Vol. 48, No. 1, 1973

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Multiplications on homogeneous spaces, nonassociative algebras and connections

Arthur Argyle Sagle and J. R. Schumi

Vol. 48 (1973), No. 1, 247–266

In this paper we show how nonassociative algebras over the real numbers arise from multiplications on certain homogeneous spaces; that is, an analytic function μ : M × M M. Then these algebras are used to obtain an invariant connection 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.

Mathematical Subject Classification 2000
Primary: 53C30
Secondary: 17E05
Received: 17 February 1971
Revised: 6 June 1972
Published: 1 September 1973
Arthur Argyle Sagle
J. R. Schumi