3-Lie bialgebras and 3-pre-Lie algebras induced by involutive derivations

TL;DR

This paper explores the structure of 3-Lie algebras with involutive derivations, establishing connections to 3-pre-Lie algebras and constructing related 3-Lie bialgebras and solutions to the classical Yang-Baxter equation.

Contribution

It introduces a method to derive compatible 3-pre-Lie algebras from 3-Lie algebras with involutive derivations and constructs associated 3-Lie bialgebras and Yang-Baxter solutions.

Findings

Existence of compatible 3-pre-Lie algebra structures from 3-Lie algebras with involutive derivations.

Construction of local cocycle 3-Lie bialgebras on semi-direct product algebras.

Explicit solutions to the 3-Lie classical Yang-Baxter equation.

Abstract

In this paper, we study the structure of 3-Lie algebras with involutive derivations. We prove that if $A$ is an $m$-dimensional 3-Lie algebra with an involutive derivation $D$, then there exists a compatible 3-pre-Lie algebra $(A, \{ , , , \}_D)$ such that $A$ is the sub-adjacent 3-Lie algebra, and there is a local cocycle $3$-Lie bialgebraic structure on the $2m$-dimensional semi-direct product 3-Lie algebra $A\ltimes_{ad^*} A^*$, which is associated to the adjoint representation $(A, ad)$. By means of involutive derivations, the skew-symmetric solution of the 3-Lie classical Yang-Baxter equation in the 3-Lie algebra $A\ltimes_{ad^*}A^*$, a class of 3-pre-Lie algebras, and eight and ten dimensional local cocycle 3-Lie bialgebras are constructed.