Reflexion spaces and homogeneous symmetric spaces

Let be S (x): yi—>x-y the left multiplication with x in M. This is an involutive map of M onto itself leaving x fixed, which may be interpreted as the reflexion in the point x. Let % be a finite dimensional Jordan algebra and / the set of invertible elements of §K. In general for x, yCi. 1 their product xy is not in ƒ, so / does not inherit a multiplicative structure from 9I. However, x-y = 2x (xy'~) —xy~ is invertible (l), and the multiplication x • y makes I a reflexion space. Every group is a reflexion space with the new product x-y = xy~x. Every set is a reflexion space with the trivial product x-y = y for all x and y. A reflexion space M where M is a connected paracompact Cmanifold and /x: MXM—+M is differentiate is called a differentiable reflexion space. The following construction gives examples. Let G be a connected Lie group, a an involutive automorphism of G and H a subgroup of G lying between the group of all fixed points of a and its identity component. Then G/His a homogeneous symmetric space and G (G/H, H) is a principal fibre bundle with base space G/H and structure group H. Let H operate on a connected manifold F on the left and let be GXHF the bundle associated with G (G/H% H) with typical fibre F (cf 2). We denote the equivalence class of (g, x) EGXF'mGXHFby g®x. In case F i s apo in t, wehave G X ^ ^ = G / i J.