Representations of simple noncommutative Jordan superalgebras I

TL;DR

This paper initiates the study of representations of simple finite-dimensional noncommutative Jordan superalgebras, establishing their reducibility properties and analyzing specific classes with new theoretical approaches.

Contribution

It provides the first systematic analysis of representations of simple noncommutative Jordan superalgebras, including reducibility results and the Kronecker factorization theorem for certain superalgebras.

Findings

Finite-dimensional representations are completely reducible for degree ≥ 3.

Superalgebras D_t(α,β,γ) and K_3(α,β,γ) are studied in detail.

Kronecker factorization theorem proved for D_t(α,β,γ).

Abstract

In this article we begin the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree $\geq 3$ we show that any finite-dimensional representation is completely reducible and, depending on the superalgebra, quasiassociative or Jordan. Then we study representations of superalgebras $D_t(\alpha,\beta,\gamma)$ and $K_3(\alpha, \beta, \gamma)$ and prove the Kronecker factorization theorem for superalgebras $D_t(\alpha,\beta,\gamma)$. In the last section we use a new approach to study noncommutative Jordan representations of simple Jordan superalgebras.