====== Square commuting hyperoctahedral quantum group ====== The **square commuting hyperoctahedral quantum groups** are a family $(H_N^{\diamond})_{N\in \N}$ of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Raum and Weber in [(:ref:RaWe16)]. Each $H_N^{\diamond}$ interpolates the [[free hyperoctahedral quantum group]] $H_N^+$ and any [[group-theoretical hyperoctahedral easy orthogonal quantum groups]] of the corresponding dimension $N$. They are a special instance of [[non-group-theoretical hyperoctahedral easy orthogonal quantum groups]]. ===== Definition ===== Given $N\in \N$, the **square commuting (hyperoctahedral) quantum group** $H_N^{\diamond}$ **for dimension** $N$ is the [[compact matrix quantum group]] $(C(H_N^{\diamond}),u)$ where $u=(u_{i,j})_{i,j=1}^N$ organizes the generators $\{u_{i,j}\}_{i,j=1}^N$ of the (unital) [[wp>Universal_C*-algebra|universal C*-algebra]] $$C(H_N^{\diamond})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u, \, uu^t=u^tu=I_N\otimes 1\,$$ $${\color{white}C(H_N^{\diamond})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{i,j,l=1}^N: i\neq j\Rightarrow u_{i,l}u_{j,l}=u_{l,i}u_{l,j}=0,$$ $${\color{white}C(H_N^{\diamond})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall a,b\in \{u_{i,j}\}_{i,j=1}^n: a^2b^2=b^2a^2\big\rangle,$$ where $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ is the complex conjugate of $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose, where $I_N$ is the identity $N\!\times \!N$-matrix and where $1$ is the unit of the universal $C^\ast$-algebra. The defining relations of $H_N^{\diamond}$ imply in particular for all $i,j=1,\ldots,N$ that $u_{i,j}^2$ is a partial isometry and that $\sum_{l=1}^N u_{i,l}^2=\sum_{l=1}^N u_{l,j}^2=1$. The square commuting quantum group $H_N^{\diamond}$ is the maximal quantum group of the descending chain $(H_N^{[\pi_k]})_{k\in \{2,\ldots,\infty\}}$ of [[non-group-theoretical hyperoctahedral easy orthogonal quantum groups]] of dimension $N$. ===== Basic Properties ===== The fundamental corepresentation matrix $u$ of $H_N^{\diamond}$ is in particular //orthogonal//. Hence, $H_N^{\diamond}$ is a compact quantum subgroup of the [[free orthogonal quantum group]] $O_N^+$. Moreover, $u$ is also //cubic// especially, implying that $H_N^{\diamond}$ is a compact quantum subgroup of the [[free hyperoctahedral quantum group]] $H_N^{+}$, the free counterpart of the hyperoctahedral group $H_N$. The fundamental corepresentation matrix of any [[group-theoretical hyperoctahedral easy orthogonal quantum groups|group-theoretical hyperoctahedral easy orthogonal quantum group]] satisfies the defining relations of $H_N^{\diamond}$. Hence, $H_N^{\diamond}$ is a compact quantum supergroup of any such quantum group. The square commuting quantum groups $(H_N^{\diamond})_{N\in \N}$ are an [[easy_quantum_group|easy]] family of compact matrix quantum groups, i.e., the intertwiner spaces of their corepresentation categories are induced by a [[category of partitions]]. More precisely, it is a [[non-group-theoretical hyperoctahedral categories of partitions|non-group-theoretical hyperoctahedral category of partitions]] that induces the corepresentation categories of $(H_N^{\diamond})_{N\in \N}$. Canonically, it is generated by the partition $\pi_2$ [(:ref:RaWe16)], where $\pi_2$ is the partition whose [[partition#word_representation|word representation]] is given by $\mathsf{a}_1\mathsf{a}_2\mathsf{a}_2\mathsf{a}_1\mathsf{a}_1\mathsf{a}_2\mathsf{a}_2\mathsf{a}_1$. ===== Representation theory ===== ===== Cohomology ===== ===== Related quantum groups ===== ===== References ===== [( :ref:BanSp09 >> author: Banica, Teodor and Speicher, Roland title: Liberation of orthogonal Lie groups year: 2009 journal: Advances in Mathematics volume: 222 issue: 4 pages: 1461--150 url: https://doi.org/10.1016/j.aim.2009.06.009 archivePrefix: arXiv eprint :0808.2628 )] [( :ref:Web12 >> author: Weber, Moritz title: On the classification of easy quantum groups year: 2013 journal: Advances in Mathematics volume: 245 pages: 500--533 url: https://doi.org/10.1016/j.aim.2013.06.019 archivePrefix: arXiv eprint :1201.4723v2 )] [( :ref:BanCuSp10 >> author: Banica, Teodor and Curran, Stephen and Speicher, Roland title: Classification results for easy quantum groups year: 2010 journal: Pacific Journal of Mathematics volume: 247 issue: 1 pages: 1-26 url: https://doi.org/10.2140/pjm.2010.247.1 archivePrefix: arXiv eprint :0906.3890v1 )] [( :ref:RaWe16 >> author: Raum, Sven and Weber, Moritz title: The full classification of orthogonal easy quantum groups year: 2016 journal: Communications in Mathematical Physics volume: 341 issue: 3 pages: 751--779 url: https://doi.org/10.1007/s00220-015-2537-z archivePrefix: arXiv eprint :1312.3857 )] [( :ref:RaWe15 >> author: Raum, Sven and Weber, Moritz title: Easy quantum groups and quantum subgroups of a semi-direct product quantum group year: 2015 journal: Journal of Noncommutative Geometry volume: 9 issue: 4 pages: 1261--1293 url: https://doi.org/10.4171/JNCG/223 archivePrefix: arXiv eprint :1311.7630v2 )] [( :ref:RaWe14 >> author: Raum, Sven and Weber, Moritz title: The combinatorics of an algebraic class of easy quantum groups year: 2014 journal: Infinite Dimensional Analysis, Quantum Probability and related topics volume: 17 issue: 3 url: https://doi.org/10.1142/S0219025714500167 archivePrefix: arXiv eprint :1312.1497v1 )]