Mikheev's construction for Mal'tsev coalgebras (Q1936273)

A construction due to \textit{P. O. Mikheev} [Algebra Logic 31, No. 2, 106--110 (1992); translation from Algebra Logika 31, No. 2, 167--173 (1992; Zbl 0798.17020)] reveals that any Mal'tsev algebra can be embedded in a Lie algebra on which a pair of automorphisms of a special form acts (later known as triality), and conversely, for a Lie algebra \(L\) with such automorphisms, it is possible to embed a Mal'tsev algebra into \(L\). The paper under review presents a dual of Mikheev's result between Mal'tsev coalgebras and Lie coalgebras with triality. Let \((L,\Delta)\) be a Lie coalgebra over a field of characteristic \(\neq 2,3\), fix automorphisms \(\sigma\) and \(\rho\) of \(L\) such that \(\sigma ^ 2 =Id\), \(\rho^ 3 =Id\), \(\sigma \rho=\rho^ 2 \sigma\) and \((\rho^ 2 +\rho+Id)(\sigma -Id)=0\). In this condition, \(L\) is called a Lie coalgebra with \textit{triality}. Assume \(M=\{x \in L : \sigma (x)=-x \}\) and \(M_\ast=\{f \in L^\ast : \sigma^\ast (f)=-f \}\), where \(\sigma^\ast\) and \(\rho^\ast\) are conjugates of the maps \(\sigma\) and \(\rho\). A main result says that \( L^\ast\) is a Lie algebra with triality \(\sigma^\ast\) and \(\rho^\ast\), and also that \(M\) can be provided with a structure of a Mal'tsev coalgebra embedded in \(L\) with dual Mal'tsev coalgebra \(M_\ast\). Conversely, given a Mal'tsev coalgebra M over a field of characteristic \(\neq 2,3\), the authors define a Lie coalgebra with indicated conditions where M is embedded.