Analytic H-spaces are
shown to be local analytic loops (satisfying the cancellation laws). Then power
associative local analytic loops are investigated and these are shown to be exactly the
class to which a local loop belongs if there is a choice of coordinate system, f, for
which the multiplication obeys V (sx,tx) = sx + tx. Here x is near 0 in Rn, each of
the numbers s, t and s + t is in [0,1] and V is the pulldown of the local loop
multiplication via f. Homomorphism of such local loops are investigated and the
set of such automorphism is shown to be isomorphic to a certain group of
linear maps. Also generalizing the Lie group-Lie algebra situation, certain
anti-commutative algebras are introduced to study these local loops. Finally
these results are applied to local loops whose multiplication is induced by a
power associative algebra. A Campbell-Hausdorff formula is shown to hold
when the algebra is alternative and is related to the inverse property in
the local loop. A relationship between S7 and simple Malcev algebras is
given.
