The hyperoctahedral quantum group.

*(English)*Zbl 1185.46046This 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)