On linear representations of analytic Moufang loops (Q1886113)

\textit{A. I. Mal'tsev} [Mat. Sb. 36(78), 569--576 (1955; Zbl 0065.00702)] and \textit{E. N. Kuz'min} [Algebra Logika 10, No. 1, 3--22 (1971; Zbl 0244.17019); English translation: Algebra Logic 10, No. 1, 1--14 (1971; Zbl 0248.17001)] established a relationship between local analytic Moufang loops and the finite-dimensional Mal'tsev algebras over the field of real numbers; this relationship is similar to the classical correspondence between local Lie groups and Lie algebras. \textit{F. S. Kerdman} [Algebra Logic 18, 325--347 (1980); translation from Algebra Logika 18, 523--555 (1979; Zbl 0457.22002)] showed that these results can be extended to the global analytic Moufang loops. In the paper under review, the author continues to study the relationship between the analytic Moufang loops and the Mal'tsev algebras. He proves two theorems: 1) Any \(G\)-module of an analytic Moufang loop is an \(A_G\)-module of its tangent algebra. Conversely, any \(A_G\)-module of a Mal'tsev algebra \(A_G\) over the field of real numbers is a \(G'\)-module of a simply connected analytic Moufang loop \(G'\) locally isomorphic to \(G\). 2) Any representation of a semisimple analytic Moufang loop is completely reducible. Any irreducible \(G\)-module of an analytic Moufang loop \(G\) is either an alternative bimodule or a Mal'tsev irreducible bimodule.