hyperoctahedral_series
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| - | The quantum groups of the hyperoctahedral series are [[group-theoretical hyperoctahedral orthogonal easy quantum groups]] and can therefore be written as semi-direct products with their diagonal subgroups [(:ref:RaWe15)]: $$C(H_N^{(s)})\cong C^\ast\langle \{a_i\}_{i=1}^n \,\vert\, \forall_{i,j=1}^n: a_i^2=1,\, (a_ia_j)^s=1,\,\forall_{i,j,k=1}^n: a_i a_k a_j=a_j a_k a_i \rangle\bowtie C(S_N)$$ | + | The quantum groups of the hyperoctahedral series are [[group-theoretical_hyperoctahedral_easy_orthogonal_quantum_groups|group-theoretical hyperoctahedral orthogonal easy quantum groups]] and can therefore be written as semi-direct products with their diagonal subgroups [(:ref:RaWe15)]: $$C(H_N^{(s)})\cong C^\ast\langle \{a_i\}_{i=1}^n \,\vert\, \forall_{i,j=1}^n: a_i^2=1,\, (a_ia_j)^s=1,\,\forall_{i,j,k=1}^n: a_i a_k a_j=a_j a_k a_i \rangle\bowtie C(S_N)$$ |
| for all $s,N\in \N$ with $s\geq 3$, where $C(S_N)$ denotes the continuous function over the symmetric group of dimension $N$ (considered as the subgroup of $\mathrm{GL}(\C,N)$ given by all [[wp>permutation matrices]]). | for all $s,N\in \N$ with $s\geq 3$, where $C(S_N)$ denotes the continuous function over the symmetric group of dimension $N$ (considered as the subgroup of $\mathrm{GL}(\C,N)$ given by all [[wp>permutation matrices]]). | ||
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| If $I$ denotes the closed two-sided ideal of $C(H_N^{(s)})$ generated by the relations $u_{i,j}u_{k,l}=u_{k,l}u_{i,j}$ for any $i,j,k,l=1,\ldots, N$, then $C(H_N^{(s)})/I$ is isomorphic to the $C^\ast$-algebra $C(H_N)$ of continuous functions on the [[wp>hyperoctahedral group]] $H_N$, the subgroup of $\mathrm{GL}(N,\C)$ given by orthogonal matrices with integer entries. Hence, $H_N^{(s)}$ is a compact quantum supergroup of $H_N$. | If $I$ denotes the closed two-sided ideal of $C(H_N^{(s)})$ generated by the relations $u_{i,j}u_{k,l}=u_{k,l}u_{i,j}$ for any $i,j,k,l=1,\ldots, N$, then $C(H_N^{(s)})/I$ is isomorphic to the $C^\ast$-algebra $C(H_N)$ of continuous functions on the [[wp>hyperoctahedral group]] $H_N$, the subgroup of $\mathrm{GL}(N,\C)$ given by orthogonal matrices with integer entries. Hence, $H_N^{(s)}$ is a compact quantum supergroup of $H_N$. | ||
| - | For every $s\in \N$ with $s\geq 3$ the quantum groups $(H_N^{(s)})_{N\in \N}$ of the hyperoctahedral series with paraemter $s$ 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 [[group-theoretical hyperoctahedral categories of partitions|group-theoretical hyperoctahedral category of partitions]] that induces the corepresentation categories of $(H_N^{(s)})_{N\in \N}$. Canonically, it is generated by the set $\{\Pabcabc,\fourpart,h_s\}$ of partitions, where $h_s$ is the partition whose [[partition#word_representation|word representation]] is given by $(ab)^s$. See also [[categories of the hyperoctahedral series]]. | + | For every $s\in \N$ with $s\geq 3$ the quantum groups $(H_N^{(s)})_{N\in \N}$ of the hyperoctahedral series with paraemter $s$ 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 [[group-theoretical hyperoctahedral categories of partitions|group-theoretical hyperoctahedral category of partitions]] that induces the corepresentation categories of $(H_N^{(s)})_{N\in \N}$. Canonically, it is generated by the set $\{\Pabcabc,\fourpart,h_s\}$ of partitions [(:ref:RaWe14)], where $h_s$ is the partition whose [[partition#word_representation|word representation]] is given by $(ab)^s$. See also [[categories of the hyperoctahedral series]]. |
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| archivePrefix: arXiv | archivePrefix: arXiv | ||
| eprint :0906.3890v1 | eprint :0906.3890v1 | ||
| + | )] | ||
| + | |||
| + | [( :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 | ||
| )] | )] | ||
hyperoctahedral_series.1580110832.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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