Representations of simple noncommutative Jordan superalgebras I

Abstract

In this article we begin the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree $\ge 3$ 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.