The hyperoctahedral quantum group. (English) Zbl 1185.46046

Banica, Teodor; Bichon, Julien; Collins, Benoît

J. Ramanujan Math. Soc. 22, No. 4, 345-384 (2007).

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.

Reviewer: Zhang Liangyun (Nanjing)

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