Representations of the Grassmann Poisson superalgebras

TL;DR

This paper classifies irreducible and unital Poisson supermodules over Grassmann Poisson superalgebras and explores their structure within related superalgebra categories, revealing their isomorphisms and reducibility properties.

Contribution

It provides a complete classification of irreducible and unital Poisson supermodules over Grassmann Poisson superalgebras and connects these structures to Jordan supermodules.

Findings

Irreducible Poisson supermodules are isomorphic to the regular supermodule or its opposite.

Unital Poisson supermodules over $G_n$ are completely reducible.

Classification of supermodules over $G_n$ in the context of dot-bracket superalgebras with Jordan brackets.

Abstract

We prove that every irreducible Poisson supermodule over the Grassmann Poisson superalgebra $G_n$ over a field of characteristic different from $2$ is isomorphic to the regular Poisson supermodule $\mathrm{Reg}\,G_n$ or to its opposite supermodule. Moreover, every unital Poisson supermodule over $G_n$ is completely reducible. If $P$ is a unital Poisson superalgebra which contains $G_n$ with the same unit then $P\cong Q\otimes G_n$ for some Poisson superalgebra $Q$. Furthermore, we classify the supermodules over $G_n$ in the category of dot-bracket superalgebras with Jordan brackets, and we prove that every irreducible Jordan supermodule over the Kantor double $\mathrm{Kan}\,G_n$ is isomorphic to the supermodule $\mathrm{Kan}\,V$, where $V$ is an irreducible dot-bracket supermodule with a Jordan bracket over $G_n$.