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Deformations and cohomologies of embedding tensors on 3-Lie algebras

Meiyan Hu, Shuai Hou, Lina Song, Yanqiu Zhou

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.

Deformations and cohomologies of embedding tensors on 3-Lie algebras

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 -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.
Paper Structure (7 sections, 21 theorems, 71 equations)

This paper contains 7 sections, 21 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Embedding tensors on $3$-Lie algebras
  3. Maurer-Cartan characterization of embedding tensors on $3$-Lie algebras
  4. The cohomology of embedding tensors
  5. Deformations of embedding tensors
  6. Formal deformations of embedding tensors on a 3-Lie algebra
  7. Order $n$ deformations of embedding tensors on a 3-Lie algebra

Key Result

Proposition 2.4

With the above notations, $(\mathfrak g\oplus V,[\cdot,\cdot,\cdot]_{\rho})$ is a $3$-Leibniz algebra, which is called the hemisemidirect product $3$-Leibniz algebra, and denoted by $\mathfrak g\ltimes_{\rho} V$.

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 50 more