Notes on Compact Quantum Groups

Abstract

We have written down a set of notes on compact quantum groups from which all the different aspects can be learned in an easy way and such that a lot of insight can be obtained without too much effort. Compact quantum groups have been studied by several authors, from different points of view. The difference lies mainly in the choice of the axioms and consequently, in the way the main results are proven. These results however are essentially the same in all these cases. In these notes, we mainly follow the approach of Woronowicz and we extensively motivate this choice. We give a complete and rather detailed treatment, starting from a simple set of axioms and obtaining the main results. We also discuss the most common examples and show how they fit into the framework. During this process, we compare with the existing other treatments.

43 pages, latex2e

References in corpus (2)

• CQG algebras: a direct algebraic approach to compact quantum groups
• Universal C*-algebraic quantum groups arising from algebraic quantum groups

Cited by in corpus (53)

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• Free actions of compact quantum group on unital C*-algebras
• A construction of finite index C*-algebra inclusions from free actions of compact quantum groups
• The spatial Rokhlin property for actions of compact quantum groups
• A New Characterisation of Idempotent States on Finite and Compact Quantum Groups
• Index theory for actions of compact Lie groups on C*-algebras
• $L_p$-improving convolution operators on finite quantum groups
• Galois extension for a compact quantum group
• Quantum Isometry Group for Spectral Triples with Real Structure
• Quantum Group of Orientation preserving Riemannian Isometries
• Admissibility Conjecture and Kazhdan's Property (T) for quantum groups
• Quantum symmetries and quantum isometries of compact metric spaces
• Rigidity of action of compact quantum groups II
• Berezin quantization of noncommutative projective varieties
• Reflective-coreflective equivalence
• Solution group representations as quantum symmetries of graphs
• Quantum Isometry Groups of Noncommutative Manifolds Obtained by Deformation Using Dual Unitary 2-Cocycles
• Existence and examples of quantum isometry group for a class of compact metric spaces
• Examples of factors which have no Cartan subalgebras
• Amenability and vanishing of L^2-Betti numbers: an operator algebraic approach
• A note on weak amenability for reduced free products of discrete quantum groups
• Berezin Quantization From Ergodic Actions of Compact Quantum Groups, and Quantum Gromov-Hausdorff Distance
• Quantum Symmetry on Potts Model
• Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors
• Mean ergodic theorem for amenable discrete quantum groups and a Wiener type theorem for compact metrizable groups
• Ergodic theory for quantum semigroups
• Applications of Foelner's condition to quantum groups
• The approximation property for locally compact quantum groups
• Quantum Isometry group of dual of finitely generated discrete groups and quantum groups
• Unitary representations of compact quantum groups
• Deformations of spectral triples and their quantum isometry groups via monoidal equivalences
• C*-algebraic partial compact quantum groups
• Invariance of KMS states on graph C*-algebras under classical and quantum symmetry
• Quantum bohr compactification of discrete quantum groups
• C${}^{\ast }$-bialgebra defined by the direct sum of Cuntz-Krieger algebras
• Quantum isometries and loose embeddings
• Quantum isometry group of dual of finitely generated discrete groups- $\text{II}$
• On Square Roots of the Haar State on Compact Quantum Groups
• Quantum symmetry groups of noncommutative tori
• Quantum information theory and Fourier multipliers on quantum groups
• Improved bound on the non zero eigenvalues of the graph Laplacian coming from quantum symmetry of vertex transitive graphs
• Connectedness and irreducibility of compact quantum groups
• A Note on Operator Biprojectivity of Compact Quantum Groups
• A note on geometric characterization of quantum isometries of classical manifolds
• An averaging trick for smooth actions of compact quantum groups on manifolds
• An example of explicit dependence of quantum symmetry on KMS states
• A note on the injectivity of action by compact quantum groups on a class of $C^{\ast }$-algebras
• On the problem of classifying simple compact quantum groups
• Equivariant ${\mathrm{C}}^{\ast }$-correspondences and compact quantum group actions on Pimsner algebras