Embedding of strongly simple Moufang loops in simple alternative algebras
A pair \((L,R)\) of maps from \(G\) into the group \(\text{Aut }M\) is called a linear representation of the quasigroup \(G\) over the class of quasigroups \(k\) if the space \(M\), equipped with the compositions \(ax=L_a x\) and \(xa=R_a x\), is a \(G\)-module over \(k\). Let \(k\) be the class of all Moufang loops, \(G\in k\) and \(M\neq 0\) be some \(G\)-module. The representation \((L,R):G\to\text{Aut }M\) is called irreducible if \(M\) and 0 are the only invariant submodules of \(M\). The author proves that: (i) every irreducible \(G\)-module \(M\) of the Moufang loop \(G\) has dimension 4 or 8; (ii) every simple nonassociative Moufang loop possessing an irreducible nontrivial representation may be embedded in a simple alternative algebra. These results are a continuation of the author's paper [Commun. Algebra 21, No. 7, 2527-2536 (1993; Zbl 0793.20065)].
scientific article; zbMATH DE number 795651