Solvable Lie algebras derived from Lie hyperalgebras

Abstract

Recently in \cite{s-n}, we have investigated Lie algebras and abelian Lie algebras derived from Lie hyperalgebras using the fundamental relations $\mathcal{L}$ and $\mathcal{A}$, respectively. In the present paper, continuing this method we obtain solvable Lie algebras from Lie hyperalgebras by ${\mathcal{S}}_n$-relations. We show that ${\bigcap }_{n\ge 1}{\mathcal{S}}_n^{\ast }$ is the smallest equivalence relation on a Lie hyperalgebra such that the quotient structure is a solvable Lie algebra. We also provide some necessary and sufficient conditions for transitivity of the relation ${\mathcal{S}}_n$ using the notion of ${\mathcal{S}}_n$-part.