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 DolbeaultDirac operators is moulded by the complex geometry of the underlying calculus. In particular, twisting the DolbeaultDirac operator by a negative Hermitian holomorphic module is shown to give a Fredholm operator if and only if the top antiholomorphic cohomology group is finitedimensional. 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 HeckenbergerKolb calculi, are presented as motivating examples.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.14007
 Bibcode:
 2019arXiv191014007D
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Differential Geometry;
 Mathematics  KTheory and Homology;
 Mathematics  Operator Algebras
 EPrint:
 This is an rewritten version of the paper, which has been divided into two separate papers. The second half will appear as a separate ArXiv entry