Wedderburn principal theorem for Jordan superalgebras I

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.