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Differential equations of smooth loops. (Russian) Zbl 0814.22001
A theory of smooth loops generalizing Lie group theory is constructed. First, a system of partial differential equations is found, describing an arbitrary smooth loop (i.e. the system has a unique solution by given initial conditions). The result is detailed for special classes of loops defined by the identities $$(x \cdot x^ m)x^ n = xx^{m + n}$$ (the so-called odules) and $$(x\cdot y^ m)y^ n = xy^{m + n}$$ (right monoalternative loops). Second, the notion of a hyperalgebra is introduced, that is a sequence (generally infinite) of binary operations of special kind given on a finite-dimensional vector space. It is proved that a hyperalgebra is connected with a smooth loop $$M$$, and conversely, every hyperalgebra $$V$$ uniquely defines a smooth loop. The result is detailed for right monoalternative loops. See also the author and P. O. Mikheev [Sov. Math., Dokl. 36, 545-548 (1988); translation from Dokl. Akad. Nauk SSSR 297, 801-804 (1987; Zbl 0659.53018)].

##### MSC:
 22A30 Other topological algebraic systems and their representations 53A60 Differential geometry of webs 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 20N05 Loops, quasigroups