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Affine homogeneous structures on analytic loops. (English) Zbl 0647.53012
Generalizing the notion of a homogeneous structure on a Riemannian manifold [see F. Tricerri and L. Vanhecke, Homogeneous structures on Riemannian manifolds (1983; Zbl 0509.53043)], the author introduces the general concept of an affine homogeneous structure and considers such structures on an analytic loop with the canonical affine connection [see the author, Hiroshima Math. J. 5, 141-179 (1975; Zbl 0304.53037)]. The main problem which is under investigation is to find homogeneous loops [see the author, loc. cit.] on a given analytic loop (G,$$\mu)$$ by changing its multiplication $$\mu$$ to a homogeneous one, provided that the unit e of $$\mu$$ is unchanged and every 1-parameter subgroup of $$\mu$$ is changed to a 1-parameter subgroup of the homogeneous loop obtained.
The loop (G,$$\mu)$$ is said to be geodesic at e if it coincides with the geodesic local loop [see M. A. Akivis, Sib. Math. J. 19, 171-178 (1978); translation from Sib. Mat. Zh. 19, 243-253 (1978; Zbl 0388.53007)] of the canonical affine connection centered at e. If (G,$$\mu)$$ is geodesic at e, then any geodesic curve c(t) passing through the unit $$e=c(0)$$ is a 1-parameter subgroup of the loop. Let (G,$$\mu)$$ be a connected and simply connected geodesic loop. If the canonical affine connection of (G,$$\mu)$$ admits an affine homogeneous structure, then, by changing every 1-parameter subgroup of $$\mu$$ to a 1-parameter subgroup of $$\mu$$ ’, the loop (G,$$\mu)$$ can be changed to a homogeneous loop (G,$$\mu$$ ’). This result allows a criterion for the existence of a change of a homogeneous loop to a homogeneous one to be found in terms of the tangent Lie triple algebras.
Reviewer: V.V.Goldberg

##### MSC:
 53A60 Differential geometry of webs 53C30 Differential geometry of homogeneous manifolds 22A30 Other topological algebraic systems and their representations