Deformations and cohomologies of embedding tensors on 3-Lie algebras

TL;DR

This paper introduces the concept of embedding tensors on 3-Lie algebras, constructs related algebraic structures, and develops a cohomology theory to analyze their deformations and extensions.

Contribution

It defines embedding tensors on 3-Lie algebras, constructs associated Lie 3-algebras and $L_{}$-algebras, and establishes a cohomology framework for studying their deformations.

Findings

Equivalent deformations have the same cohomology class

Obstruction classes determine extendability of deformations

Cohomology groups classify deformation properties

Abstract

In this paper, first we introduce the notion of an embedding tensor on a 3-Lie algebra, which naturally induces a 3-Leibniz algebra. Using the derived bracket, we construct a Lie 3-algebra, whose Maurer-Cartan elements are embedding tensors. Consequently, we obtain the $L_{\infty}$-algebra that governs deformations of embedding tensors. We define the cohomology theory for embedding tensors on 3-Lie algebras. As applications, we show that if two formal deformations of an embedding tensor on a 3-Lie algebra are equivalent, then their infinitesimals are in the same cohomology class in the second cohomology group. Moreover, an order n deformation of an embedding tensor is extendable if and only if the obstruction class, which is in the third cohomology group, is trivial.