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Non-abelian extensions and Wells exact sequences of Lie-Yamaguti algebras

Qinxiu Sun, Zhen Li

Abstract

The goal of the present paper is to investigate non-abelian extensions of Lie-Yamaguti algebras and explore extensibility of a pair of automorphisms about a non-abelian extension of Lie-Yamaguti algebras. First, we study non-abelian extensions of Lie-Yamaguti algebras and classify the non-abelian extensions in terms of non-abelian cohomology groups. Next, we characterize the non-abelian extensions in terms of Maurer-Cartan elements. Moreover, we discuss the equivalent conditions of the extensibility of a pair of automorphisms about a non-abelian extension of Lie-Yamaguti algebras, and derive the fundamental sequences of Wells in the context of Lie-Yamaguti algebras. Finally, we discuss the previous results in the case of abelian extensions of Lie-Yamaguti algebras.

Non-abelian extensions and Wells exact sequences of Lie-Yamaguti algebras

Abstract

The goal of the present paper is to investigate non-abelian extensions of Lie-Yamaguti algebras and explore extensibility of a pair of automorphisms about a non-abelian extension of Lie-Yamaguti algebras. First, we study non-abelian extensions of Lie-Yamaguti algebras and classify the non-abelian extensions in terms of non-abelian cohomology groups. Next, we characterize the non-abelian extensions in terms of Maurer-Cartan elements. Moreover, we discuss the equivalent conditions of the extensibility of a pair of automorphisms about a non-abelian extension of Lie-Yamaguti algebras, and derive the fundamental sequences of Wells in the context of Lie-Yamaguti algebras. Finally, we discuss the previous results in the case of abelian extensions of Lie-Yamaguti algebras.
Paper Structure (7 sections, 23 theorems, 130 equations)

This paper contains 7 sections, 23 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Lie-Yamaguti algebras
  3. Non-abelian extensions and non-abelian (2,3)-cocycles of Lie-Yamaguti algebras
  4. Non-abelian extensions in terms of Maurer-Cartan elements
  5. Extensibility of a pair of Lie-Yamaguti algebra automorphisms
  6. Wells exact sequences for Lie-Yamaguti algebras
  7. Particular case: abelian extensions of Lie-Yamaguti algebras

Key Result

Proposition 2.3

Let $(\mathfrak g,[ \ , \ ]_{\mathfrak g},\{ \ , \ , \ \}_{\mathfrak g})$ be a Lie-Yamaguti algebra and $V$ a vector space. Assume that $\mu:\mathfrak g\longrightarrow {gl}(V)$ is a linear map and $\theta,D:\mathfrak g\wedge \mathfrak g\longrightarrow {gl}(V)$ are bilinear maps. Then $(V,\mu,\t and for all $x,y,z\in T,u,v,w\in V$. The Lie-Yamaguti algebra $(\mathfrak g\oplus V,[ \ , \ ],\

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Example 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 3.5
  • proof
  • ...and 36 more