Reductive Lie-admissible algebras applied to H-spaces and connections

An algebra A with multiplication xy is Lie-admissible if the vector space A with new multiplication (x,y) = xy-yx is a Lie algebra; we denote this Lie algebra by A/sup -/. Thus, an associative algebra is Lie-admissible but a Cayley algebra is not Lie-admissible. In this paper we show how Lie-admissible algebras arise from Lie groups and their application to differential geometry on Lie groups via the following theorem. Let A be an n-dimensional Lie-admissible algebra over the reals. Let G be a Lie group with multiplication function ..mu.. and with Lie algebra g which is isomorphic to A/sup -/. Then there exiss a corrdinate system at the identify e in G which represents ..mu.. by a function F:gxg..-->..g defined locally at the origin, such that the second derivative, F/sup 2/, at the origin defines on the vector space g the structure of a nonassociative algebra (g, F/sup 2/). Furthermore this algebra is isomorphic to A and (g, F/sup 2/)/sup -/ is isomorphic to A/sup -/. Thus roughly, any Lie-admissible algebra is isomorphic to an algebra obtained from a Lie algebra via a change of coordinates in the Lie group. Lie algebras arise by using canonical coordinates and the Campbell-Hausdorffmore » formula. Applications of this show that any G-invariant psuedo-Riemannian connection on G is completely determined by a suitable Lie-admissible algebra. These results extend to H-spaces, reductive Lie-admissible algebras and connections on homogeneous H-spaces. Thus, alternative and other non-Lie-admissible algebras can be utilized.« less