Nonabelian embedding tensors

TL;DR

This paper introduces the concept of nonabelian embedding tensors and their algebraic structures, providing a framework for understanding their deformations and cohomology in a generalized algebraic context.

Contribution

It defines nonabelian embedding tensors, introduces Leibniz-Lie algebras as their underlying structure, and constructs a differential graded Lie algebra governing their deformations.

Findings

Defined nonabelian embedding tensors and Leibniz-Lie algebras.

Constructed a differential graded Lie algebra for deformations.

Characterized linear deformations via second cohomology group.

Abstract

In this paper, first we introduce the notion of a nonabelian embedding tensor, which is a nonabelian generalization of an embedding tensor. Then we introduce the notion of a Leibniz-Lie algebra, which is the underlying algebraic structure of a nonabelian embedding tensor, and can also be viewed as a nonabelian generalization of a Leibniz algebra. Next using the derived bracket, we construct a differential graded Lie algebra, whose Maurer-Cartan elements are exactly nonabelian embedding tensors. Consequently, we obtain the differential graded Lie algebra that governs deformations of a nonabelian embedding tensor. Finally, we define the cohomology of a nonabelian embedding tensor and use the second cohomology group to characterize linear deformations.