Related papers: On the problem of classifying simple compact quant…
We classify compact quantum groups associated to noncrossing partitions coloured with two elements $x$ and $y$ which are their own inverses. Together with the work of P. Tarrago and M. Weber, this completes the classification of all…
We introduce the analog of Bohr compactification for discrete quantum groups on C*-algebra level. The cases of unimodular and general C*-algebraic discrete quantum groups are treated separately. The passage from the former case to the…
We give a general definition of classical and quantum groups whose representation theory is "determined by partitions" and study their structure. This encompasses many examples of classical groups for which Schur-Weyl duality is described…
Bornological quantum groups were introduced by Voigt in order to generalize the theory of algebraic quantum groups in the sense of van Daele. In particular the class of bornological quantum groups contains all classical locally compact…
This is a self-contained review on the theory of quantum group and its applications to modern physics. A brief introduction is given to the Yang-Baxter equation in integrable quantum field theory and lattice statistical physics. The quantum…
This paper is an adaptation of a chapter from an upcoming monograph on noncommutative geometry and quantum groups. We present examples of non compact quantum groups which are deformations of low dimensional Lie groups. The paper is of…
The complexity of quantum computation remains poorly understood. While physicists attempt to find ways to create quantum computers, we still do not have much evidence one way or the other as to how useful these machines will be. The tools…
We prove various finite de Finetti theorems for non-commutative distributions which are invariant under the free easy quantum group actions. This complements the free de Finetti theorems by Banica, Curran and Speicher, which mostly focus on…
Present-day quantum field theory can be regularized by a decomposition into quantum simplices. This replaces the infinite-dimensional Hilbert space by a high-dimensional spinor space and singular canonical Lie groups by regular spin groups.…
We propose a definition of compact quantum groupoids in the setting of C*-algebras, associate to such a quantum groupoid a regular C*-pseudo-multiplicative unitary, and use this unitary to construct a dual Hopf C*-bimodule and to pass to a…
We study the fusion semirings arising from easy quantum groups. We classify all the possible free ones, answering a question of T. Banica and R. Vergnioux : these are exactly the fusion rings of quantum groups without any nontrivial…
An algebraic quantum group is a multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication…
In this paper, we give a construction of a (C*-algebraic) quantum Heisenberg group. This is done by viewing it as the dual quantum group of the specific non-compact quantum group (A,\Delta) constructed earlier by the author. Our definition…
This paper is meant to be an informal introduction to Quantum Groups, starting from its origins and motivations until the recent developments. We call in particular the attention on the newly descovered relationship among quantum groups,…
This article surveys the application of the representation theory of loop groups to simple models in quantum field theory and to certain integrable systems. The common thread in the discussion is the construction of quantum fields using…
We introduce a non commutative analog of the Bohr compactification. Starting from a general quantum group G we define a compact quantum group bG which has a universal property such as the universal property of the classical Bohr…
In this paper, we summarize the work on the characterization of finite simple groups and the study on finite groups with the set of element orders and two orders (the order of group and the set of element orders). Some related topics, and…
We study sequences of noncommutative random variables which are invariant under "quantum transformations" coming from an orthogonal quantum group satisfying the "easiness" condition axiomatized in our previous paper. For 10 easy quantum…
By finite quantum groups we mean Lusztig's finite-dimensional pointed Hopf algebras called quantum Frobenius Kernels [9, 10], and their natural generalizations due to Andruskiewitsch and Schneider [2, 3]. For a Hopf algebra $H$ in a special…
Quantum computing opens exciting opportunities for kernel-based machine learning methods, which have broad applications in data analysis. Recent works show that quantum computers can efficiently construct a model of a classifier by…