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Linear representations of analytic Moufang loops. (English) Zbl 0868.22002
The author continues his investigations [Commun. Algebra 21, 2527-2536 (1993; Zbl 0793.20065)] and E. N. Kuzmin’s results [Algebra Logika 10, 3-22 (1971; Zbl 0244.17019)]. It is known that the tangent algebra of a local analytic Moufang loop (LAML) is a Malcev algebra. Kuzmin proved that every finite-dimensional real Malcev algebra is the tangent algebra of some LAML. The main theorem of this paper is: Every $$G$$-module of a LAML $$G$$ in the class $${\mathcal K}$$ of all LAMLs is a Malcev $$A_G$$-module of its tangent algebra $$A_G$$ and, conversely, every finite-dimensional Malcev $$A$$-module of a finite-dimensional real Malcev algebra $$A$$ is a $$G$$-module of some LAML $$G$$ in the class $${\mathcal K}$$.
##### MSC:
 22A30 Other topological algebraic systems and their representations 20N05 Loops, quasigroups 17D10 Mal’tsev rings and algebras