Prime alternative superalgebras of arbitrary characteristic

Abstract

Simple nonassociative alternative superalgebras are classified. Any such superalgebra either is trivial (i.e., has zero odd part) or has characteristic 2 or 3 and is isomorphic over its center to a superalgebra of one of the following five types: in characteristic 3, these are two superalgebras of dimensions 3 and 6 and a “twisted superalgebra of vector type,” which either is infinite-dimensional or has dimension 2·3n; in characteristic 2, those are either a Cayley-Dixon algebra with a grading induced by the Cayley-Dixon process or a “double Cayley-Dixon algebra.” Under certain constraints on the structure of even parts, we also give a description of prime nonassociative alternative nontrivial superalgebras in terms of central orders of simple superalgebras. The simple superalgebras of dimensions 3 and 6 are then used to construct simple Jordan superalgebras of characteristic 3 and of dimensions 12 and 21, respectively.