie superalgebras. Certain cocycles in the restricted cohomology represent obstructions to symplectic double extension, which we fully describe. We found a necessary condition for which a restricted quasi-Frobenius Lie superalgebras is a symplectic double extension of a smaller restricted Lie superalgebra. The constructions are illustrated with a few examples.