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 ${\rm dim}_k (V)$ in terms of a non-abelian cohomological type object ${\mathcal J}_{A} \, (V, \, A)$. Any such algebra is isomorphic to a newly introduced object called \emph{unified product} $A \, \natural \, V$. 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 ${\rm H}^2_{\rm nab} \, (V, \, A )$ 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 ${\rm H}^2_{\rm nab} \, (k, \, k^n)$ is identified with the set of all matrices $D\in M_n(k)$ satisfying $2\, D^3 - 3 \, D^2 + D = 0$.