Wedderburn principal theorem for Jordan superalgebras I

TL;DR

This paper investigates the structure of finite dimensional Jordan superalgebras over an algebraically closed field of characteristic zero, establishing conditions under which a Wedderburn Principal Theorem analogue holds, with specific restrictions on modules.

Contribution

It extends the Wedderburn Principal Theorem to certain Jordan superalgebras by identifying necessary restrictions on irreducible subsuperbimodules, supported by counterexamples.

Findings

WPT analogue holds under specific restrictions

Counterexamples demonstrate the necessity of restrictions

Classification of Jordan superalgebras with solvable radical

Abstract

We consider finite dimensional Jordan superalgebras $\jor$ over an algebraically closed field of characteristic 0, with solvable radical $\rad$ such that $\radd=0$ and $\jor/\rad$ is a simple Jordan superalgebra of one of the following types: Kac $\kac$, Kaplansky $\mathcal{K}_3$ superform or $\algdt$. We prove that an analogue of the Wedderburn Principal Theorem (WPT) holds if certain restrictions on the types of irreducible subsuperbimodules of $N$ are imposed, where $N$ is considered as a $J$-superbimodule, and $J$ is a simple Jordan superalgebra. Using counterexamples, it is shown that the imposed restrictions are essential.