Embedding tensors on 3-Leibniz algebras and their derived algebraic structures and deformations

TL;DR

This paper introduces new algebraic structures called 3-tri-Leibniz algebras and embedding tensors, explores their interrelations, and studies their deformations through cohomology, advancing the understanding of algebraic and deformation theory.

Contribution

It defines 3-tri-Leibniz algebras and embedding tensors, establishes their connections, and investigates their deformations via cohomology, providing new insights into algebraic structures.

Findings

Embedding tensors induce 3-tri-Leibniz algebras.

Any 3-tri-Leibniz algebra can embed into an averaging 3-Leibniz algebra.

Linear deformations of embedding tensors are characterized by first cohomology.

Abstract

In this paper, first we introduce the notions of 3-tri-Leibniz algebras and embedding tensors on 3-Leibniz algebras. We show that an embedding tensor gives rise to a 3-tri-Leibniz algebra. Conversely, a 3-tri-Leibniz algebra gives rise to a 3-Leibniz algebra and a representation such that the quotient map is an embedding tensor. Furthermore, any 3-tri-Leibniz algebra can be embedded into an averaging 3-Leibniz algebra. Next, we introduce the notion of 3-tri-Leibniz dialgebras and demonstrate that homomorphic embedding tensors inherently induce 3-tri-Leibniz dialgebras. Finally, we study the linear deformations of embedding tensors by defining first cohomology.