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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 \({\mathcal K}\).
22A30 Other topological algebraic systems and their representations
20N05 Loops, quasigroups
17D10 Mal’tsev rings and algebras