Quadratic and symplectic structures on 3-(Hom)-$\rho$-Lie algebras

TL;DR

This paper extends quadratic and symplectic structures to 3-(Hom)-ρ-Lie algebras, introduces 3-pre-(Hom)-ρ-Lie algebras, and explores their properties and representations.

Contribution

It generalizes key geometric structures to a new algebraic setting and introduces 3-pre-(Hom)-ρ-Lie algebras with their representations.

Findings

Properties of quadratic and symplectic structures on 3-(Hom)-ρ-Lie algebras

Introduction of 3-pre-(Hom)-ρ-Lie algebras

Definition of representations for these algebras

Abstract

Our purpose in this paper is the generalization of the notions of quadratic and symplectic structures to the case of 3-(Hom)-$\rho$-Lie algebras. We describe some properties of them by expressing the related lemmas and theorems. Also, we introduce the concept of 3-pre-(Hom)-$\rho$-Lie algebras and define their representation.