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The hyperoctahedral quantum group. (English) Zbl 1185.46046
This paper mainly considers the hypercube in $$\mathbb{R}^n$$, and shows that its quantum symmetry group is a $$q$$-deformation of $$O_n$$ at $$q=-1$$. First, the authors discuss the integration problem, with a presentation of some previously known results, and with an explicit computation for $$H_n$$. Secondly, they show that $$O_n^{-1}$$ is the quantum symmetry group of the hypercube in $$\mathbb{R}^{n}$$. Thus $$O_n^{-1}$$ can be regarded as a noncommutative version of $$H_n$$. Thirdly, the free version $$H_n^+$$ is constructed, as quantum symmetry group of the graph formed by $$n$$ segments. The authors perform a detailed combinatorial study of $$H_n^+$$, the main result being the asymptotic freeness of diagonal coefficients. Finally, the authors extend a part of the results to a certain class of quantum groups which are called free. They also give some other examples of free quantum groups, by using a free product construction, and discuss the classification problem.

##### MSC:
 46L65 Quantizations, deformations for selfadjoint operator algebras 17B37 Quantum groups (quantized enveloping algebras) and related deformations 46L54 Free probability and free operator algebras 58B32 Geometry of quantum groups 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
##### Keywords:
freeness; hypercube; Bessel function; quantum symmetry group
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