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Loops, their cores and symmetric spaces. (English) Zbl 0909.20052
Some relations between global symmetric spaces, global smooth Bol and Moufang loops and the corresponding 3-webs are investigated. Properties of groups generated by the left and right translations of differentiable connected Moufang loops are described and some propositions for symmetric space structure connected with a differentiable Bol loop are proved. One corollary of the theory is that every differentiable connected Moufang loop is analytic. Some well-known local results of the Bol loop theory are obtained in an original way. In particular, generalization of the classical Hausdorff-Campbell formulae for the class of the left alternative local loops is found. Every connected differentiable Bol loop satisfying the identity \(x(y^2x)=y(x^2y)\) must be an abelian group.

MSC:
20N05 Loops, quasigroups
53C35 Differential geometry of symmetric spaces
22A30 Other topological algebraic systems and their representations
53A60 Differential geometry of webs
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