Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the flag structure and classification of the holomorphic curves on C*-algebras

Zhimeng Chen, Jing Xu

Abstract

In this note, we will define the formulas of curvature and it's covariant derivatives for holomorphic curves on C*-algebras for the multivariable case. As applications, the unitarily and similarly classification theorems for holomorphic bundle and commuting operator tuples in Cowen-Douglas class are given.

On the flag structure and classification of the holomorphic curves on C*-algebras

Abstract

In this note, we will define the formulas of curvature and it's covariant derivatives for holomorphic curves on C*-algebras for the multivariable case. As applications, the unitarily and similarly classification theorems for holomorphic bundle and commuting operator tuples in Cowen-Douglas class are given.
Paper Structure (7 sections, 20 theorems, 152 equations)

This paper contains 7 sections, 20 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. The curvature of extended holomorphic curves on $C^{*}$-algebra
  3. Extended holomorphic curves
  4. The Cowen-Douglas class
  5. The curvature and connection
  6. Similarity and unitary equivalent of extended holomorphic curvature
  7. The flag structure of the class $\mathcal{FB}_{n}(\Omega)$

Key Result

Lemma 2.5

Let $\mathcal{I}\in\mathcal{I}_{n}(\Omega,\mathcal{U})$ be an extended holomorphic curve on $C^{*}$-algebras. Then the covariant derivative $\mathscr{K}_{I,J}(\mathcal{I})$ of the curvature $\mathscr{K}(\mathcal{I})$ of $\mathcal{I}$ has the following properties:

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Theorem 2.7
  • proof
  • Corollary 2.8
  • ...and 36 more