Nonabelian embedding tensors on 3-Lie algebras and 3-Leibniz-Lie algebras

TL;DR

This paper introduces nonabelian embedding tensors on 3-Lie algebras, defines associated 3-Leibniz-Lie algebras, develops their cohomology theory, and explores their deformations and connections to Lie algebras.

Contribution

It defines nonabelian embedding tensors on 3-Lie algebras and introduces 3-Leibniz-Lie algebras as their underlying structure, advancing the algebraic theory.

Findings

Developed cohomology for nonabelian embedding tensors

Characterized infinitesimal deformations via first cohomology

Studied induced tensors from Lie algebras

Abstract

In this paper, first we introduce the notion of a nonabelian embedding tensor on the 3-Lie algebra. Then, we introduce the notion of a 3-Leibniz-Lie algebra, which is the underlying algebraic structure of a nonabelian embedding tensor on the 3-Lie algebra, and can also be viewed as a nonabelian generalization of a 3-Leibniz algebra. Next we develop the cohomology of nonabelian embedding tensors on 3-Lie algebras with coefficients in a suitable representation and use the first cohomology group to characterize infinitesimal deformations. Finally, we investigate nonabelian embedding tensors on 3-Lie algebras induced by Lie algebras.