====== Group-theoretical hyperoctahedral easy orthogonal quantum groups ====== The **group-theoretical hyperoctahedral easy orthogonal quantum groups** are a two-parameter family of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Raum and Weber and classified combinatorially in [(:ref:RaWe14)] and algebraically in [(:ref:RaWe15)]. The first parameter is the matrix dimension $N\in \N$ of the fundamental corepresentation. The second one runs through all strongly symmetric reflection groups on $N$ generators excepting $\Z_2$ and the trivial group. They are particular examples of the more general class of [[group-theoretical quantum groups]]. ===== Definition ===== For every $N\in \N$ let $\Z_2^{\ast N}$ be the [[wp>free product]] group of $N$ many copies of thy cyclic group $\Z_2\equiv \Z/2\Z$ of order $2$ and for every $k\in\{1,\ldots,N\}$ let $a_k$ be the image of $1\equiv 1+2\Z$ under the embedding of $\Z_2$ as the $k$-th free factor in $\Z_2^{\ast N}$. Then, $\{a_1,\ldots,a_N\}$ generates $\Z_2^{\ast N}$ and any group endomorphism of $\Z_2^{\ast N}$ is uniquely determined by its restriction to $\{a_1,\ldots,a_N\}$. The **strong symmetric semigroup** $\mathrm{sS}_N$ is the subsemigroup of the semigroup $\mathrm{End}(\Z_2^{\ast N})$ of group endomorphisms of $\Z_2^{\ast N}$ generated by the endomorphisms defined by $a_k\mapsto a_{i(k)}$ for all $k\in \{1,\ldots,N\}$ for all mappings $i:\{1,\ldots,N\}\to\{1,\ldots,N\}$. In other words, $\mathrm{sS}_N$ is given by all identifications of letters in words in an alphabet of $N$ letters from $\Z_2$. A **strongly symmetric reflection group (on** $N$ **generators)** is now a quotient group of $\Z_2^{\ast N}$ by a normal subgroup $A$ which is invariant under the action of $\mathrm{sS}_N$, the latter condition meaning $\varphi(w)\in A$ for all $w\in A$ and $\varphi\in \mathrm{sS}_N$. Given $N\in \N$ and a strongly symmetric reflection group $\Z^{\ast N}/A$ on $N$ generators __other than__ $\Z_2$ __and the trivial group__, the **group-theoretical hyperoctahedral easy orthogonal quantum group with parameter** $\Z^{\ast N}/A$ **(for dimension** $N$**)** is the [[compact matrix quantum group]] $(C(H_N^{}),u)$ 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]] $$C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u, \, uu^t=u^tu=I_N\otimes 1\,$$ $${\color{white}C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{i,j,l=1}^N: i\neq j\Rightarrow u_{i,l}u_{j,l}=u_{l,i}u_{l,j}=0,$$ $${\color{white}C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{i,j,k,l=1}^N: u_{i,j}^2u_{k,l}=u_{k,l}u_{i,j}^2,$$ $${\color{white}C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{m\in \N}:\, \forall_{t_1,\ldots,t_m=1}^N: \, a_{t_1}\ldots a_{t_m}\in A \Rightarrow$$ $${\color{white}C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,} \forall_{i_1,j_1,\ldots,i_m,j_m=1}^N: (\forall_{p,q=1}^m: i_p=i_q\Leftrightarrow j_p=j_q\Leftrightarrow t_p=t_q)\Rightarrow$$ $${\color{white}C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,} u_{i_1,j_1}u_{i_2,j_2}\ldots u_{i_m,j_m}=u_{i_1,j_1}^2u_{i_2,j_2}^2\ldots u_{i_m,j_m}^2\big\rangle,$$ 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 $H_N^{}$ imply in particular for all $i,j=1,\ldots,N$ that $u_{i,j}^2$ is a projection and that $\sum_{l=1}^N u_{i,l}^2=\sum_{l=1}^N u_{l,j}^2=1$. Another characterization of $H_N^{}$ is given by writing it as a [[semi-direct product]] with its [[diagonal subgroup of a compact matrix quantum group|diagonal subgroup]] [(:ref:RaWe15)], which is precisely the strongly symmetric reflection group $\Z^{\ast N}/A$: $$C(H_N^{})\cong C^\ast(\Z^{\ast N}/A)\bowtie C(S_N),$$ where $C^\ast(\Z^{\ast N}/A)$ is the [[wp>Group_algebra#The_group_C*-algebra_C*(G)|group C*-algebra]] of $\Z^{\ast N}/A$ and where $C(S_N)$ denotes the continuous functions over the symmetric group of dimension $N$ (considered as the subgroup of $\mathrm{GL}(\C,N)$ given by all [[wp>permutation matrices]]). Allowing also $\Z_2$ and the trivial group in the above definition yields the two group-theoretical __non-hyperoctahedral__ easy orthogonal quantum groups $S_N'$ and $S_N$, the [[modified symmetric group]] and the symmetric group $S_N$. ===== Basic Properties ===== The fundamental corepresentation matrix $u$ of $H_N^{}$ is in particular //orthogonal//. Hence, $H_N^{}$ is a compact quantum subgroup of the [[free orthogonal quantum group]] $O_N^+$. Moreover, $u$ is also //cubic// especially, implying that $H_N^{}$ is a compact quantum subgroup of the [[free hyperoctahedral quantum group]] $H_N^{+}$, the free counterpart of the hyperoctahedral group $H_N$. ===== 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 )]