# Classification results for easy quantum groups

@article{Banica2009ClassificationRF, title={Classification results for easy quantum groups}, author={Teodor Banica and Stephen J. Curran and Roland Speicher}, journal={Pacific Journal of Mathematics}, year={2009}, volume={247}, pages={1-26} }

We study the orthogonal quantum groups satisfying the "easiness" assumption axiomatized in our previous paper, with the construction of some new examples and with some partial classification results. The conjectural conclusion is that the easy quantum groups consist of the previously known 14 examples, plus a hypothetical multiparameter "hyperoctahedral series", related to the complex reflection groups H s n = ℤ s z S n . We also discuss the general structure and the computation of asymptotic…

## 46 Citations

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We consider several orthogonal quantum groups satisfying the “easiness” assumption axiomatized in our previous paper. For each of them we discuss the computation of the asymptotic law of Tr(uk) with…

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## References

SHOWING 1-10 OF 41 REFERENCES

### Stochastic aspects of easy quantum groups

- Mathematics
- 2009

We consider several orthogonal quantum groups satisfying the “easiness” assumption axiomatized in our previous paper. For each of them we discuss the computation of the asymptotic law of Tr(uk) with…

### Free products of compact quantum groups

- Mathematics
- 1995

We construct and study compact quantum groups from free products ofC*-algebras. In this connection, we discover two mysterious classes of natural compact quantum groups,Au(m) andAo(m). TheAu(m)'s…

### Fusion rules for quantum reflection groups

- Mathematics
- 2008

We find the fusion rules for the quantum analogues of the complex reflection groups $H_n^s=\mathbb Z_s\wr S_n$. The irreducible representations can be indexed by the elements of the free monoid…

### Le Groupe Quantique Compact Libre U(n)

- Mathematics
- 1997

Abstract:The free analogues of U(n) in Woronowicz' theory [Wo2] are the compact matrix quantum groups introduced by Wang and Van Daele. We classify here their irreducible representations. Their…

### Quantum permutation groups: a survey

- Mathematics
- 2006

This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties;…

### The boundary of universal discrete quantum groups, exactness, and factoriality

- Mathematics
- 2005

We study the C -algebras and von Neumann algebras associated with the universal discrete quantum groups. They give rise to full prime factors and simple exact C -algebras. The main tool in our work…

### Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups

- Mathematics
- 2005

We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These…

### Quantum Groups

- Mathematics
- 1993

This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions…

### THE HYPEROCTAHEDRAL QUANTUM GROUP

- Mathematics
- 2007

We consider the hypercube in R n , and show that its quantum symmetry group is a q-deformation of On at q = 1. Then we consider the graph formed by n segments, and show that its quantum symmetry…