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On Nijenhuis Lie triple systems

Shuangjian Guo, Bibhash Mondal, Ripan Saha

Abstract

In this paper, we investigate the mathematical structure of Nijenhuis Lie triple systems, an extension of classical Lie triple systems augmented with the Nijenhuis operator. Our study focuses on the cohomology of Nijenhuis Lie triple systems and demonstrates how abelian extensions of Nijenhuis Lie triple systems are related to cohomology groups. Additionally, we define Nijenhuis Lie triple 2-systems and also classify `strict' and `skeletal' Nijenhuis Lie triple 2-systems in terms of crossed modules and the cohomology of Nijenhuis Lie triple systems.

On Nijenhuis Lie triple systems

Abstract

In this paper, we investigate the mathematical structure of Nijenhuis Lie triple systems, an extension of classical Lie triple systems augmented with the Nijenhuis operator. Our study focuses on the cohomology of Nijenhuis Lie triple systems and demonstrates how abelian extensions of Nijenhuis Lie triple systems are related to cohomology groups. Additionally, we define Nijenhuis Lie triple 2-systems and also classify `strict' and `skeletal' Nijenhuis Lie triple 2-systems in terms of crossed modules and the cohomology of Nijenhuis Lie triple systems.
Paper Structure (6 sections, 11 theorems, 58 equations)

This paper contains 6 sections, 11 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaires
  3. Nijenhuis Lie triple systems and representations
  4. Cohomology of Nijenhuis Lie triple systems
  5. Abelian extension of Nijenhuis Lie triple systems
  6. Skeletal Nijenhuis Lie triple 2-systems and crossed modules

Key Result

Proposition 3.3

(CM) Let $(T_N , [\cdot, \cdot, \cdot])$ be a Nijenhuis Lie triple system. Then $(T, [\cdot, \cdot, \cdot]_N)$ is a Lie triple system, where and $N$ is a homomorphism from $(T, [\cdot, \cdot, \cdot]_N)$ and $(T, [\cdot, \cdot, \cdot])$.

Theorems & Definitions (39)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Example 3.4
  • Definition 3.5
  • ...and 29 more