$(\mu, \rho, \beta)$-Extension of $3$-Lie algebras

TL;DR

This paper investigates how to extend two 3-Lie algebras into a larger algebra and determines the conditions under which derivations from the original algebras can be extended to the combined structure.

Contribution

It introduces a new extension framework for 3-Lie algebras and establishes necessary and sufficient conditions for derivation extension.

Findings

Conditions for the extension to be a 3-Lie algebra

Necessary and sufficient conditions for derivation extension

Framework for combining 3-Lie algebras

Abstract

We study an extension algebra $A$ from two given $3$-Lie algebras $M$ and $H$, and discuss the extensibility of a pair of derivations, one from the derivation algebra of $M$ and the other from that of $H$, to a derivation of $A$. In particular, we give conditions for such an extension to be a $3$-Lie algebra, and provide necessary and sufficient conditions of the pair of derivations to be extendable.