Vol. 91, No. 1, 1980

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Analytic H-spaces, Campbell-Hausdorff formula, and alternative algebras

John P. Holmes and Arthur Argyle Sagle

Vol. 91 (1980), No. 1, 105–134

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.

Mathematical Subject Classification 2000
Primary: 17D05
Secondary: 55P99
Received: 12 July 1978
Revised: 6 December 1979
Published: 1 November 1980
John P. Holmes
Arthur Argyle Sagle