Modularity in Malcev algebras (Q795914)

If \(H\) is a subalgebra of a Malcev algebra \(M\) we define \(H_M\), the core of \(H\) in \(M\), as the largest ideal of \(M\) contained in \(H\). We say that \(H\) is core-free in \(M\) if \(H_M=0\). In this paper we study analogous concepts to those of quasi-ideal and modular subalgebra in Lie algebras for Malcev algebras in order to make use of them to study the structure of Malcev algebras which have a nontrivial core-free modular subalgebra. We state the main results of this paper: (i) If \(Q\) is a quasi-ideal of a Malcev algebra \(M\) (finite or infinite dimensional over a field of characteristic not two such that \(Q\) is not an ideal of \(M\), then \(M/Q_M\) is a Lie algebra. - (ii) If \(H\) is a modular subalgebra of a finite-dimensional Malcev algebra \(M\) over a field of characteristic zero such that \(H\) is not an ideal of \(M\), then \(M/H_M\) is a Lie algebra.