On Poisson superalgebras in characteristic 2

TL;DR

This paper explores the structure and properties of Poisson superalgebras in characteristic 2, including their representations, cohomology, deformations, and connections with Lie-Rinehart superalgebras, providing new insights and examples.

Contribution

It introduces Lie-Rinehart superalgebras in characteristic 2 and establishes their relation to Poisson superalgebras, along with computations and initial studies of pre-Poisson superalgebras.

Findings

Universal enveloping algebra of a Poisson superalgebra matches that of an associated Lie-Rinehart superalgebra.

Computed explicit examples of Poisson superalgebras in characteristic 2.

Initiated the study of pre-Poisson superalgebras in this setting.

Abstract

This paper is devoted to the study of Poisson superalgebras over fields of characteristic $2$. We investigate their representations, semidirect products, cohomology, formal deformations, and universal enveloping algebras. We also introduce Lie-Rinehart superalgebras in characteristic $2$ and clarify their connections with Poisson superalgebras. In particular, we show that the universal enveloping algebra of a Poisson superalgebra coincides with the universal enveloping algebra of an associated Lie-Rinehart superalgebra. We also compute examples and initiate the study of pre-Poisson superalgebras.