half-liberated_hyperoctahedral_quantum_group
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half-liberated_hyperoctahedral_quantum_group [2020/01/22 09:17] amang created |
half-liberated_hyperoctahedral_quantum_group [2021/11/23 11:56] (current) |
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| If $I$ denotes the closed two-sided ideal of $C(H_N^{\ast})$ 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 $C(H_N^{\ast})/I$ 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, $H_N^{\ast}$ is a compact quantum supergroup of $H_N$. | If $I$ denotes the closed two-sided ideal of $C(H_N^{\ast})$ 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 $C(H_N^{\ast})/I$ 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, $H_N^{\ast}$ is a compact quantum supergroup of $H_N$. | ||
| - | The half-liberated hyperoctahedral quantum groups $(H_N^{\ast})_{N\in \N}$ 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 the [[category of partitions with blocks of even size and even distances between legs]] that induces the corepresentation categories of $(H_N^{\ast})_{N\in \N}$. Canonically, it is generated by the set $\{\Pabcabc,\fourpart\}$ of partitions. | + | The half-liberated hyperoctahedral quantum groups $(H_N^{\ast})_{N\in \N}$ 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 the [[category of partitions with blocks of even size and even distances between legs|category of partitions with blocks of even size and parity-balanced legs]] that induces the corepresentation categories of $(H_N^{\ast})_{N\in \N}$. Canonically, it is generated by the set $\{\Pabcabc,\fourpart\}$ of partitions. |
half-liberated_hyperoctahedral_quantum_group.txt ยท Last modified: 2021/11/23 11:56 (external edit)
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