##
**Compact quantum groups and their representation categories.**
*(English)*
Zbl 1316.46003

Cours Spécialisés (Paris) 20. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-777-3/hbk). iv, 169 p. (2013).

The aim of the book under review is an introduction to compact quantum groups and their representations in the analytical setting of operator algebras while emphasizing the categorical point of view when constructing and investigating concrete examples. The book is divided into two parts which both consist of two chapters. The first part is devoted to the general theory and the second part is more specialized. In the following the contents of the book will be described in more detail.

In the first chapter the authors collect the basic definitions and examples of compact quantum groups and their duals, the discrete quantum groups. Then the Haar state of a compact quantum group, the quantum analogue of the Haar measure of a compact group, is introduced, and the unitary representations of a compact quantum group on a Hilbert space are investigated. In particular, the quantum analogue of the Peter-Weyl theorem is proved. It turns out that the theory of quantum compact groups is very similar to the theory of compact groups. An additional technical difficulty arises from the modular properties of the Haar state of a compact quantum group, which originate from the fact that the Haar state is not necessarily central.

In the second chapter compact quantum groups are studied from a categorical point of view. The main result is Woronowicz’s quantum analogue of Tannaka-Krein duality for compact groups which shows how a compact quantum group can be reconstructed from the \(C^*\)-tensor category of its finite dimensional unitary representations. This can be used to construct compact quantum groups from certain \(C^*\)-subcategories of the \(C^*\)-tensor category of the finite dimensional unitary representations of a Hopf \(*\)-algebra. In particular, by using the Drinfeld-Jimbo \(q\)-deformations of universal enveloping algebras, the deformation \(\mathrm{SU}_q(2)\) of \(\mathrm{SU}(2)\) introduced in the first chapter can be generalized to all simply connected semisimple compact Lie groups. The representation category of \(\mathrm{SU}_q(2)\) is studied in detail and a characterization of those quantum groups that are monoidally equivalent to \(\mathrm{SU}_q(2)\) is obtained. Moreover, it is shown that every braided \(C^*\)-tensor category with conjugates has a canonical ribbon structure. In the final section the authors prove that the fusion rules of a compact quantum group \(G_q\) detect whether its GNS-representation restricts to an isomorphism between the universal form and the reduced form of \(G_q\).

Chapter 3 is devoted to the cohomology of discrete quantum groups in low degrees. The main goal of this chapter is to prove that for any Drinfeld-Jimbo \(q\)-deformation \(G_q\) of a simply connected semisimple compact Lie group \(G\) the \(G_q\)-invariant second cohomology group \(H_{G_q}^2(\hat{G}_q;\mathbb{C}^*)\) is isomorphic to the second cohomology group \(H^2(\widehat{Z(G)};\mathbb{C}^*)\), where \(\widehat{Z(G)}\) denotes the dual of the center of \(G\). Note that \(H_{G_q}^2 (\hat{G}_q;\mathbb{C}^*)\) can be identified with the group of all monoidal auto-equivalences of the \(C^*\)-tensor category of the finite dimensional unitary representations of \(G_q\) that, up to natural monoidal isomorphisms, are isomorphic to the identity functor.

In the final chapter it is shown that the \(C^*\)-tensor category of finite dimensional unitary representations of a simply connected semisimple compact Lie group \(G\) is equivalent to the corresponding category of its \(q\)-deformation \(G_q\) if the trivial associativity morphisms in the latter category are replaced by the monodromy of the Knizhnik-Zamolodchikov equations. Then a version of the Drinfeld-Kohno theorem is established and some implications for operator algebras are discussed.

The book originated from lectures of the first author and should be quite useful for graduate students specializing in operator algebras and related fields as a first introduction to compact quantum groups, their representations, and their cohomology.

In the first chapter the authors collect the basic definitions and examples of compact quantum groups and their duals, the discrete quantum groups. Then the Haar state of a compact quantum group, the quantum analogue of the Haar measure of a compact group, is introduced, and the unitary representations of a compact quantum group on a Hilbert space are investigated. In particular, the quantum analogue of the Peter-Weyl theorem is proved. It turns out that the theory of quantum compact groups is very similar to the theory of compact groups. An additional technical difficulty arises from the modular properties of the Haar state of a compact quantum group, which originate from the fact that the Haar state is not necessarily central.

In the second chapter compact quantum groups are studied from a categorical point of view. The main result is Woronowicz’s quantum analogue of Tannaka-Krein duality for compact groups which shows how a compact quantum group can be reconstructed from the \(C^*\)-tensor category of its finite dimensional unitary representations. This can be used to construct compact quantum groups from certain \(C^*\)-subcategories of the \(C^*\)-tensor category of the finite dimensional unitary representations of a Hopf \(*\)-algebra. In particular, by using the Drinfeld-Jimbo \(q\)-deformations of universal enveloping algebras, the deformation \(\mathrm{SU}_q(2)\) of \(\mathrm{SU}(2)\) introduced in the first chapter can be generalized to all simply connected semisimple compact Lie groups. The representation category of \(\mathrm{SU}_q(2)\) is studied in detail and a characterization of those quantum groups that are monoidally equivalent to \(\mathrm{SU}_q(2)\) is obtained. Moreover, it is shown that every braided \(C^*\)-tensor category with conjugates has a canonical ribbon structure. In the final section the authors prove that the fusion rules of a compact quantum group \(G_q\) detect whether its GNS-representation restricts to an isomorphism between the universal form and the reduced form of \(G_q\).

Chapter 3 is devoted to the cohomology of discrete quantum groups in low degrees. The main goal of this chapter is to prove that for any Drinfeld-Jimbo \(q\)-deformation \(G_q\) of a simply connected semisimple compact Lie group \(G\) the \(G_q\)-invariant second cohomology group \(H_{G_q}^2(\hat{G}_q;\mathbb{C}^*)\) is isomorphic to the second cohomology group \(H^2(\widehat{Z(G)};\mathbb{C}^*)\), where \(\widehat{Z(G)}\) denotes the dual of the center of \(G\). Note that \(H_{G_q}^2 (\hat{G}_q;\mathbb{C}^*)\) can be identified with the group of all monoidal auto-equivalences of the \(C^*\)-tensor category of the finite dimensional unitary representations of \(G_q\) that, up to natural monoidal isomorphisms, are isomorphic to the identity functor.

In the final chapter it is shown that the \(C^*\)-tensor category of finite dimensional unitary representations of a simply connected semisimple compact Lie group \(G\) is equivalent to the corresponding category of its \(q\)-deformation \(G_q\) if the trivial associativity morphisms in the latter category are replaced by the monodromy of the Knizhnik-Zamolodchikov equations. Then a version of the Drinfeld-Kohno theorem is established and some implications for operator algebras are discussed.

The book originated from lectures of the first author and should be quite useful for graduate students specializing in operator algebras and related fields as a first introduction to compact quantum groups, their representations, and their cohomology.

Reviewer: Jörg Feldvoss (Mobile)

### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46L65 | Quantizations, deformations for selfadjoint operator algebras |

46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |

22E46 | Semisimple Lie groups and their representations |

16T20 | Ring-theoretic aspects of quantum groups |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

20G42 | Quantum groups (quantized function algebras) and their representations |

18D99 | Categorical structures |