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Representations and cohomology of a matched pair of Lie algebras, and $L_\infty$-algebras

Anusuiya Baishya, Apurba Das

Abstract

The notion of a matched pair of Lie algebras was introduced in the study of Lie bialgebras and Poisson-Lie groups. In this paper, we introduce representations and cohomology of a matched pair of Lie algebras. We show that there is a morphism from the cohomology of a Lie bialgebra to the cohomology of the corresponding matched pair of Lie algebras. Our cohomology is also useful to study infinitesimal deformations and abelian extensions. In the last part, we define the notion of a matched pair of $L_\infty$-algebras and construct the corresponding bicrossed product. Finally, we show that a matched pair of skeletal $L_\infty$-algebras is closely related to the cohomology introduced in the paper.

Representations and cohomology of a matched pair of Lie algebras, and $L_\infty$-algebras

Abstract

The notion of a matched pair of Lie algebras was introduced in the study of Lie bialgebras and Poisson-Lie groups. In this paper, we introduce representations and cohomology of a matched pair of Lie algebras. We show that there is a morphism from the cohomology of a Lie bialgebra to the cohomology of the corresponding matched pair of Lie algebras. Our cohomology is also useful to study infinitesimal deformations and abelian extensions. In the last part, we define the notion of a matched pair of -algebras and construct the corresponding bicrossed product. Finally, we show that a matched pair of skeletal -algebras is closely related to the cohomology introduced in the paper.
Paper Structure (11 sections, 22 theorems, 140 equations)

This paper contains 11 sections, 22 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Lie algebras and matched pair of Lie algebras
  3. Representations of a matched pair of Lie algebras
  4. Cohomology of a matched pair of Lie algebras
  5. Maurer-Cartan characterization and cohomology of a matched pair of Lie algebras
  6. Cohomology with coefficients in a representation
  7. Infinitesimal deformations and abelian extensions
  8. Infinitesimal deformations
  9. Abelian extensions
  10. Matched pairs of $L_\infty$-algebras
  11. Matched pairs of skeletal $L_\infty$-algebras

Key Result

Theorem 2.1

(Nijenhuis-Richardson) Let $\mathfrak{g}$ be a vector space. (i) Then the Nijenhuis-Richardson bracket $[~,~]_\mathrm{NR}$ makes the graded space $\oplus_{n \geq 0} \mathrm{Hom} (\wedge^{n+1} \mathfrak{g}, \mathfrak{g})$ of all skew-symmetric multilinear maps on $\mathfrak{g}$ into a graded Lie alge

Theorems & Definitions (60)

  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 2.6
  • Remark 2.7
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • ...and 50 more