Simple Special Jordan Superalgebras with Associative Nil-Semisimple Even Part

Abstract

We describe unital simple special Jordan superalgebras with associative nil-semisimple even part. In every such superalgebra J=A+M, either M is an associative and commutative A-module, or the associator space (A,A,M) coincides with M. In the former case, if JJ is not a superalgebra of the non-degenerate bilinear superform then its even part A is a differentiably simple algebra and its odd part M is a finitely generated projective A-module of rank 1. Multiplication in M is defined by fixed finite sets of derivations and elements of A. If, in addition, M is one-generated then the initial superalgebra is a twisted superalgebra of vector type. The condition of being one-generated for M is satisfied, for instance, if A is local or isomorphic to a polynomial algebra. We also give a description of superalgebras for which (A,A,M)≠0 and M⋂[A,M]≠0, where [,] is a commutator in the associative enveloping superalgebra of J. It is shown that such each infinite-dimensional superalgebra may be obtained from a simple Jordan superalgebra whose odd part is an associative module over the even.