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

Wen Teng, Xiansheng Dai

TL;DR

The paper develops a framework for nonabelian embedding tensors on $3$-Lie algebras by introducing $3$-Leibniz-Lie algebras as the underlying algebraic structure and a descendent $3$-Leibniz algebra on the embedding tensor’s domain. It constructs a cohomology theory with a coboundary operator $oldsymbol riangle_ abla$ whose first cohomology group $ rak H^1_ abla(H,L)$ classifies infinitesimal deformations of the embedding tensor $ abla:H o L$, and shows how equivalence of deformations corresponds to coboundaries. The paper then proves that nonabelian embedding tensors on $3$-Lie algebras can be induced from Lie algebras via trace maps, giving explicit constructions such as the $3$-Lie algebra $L_oldsymbol{ au}$ and the coherent actions $ ho_oldsymbol{ au}$, thereby linking higher Lie structures with cohomological deformation theory. Overall, the work extends embedding-tensor theory from Lie algebras to the $3$-Lie setting and provides a cohesive algebraic and cohomological toolkit for higher gauge theories and related applications.

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.

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

TL;DR

The paper develops a framework for nonabelian embedding tensors on -Lie algebras by introducing -Leibniz-Lie algebras as the underlying algebraic structure and a descendent -Leibniz algebra on the embedding tensor’s domain. It constructs a cohomology theory with a coboundary operator whose first cohomology group classifies infinitesimal deformations of the embedding tensor , and shows how equivalence of deformations corresponds to coboundaries. The paper then proves that nonabelian embedding tensors on -Lie algebras can be induced from Lie algebras via trace maps, giving explicit constructions such as the -Lie algebra and the coherent actions , thereby linking higher Lie structures with cohomological deformation theory. Overall, the work extends embedding-tensor theory from Lie algebras to the -Lie setting and provides a cohesive algebraic and cohomological toolkit for higher gauge theories and related applications.

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.
Paper Structure (5 sections, 15 theorems, 56 equations)

This paper contains 5 sections, 15 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Nonabelian embedding tensors on 3-Lie algebras
  3. 3-Leibniz-Lie algebras
  4. Cohomology theory of nonabelian embedding tensors on 3-Lie algebras
  5. Nonabelian embedding tensors on 3-Lie algebras induced by Lie algebras

Key Result

Proposition 2.5

Let $(L, [-,-,- ]_L)$ and $(H, [-,-,- ]_H)$ be two 3-Lie algebras and $\rho: \wedge^2 L \rightarrow \mathrm{End}(H)$ a bilinear map. Then $(H, [-,-,-]_H; \rho^{\dag})$ is a coherent action of $L$ if and only if $L \oplus H$ is a 3-Leibniz algebra under the following map: for any $l_1, l_2, l_3\in L$ and $h_1,h_2,h_3\in H$. $(L \oplus H, [-,-,-]_{\rho})$ is called the nonabelian hemisemidirect pr

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • Example 2.8
  • Theorem 2.9
  • ...and 31 more