Representations of the Grassmann Poisson superalgebras

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 $G_n$.