Solvable Lie algebras derived from Lie hyperalgebras

TL;DR

This paper extends the study of Lie hyperalgebras by introducing $ ext{S}_n$-relations to derive solvable Lie algebras, establishing the minimal relation for solvability and conditions for relation transitivity.

Contribution

It introduces $ ext{S}_n$-relations to obtain solvable Lie algebras from hyperalgebras and characterizes the minimal relation needed for solvability.

Findings

$igcap_{n ext{≥}1} ext{S}_n^*$ is the smallest relation for solvable quotient

Provided necessary and sufficient conditions for $ ext{S}_n$-relation transitivity

Established a method to derive solvable Lie algebras from 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 $\bigcap_{n\geq 1}\mathcal{S}^*_n$ 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.