Embeddability of Mal'tsev algebras in Lie algebras
Let \(L\) be a Lie algebra over a field of characteristic \(\ne 2,3\), and let \(\sigma\), \(\rho\) be nonidentity automorphisms of \(L\) such that \(\sigma^2= \text{Id}\), \(\rho^3= \text{Id}\), \(\sigma\rho= \rho^2\sigma\). Then \(H= \{x\in L\mid \sigma(x)=x\}\) is a subalgebra of \(L\) and \(M= \{x\in L\mid \sigma(x) =-x\}\) is a subspace. Suppose that \((\rho^2+ \rho+ \text{Id})(x)=0\) for \(x\in M\). Then \(L=M+ \rho(H)\) (vector space direct sum). Let \(\pi: L\to M\) be the projection with kernel \(\rho(H)\). If multiplication is defined on \(M\) by \(x\cdot y= \pi([ x,y ])\), where \([x,y]\) is the multiplication in \(L\), then \(M(\cdot)\) is a Mal'tsev algebra. Conversely, given any Mal'tsev algebra \(M(\cdot)\) over a field of characteristic \(\ne 2,3\), then there is a Lie algebra \(L\) which satisfies the indicated conditions and has \(M(\cdot)\) as the Mal'tsev algebra just constructed.
scientific article; zbMATH DE number 536658