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Cohomology and Homotopy of Lie triple systems

Haobo Xia, Yunhe Sheng, Rong Tang

Abstract

In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that $2$-term homotopy Lie triple systems is equivalent to Lie triple $2$-systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple $2$-systems. We show that skeletal Lie triple $2$-systems can be classified the third cohomology group, and strict Lie triple $2$-systems are equivalent to crossed modules of Lie triple systems.

Cohomology and Homotopy of Lie triple systems

Abstract

In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that -term homotopy Lie triple systems is equivalent to Lie triple -systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple -systems. We show that skeletal Lie triple -systems can be classified the third cohomology group, and strict Lie triple -systems are equivalent to crossed modules of Lie triple systems.
Paper Structure (8 sections, 13 theorems, 76 equations)

This paper contains 8 sections, 13 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. The controlling algebra of Lie triple systems
  3. Homotopy Nambu algebras and homotopy Lie triples systems
  4. 2-term homotopy Lie triple systems and Lie triple $2$-systems
  5. 2-term homotopy Lie triple systems
  6. Lie triple 2-systems
  7. The equivalence between 2-term homotopy Lie triple systems and Lie triple $2$-systems
  8. Skeletal and strict Lie triple $2$-systems

Key Result

Theorem 2.2

Rot Maurer-Cartan elements of the graded Lie algebra $(C^*(\mathfrak g,\mathfrak g),\left\llbracket \cdot,\cdot\right\rrbracket )$ are the Nambu algebra structures on the vector space $\mathfrak g$.

Theorems & Definitions (38)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Definition 2.6
  • Example 2.7
  • Theorem 2.8
  • ...and 28 more