Representations of quadratic Jordan algebras

Abstract:

Although representations do not play as much of a role in the theory of Jordan algebras as they do in the associative or Lie theories, they are important in considering Wedderburn splitting theorems and other applications. In this paper we develop a representation theory for quadratic Jordan algebras over an arbitrary ring of scalars, generalizing the usual theory for linear Jordan algebras over a field of characteristic $\ne 2$. We define multiplication algebras and representations, characterize these abstractly as quadratic specializations, and relate them to bimodules. We obtain first and second cohomology groups with the usual properties. We define a universal object for quadratic specializations and show it is finite dimensional for a finite-dimensional algebra. The most important examples of quadratic representations, those obtained from commuting linear representations, are discussed and examples are given of new “pathological” representations which arise only in characteristic 2.