Embedding tensors on 3-Leibniz algebras and their derived algebraic structures and deformations
Wen Teng, Shuangjian Guo
TL;DR
The paper extends embedding-tensor theory to the realm of higher arity by introducing 3-tri-Leibniz algebras and studying embedding tensors on 3-Leibniz algebras. It establishes that embedding tensors yield 3-tri-Leibniz structures and that any 3-tri-Leibniz algebra arises from a 3-Leibniz algebra with a representation, with the quotient map forming an embedding tensor; it also shows that every 3-tri-Leibniz algebra can be embedded into an averaging 3-Leibniz algebra. The work then develops a dialgebra analogue (3-tri-Leibniz dialgebras) via homomorphic embedding tensors and actions, and connects these to semi-direct products and crossed modules. Finally, it develops a deformation theory for embedding tensors using first cohomology, characterizing trivial deformations through Nijenhuis elements and establishing a bijection between deformation classes and HH_{T}^{1}(V, g), thereby providing a cohomological framework for higher-arity tensor embeddings with potential applications to higher gauge theories.
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.