Multiplications on homogeneous spaces, nonassociative algebras and connections

Abstract

In this paper we show how nonassociative algebras over the real numbers arise from multiplications on certain homogeneous spaces; that is, an analytic function : M x M -> M. Then these algebras are used to obtain an invariant connection V on the homogeneous space and we give some applications of nonassociative algebras to these topics. Conversely every finite dimensional nonassociative algebra over the real numbers arises from an invariant connection and a local multiplication on a homogeneous space. Thus, analogous to the theory of Lie groups and Lie algebras, much of the basic theory of nonassociative algebras can be formulated in terms of multiplications and connections and conversely.