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Compact Quantum Homogeneous Kähler Spaces

Biswarup Das, Réamonn Ó Buachalla, Petr Somberg

Abstract

Noncommutative Kähler structures were recently introduced as an algebraic framework for studying noncommutative complex geometry on quantum homogeneous spaces. In this paper, we introduce the notion of a \emph{compact quantum homogeneous Kähler space} which gives a natural set of compatibility conditions between covariant Kähler structures and Woronowicz's theory of compact quantum groups. Each such object admits a Hilbert space completion possessing a remarkably rich yet tractable structure. The analytic behaviour of the associated Dolbeault-Dirac operators is moulded by the complex geometry of the underlying calculus. In particular, twisting the Dolbeault-Dirac operator by a negative Hermitian holomorphic module is shown to give a Fredholm operator if and only if the top anti-holomorphic cohomology group is finite-dimensional. In this case, the operator's index coincides with the twisted holomorphic Euler characteristic of the underlying noncommutative complex structure. The irreducible quantum flag manifolds, endowed with their Heckenberger-Kolb calculi, are presented as motivating examples.

Compact Quantum Homogeneous Kähler Spaces

Abstract

Noncommutative Kähler structures were recently introduced as an algebraic framework for studying noncommutative complex geometry on quantum homogeneous spaces. In this paper, we introduce the notion of a \emph{compact quantum homogeneous Kähler space} which gives a natural set of compatibility conditions between covariant Kähler structures and Woronowicz's theory of compact quantum groups. Each such object admits a Hilbert space completion possessing a remarkably rich yet tractable structure. The analytic behaviour of the associated Dolbeault-Dirac operators is moulded by the complex geometry of the underlying calculus. In particular, twisting the Dolbeault-Dirac operator by a negative Hermitian holomorphic module is shown to give a Fredholm operator if and only if the top anti-holomorphic cohomology group is finite-dimensional. In this case, the operator's index coincides with the twisted holomorphic Euler characteristic of the underlying noncommutative complex structure. The irreducible quantum flag manifolds, endowed with their Heckenberger-Kolb calculi, are presented as motivating examples.

Paper Structure

This paper contains 41 sections, 27 theorems, 113 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Summary of the Paper
  3. Preliminaries on Hermitian Structures
  4. Differential Calculi and Complex Structures
  5. Hermitian and Kähler Structures
  6. A Representation of $\frak{sl}_2$
  7. Holomorphic Modules
  8. Hermitian Modules
  9. Chern Connections and Positive Hermitian Holomorphic Modules
  10. Inner Products, and Twisted Dolbeault--Dirac and Laplace Operators
  11. Covariant Differential Calculi and Hermitian Structures
  12. Compact Quantum Homogeneous Kähler Spaces
  13. Square Integrable Forms
  14. CQH-Hermitian Spaces
  15. Morphisms as Bounded Operators
  16. ...and 26 more sections

Key Result

Proposition 3.2

For a quantum homogeneous space $\mathbf{H} = (B \subseteq A, \Omega^{\bullet}, \Omega^{(\bullet,\bullet)},\sigma)$, and a covariant Hermitian module ${\mathcal{F}}$, if $A$ is finitely generated, then the Hilbert space $L^2(\Omega^{\bullet} \otimes_B {\mathcal{F}} )$ is separable.

Theorems & Definitions (64)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 3.1
  • Proposition 3.2
  • proof
  • ...and 54 more