Unified products for Jordan algebras. Applications

Abstract

Given a Jordan algebra A and a vector space V, we describe and classify all Jordan algebras containing A as a subalgebra of codimension dimk(V) in terms of a non-abelian cohomological type object JA(V,A). Any such algebra is isomorphic to a newly introduced object called unified productA♮V. The crossed/twisted product of two Jordan algebras are introduced as special cases of the unified product and the role of the subsequent problem corresponding to each such product is discussed. The non-abelian cohomology Hnab2(V,A) associated to two Jordan algebras A and V which classifies all extensions of V by A is also constructed. Several applications and examples are given: we prove that Hnab2(k,kn) is identified with the set of all matrices D∈Mn(k) satisfying 2D3−3D2+D=0, where we consider the abelian Jordan algebra structure on k and kn.