As a generalization of certain
results for Lie groups it is shown that an n-dimensional H-space (M,μ)
with identity e has a coordinate system at e in which μ can be represented
by a function F : Rn× Rn→ Rn which is analytic at (0,0) and that the
second derivative of F induces a bilinear anti-commutative multiplication α
on Rn. In this way an algebra (Rn,α) analogous to the Lie algebra of a
Lie group is obtained and all such algebras are shown to be isomorphic. If
M = G∕H is a reductive homogeneous space, then these results generalize the
Lie group-Lie algebra correspondence and the algebra (Rn,α) induces a
G-invariant connection on G∕H. Relative to this connection it is shown that an
automorphism of (G∕H,μ) is an affine map and induces an algebra automorphism of
(Rn,α). Also the connection is irreducible if (G∕H,μ) has no proper invariant
subsystems (the analog of normal subgroups). In the case where G∕H has a
Riemannian structure, it may happen that there are no local isometries among the
coordinate maps which give rise to anti-commutative multiplications on
Rn.
