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Local analytical loops with the identity of right alternativity. (Russian) Zbl 0572.20056
Problems of the theory of webs and quasi-groups, Collect. sci. Works, Kalinin 1985, 72-75 (1985).
[For the entire collection see Zbl 0567.00007.]
Let (M,\(\times,e)\) be a local analytical loop with the right alternativity identity: \((a\times b)\times b=a\times (b\times b)\). In M, in a neighborhood of e, it is possible to define a local multiplication \(R\times M\to N\), \(((t,a)\to t_ ea)\) such that \((M,\times,e,\{t_ e\}_{t\in R})\) is a \(C^{\omega}\)-odule [see the first author, Dokl. Akad. Nauk SSSR 233, 800-803 (1977; Zbl 0375.53021) and S. Kobayashi and K. Nomizu, Foundations of differential geometry (1963; Zbl 0119.375)] with the identity of right alternativity, i.e. \((a\times (t_ eb))\times (s_ eb)=a\times (t+s)_ eb)\). In particular, for a,b\(\in M\), which are close enough to e, and any integers m and n, the following identity holds: \((a\times b^ m)\times b^ k=a\times b^{m+k}\).
Reviewer: V.V.Goldberg

20N05 Loops, quasigroups
17D15 Right alternative rings
53A60 Differential geometry of webs