On classification and deformations of Lie-Rinehart superalgebras

TL;DR

This paper classifies Lie-Rinehart superalgebras over characteristic zero fields in small dimensions, and develops a cohomology-based deformation theory to understand their structural variations.

Contribution

It provides a complete classification of low-dimensional Lie-Rinehart superalgebras and introduces a cohomology framework for their formal deformations.

Findings

Classified all Lie-Rinehart superalgebras with dim(A) ≤ 2 and dim(L) ≤ 4.

Constructed a cohomology complex for these superalgebras.

Developed a theory of formal deformations based on cohomology.

Abstract

The purpose of this paper is to study Lie-Rinehart superalgebras over characteristic zero fields, which are consisting of a supercommutative associative superalgebra $A$ and a Lie superalgebra $L$ that are compatible in a certain way. We discuss their structure and provide a classification in small dimensions. We describe all possible pairs defining a Lie-Rinehart superalgebra for $\dim(A)\leq 2$ and $\dim(L)\leq 4$. Moreover, we construct a cohomology complex and develop a theory of formal deformations based on formal power series and this cohomology.