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Enveloping algebras of Malcev algebras. (English) Zbl 1224.17034

Summary: We first discuss the construction by J. M. Pérez-Izquierdo and I. P. Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras (both nonassociative and alternative) of the 4-dimensional solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We include a proof (due to I. P. Shestakov) that the universal alternative enveloping algebra of the real 7-dimensional simple Malcev algebra is isomorphic to the 8-dimensional division algebra of real octonions. We conclude with some brief remarks on tangent algebras of analytic Bol loops and monoassociative loops.

MSC:

17D10 Mal’tsev rings and algebras
17D05 Alternative rings
17D15 Right alternative rings
20N05 Loops, quasigroups
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