Representations of the Grassmann Poisson superalgebras

TL;DR

The paper classifies irreducible Poisson G_n-modules, showing they are either the regular module or its opposite, and proves a coordinatization theorem for Poisson algebras containing G_n. It then builds a bridge between Poisson, Jordan, and contact Lie structures using the Kantor double Kan and the Kan functor, providing a parametrized family of irreducible modules G_n(β) for the G_n-based categories and demonstrating that irreducible Jordan Kan G_n-modules arise from irreducible dot-bracket G_n-modules. Together, these results reveal a nuanced landscape of representations across multiple bracket frameworks and resolve certain conjectures by exhibiting parameter-dependent families instead of a unique Kan-image of Reg G_n.

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 $G_n$.

Theorems & Definitions (13)

• Proposition 2.1
• Corollary 2.2
• Theorem 3.1
• Theorem 4.1
• Proposition 6.1
• Lemma 6.2
• Lemma 6.3
• Corollary 6.4
• Lemma 6.5
• Lemma 6.6