Representations of the Grassmann Poisson superalgebras

2025/11/29 by I. P. Shestakov, Shestakov, Ivan, Ualbai Umirbaev +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2512.00585

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 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≅ Q⊗ 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 Kan G_n is isomorphic to the supermodule Kan V, where V is an irreducible dot-bracket supermodule with a Jordan bracket over G_n.

Citations

• On a problem by Nathan Jacobson
• Irreductible Representations of the Simple Jordan Superalgebra of Grassmann Poisson bracket

Related

• Irreductible Representations of the Simple Jordan Superalgebra of Grassmann Poisson bracket
• On Poisson superalgebras in characteristic 2
• Poisson superbialgebras
• On the simple transposed Poisson algebras and Jordan superalgebras
• Transposed Poisson superalgebra