free_symmetric_quantum_group
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free_symmetric_quantum_group [2020/01/02 16:50] amang [Basic Properties] |
free_symmetric_quantum_group [2021/11/23 11:56] (current) |
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| The fundamental corepresentation matrix $u$ of $S_N^+$ is in particular //orthogonal//. Hence, $S_N^+$ is a compact quantum subgroup of the [[free orthogonal quantum group]] $O_N^+$. | The fundamental corepresentation matrix $u$ of $S_N^+$ is in particular //orthogonal//. Hence, $S_N^+$ is a compact quantum subgroup of the [[free orthogonal quantum group]] $O_N^+$. | ||
| - | If $I$ denotes the closed two-sided ideal of $C(S_N^+)$ 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 $C(S_N^+)/I$ is isomorphic to the $C^\ast$-algebra $C(S_N)$ of continuous functions on the [[wp>symmetric group]] $S_N$, the latter interpreted as the subgroup of $\mathrm{GL}(n,\C)$ given by all [[wp>permutation matrix|permutation matrices]]. Hence, $S_N^+$ is a compact quantum supergroup of $S_N$. | + | If $I$ denotes the closed two-sided ideal of $C(S_N^+)$ 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 $C(S_N^+)/I$ is isomorphic to the $C^\ast$-algebra $C(S_N)$ of continuous functions on the [[wp>symmetric group]] $S_N$, the latter interpreted as the subgroup of $\mathrm{GL}(N,\C)$ given by all [[wp>permutation matrix|permutation matrices]]. Hence, $S_N^+$ is a compact quantum supergroup of $S_N$. |
| The free symmetric quantum groups $(S_N^+)_{N\in \N}$ 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 $\mathrm{NC}$ of [[category_of_all_non-crossing_partitions|all non-crossing partitions]] that induces the corepresentation categories of $(S_N^+)_{N\in \N}$. The canonical generating set of partitions of $\mathrm{NC}$ is $\{\fourpart,\singleton\}$. | The free symmetric quantum groups $(S_N^+)_{N\in \N}$ 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 $\mathrm{NC}$ of [[category_of_all_non-crossing_partitions|all non-crossing partitions]] that induces the corepresentation categories of $(S_N^+)_{N\in \N}$. The canonical generating set of partitions of $\mathrm{NC}$ is $\{\fourpart,\singleton\}$. | ||
free_symmetric_quantum_group.1577983809.txt.gz ยท Last modified: 2021/11/23 11:56 (external edit)
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