Berezin Quantization From Ergodic Actions of Compact Quantum Groups, and Quantum Gromov-Hausdorff Distance

Abstract

Over the last 25 years, the notion of "fuzzy spaces" has become ubiquitous in the high-energy physics literature. These are finite dimensional noncommutative approximations of the algebra of functions on a classical space. The most well known examples come from the Berezin quantization of coadjoint orbits of compact semisimple Lie groups. We develop a theory of Berezin quantization for certain quantum homogeneous spaces coming from ergodic actions of compact quantum groups. This allows us to construct fuzzy versions of these quantum homogeneous spaces. We show that the finite dimensional approximations converge to the homogeneous space in a continuous field of operator systems, and in the quantum Gromov-Hausdorff distance of Rieffel. We apply the theory to construct a fuzzy version of an ellipsoid which is naturally endowed with an orbifold structure, as well as a fuzzy version of the $\theta$-deformed coset spaces $C(G/H{)}_{\hslash \theta }$ of Varilly. In the process of the latter, we show that our Berezin quantization commutes with Rieffel's deformation quantization for actions of ${\mathbb{R}}^d$.

Ph.D Thesis

References in corpus (23)

• Symmetries of quantum spaces. Subgroups and quotient spaces of quantum $SU(2)$ and $SO(3)$ groups
• Metrics on states from actions of compact groups
• Co-Amenability of compact quantum groups
• Notes on Compact Quantum Groups
• Matrix algebras converge to the sphere for quantum Gromov--Hausdorff distance
• Quantum symmetry groups of noncommutative spheres
• Leibniz seminorms for "Matrix algebras converge to the sphere"
• On idempotent states on quantum groups
• Simulation of a scalar field on a fuzzy sphere
• Fusion rules for representations of compact quantum groups
• A universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces
• Coherent States for Quantum Compact Groups
• Compact and discrete subgroups of algebraic quantum groups I
• The noncommutative Choquet boundary III: Operator systems in matrix algebras
• Duality for actions of weak Kac algebras and crossed product inclusions of II_1 factors
• Classification of minimal actions of a compact Kac algebra with amenable dual
• Quantum Isometry Groups of 0- Dimensional Manifolds
• Quantum symmetries and quantum isometries of compact metric spaces
• Complete Positivity of Rieffel's Deformation Quantization by Actions of ${\mathbb{R}}^d$
• Unified Gauge Theories and Reduction of Couplings: from Finiteness to Fuzzy Extra Dimensions
• Simulations of a supersymmetry inspired model on a fuzzy sphere
• q-Fuzzy spheres and quantum differentials on B_q[SU_2] and U_q(su_2)
• Matricial quantum Gromov-Hausdorff distance

Cited by in corpus (2)

• Convergence of Fourier truncations for compact quantum groups and finitely generated groups
• Berezin quantization of noncommutative projective varieties