Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cohomologies of difference Lie groups and van Est theorem

Jun Jiang, Yunnan Li, Yunhe Sheng

Abstract

A difference Lie group is a Lie group equipped with a difference operator, equivalently a crossed homomorphism with respect to the adjoint action. In this paper, first we introduce the notion of a representation of a difference Lie group, and establish the relation between representations of difference Lie groups and representations of difference Lie algebras via differentiation and integration. Then we introduce a cohomology theory for difference Lie groups and justify it via the van Est theorem. Finally, we classify abelian extensions of difference Lie groups using the second cohomology group as applications.

Cohomologies of difference Lie groups and van Est theorem

Abstract

A difference Lie group is a Lie group equipped with a difference operator, equivalently a crossed homomorphism with respect to the adjoint action. In this paper, first we introduce the notion of a representation of a difference Lie group, and establish the relation between representations of difference Lie groups and representations of difference Lie algebras via differentiation and integration. Then we introduce a cohomology theory for difference Lie groups and justify it via the van Est theorem. Finally, we classify abelian extensions of difference Lie groups using the second cohomology group as applications.
Paper Structure (11 sections, 17 theorems, 88 equations)

This paper contains 11 sections, 17 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Representations of difference Lie groups and difference Lie algebras
  3. Representations of difference Lie groups
  4. Differentiation and integration of representations
  5. Cohomologies of difference Lie groups
  6. Cohomologies of difference operators on Lie groups
  7. Cohomologies of difference Lie groups
  8. The van Est theorem for cohomologies of difference Lie groups
  9. Cohomologies of difference Lie algebras
  10. The van Est theorem
  11. Abelian extensions of difference Lie groups

Key Result

Lemma 2.7

Let $(G, \mathcal{D})$ be a difference Lie group. Then $\mathcal{D}(e_G)=e_G$, where $e_G$ is the unit of $G$, and

Theorems & Definitions (51)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 41 more