easy_quantum_group
Differences
This shows you the differences between two versions of the page.
| Next revision | Previous revision | ||
|
easy_quantum_group [2020/02/06 08:50] amang created |
easy_quantum_group [2021/11/23 11:56] (current) |
||
|---|---|---|---|
| Line 11: | Line 11: | ||
| where $(e_i)_{i=1}^N$ is the standard basis of $\C^N$ and where for all $i_1,\ldots,i_\ell\in N$ the symbol $\delta_p(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ is $1$ if the kernel, i.e., the induced partition with $k$ upper and $\ell$ lower points, of $(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ refines $p$ and is $0$ otherwise. | where $(e_i)_{i=1}^N$ is the standard basis of $\C^N$ and where for all $i_1,\ldots,i_\ell\in N$ the symbol $\delta_p(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ is $1$ if the kernel, i.e., the induced partition with $k$ upper and $\ell$ lower points, of $(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ refines $p$ and is $0$ otherwise. | ||
| - | ===== Systematization ===== | + | ===== Taxonomy ===== |
| + | There are several systems to divide the class of of easy orthogonal quantum groups into cases. Let $G\cong(G(G),u)$ be an easy orthogonal quantum group and let $\Cscr\subseteq\Pscr$ be the category of partitions generating its corepresentation category. We say that $G$ is | ||
| + | |||
| + | * Classical case distinction: | ||
| + | * **orthogonal** or **case** $O$: if $\singleton\notin \Cscr$ and $\fourpart\notin \Cscr$, | ||
| + | * **bistochastic** or **case** $B$: if $\singleton\in \Cscr$ and $\fourpart\notin \Cscr$, | ||
| + | * **symmetric** or **case** $S$: if $\singleton\in \Cscr$ and $\fourpart\in \Cscr$, | ||
| + | * **hyperoctahedral** or **case** $H$: if $\singleton\notin \Cscr$ and $\fourpart\in \Cscr$, | ||
| + | * Liberty distinction: | ||
| + | * **free** or **liberated**: if all partitions of $\Cscr$ are [[category_of_all_non-crossing_partitions|non-crossing]], | ||
| + | * **half-liberated**: if $\Pabcabc\in \Cscr$ and $\Pabab\notin\Cscr$, | ||
| + | * **classical** or **commutative** or **group case**: if $\Pabab\in\Cscr$, | ||
| + | * Group theoreticity distinction: | ||
| + | * **group-theoretical**: if $\Paabaab\in\Cscr$, | ||
| + | * **non-group-theoretical**: if $\Paabaab\notin\Cscr$. | ||
| ===== Classification ===== | ===== Classification ===== | ||
| ===== References ===== | ===== References ===== | ||
| - | [(ref:BS09>> | + | |
| - | title : Liberation of orthogonal Lie groups | + | [( :ref:BanSp09 >> |
| - | journal : Advances in Mathematics | + | author: Banica, Teodor and Speicher, Roland |
| - | volume : 222 | + | title: Liberation of orthogonal Lie groups |
| - | number : 4 | + | year: 2009 |
| - | pages : 1461--1501 | + | journal: Advances in Mathematics |
| - | year : 2009 | + | volume: 222 |
| - | url : http://dx.doi.org/10.1016/j.aim.2009.06.009 | + | issue: 4 |
| - | author : Teodor Banica and Roland Speicher | + | pages: 1461--150 |
| + | url: https://doi.org/10.1016/j.aim.2009.06.009 | ||
| + | archivePrefix: arXiv | ||
| + | eprint :0808.2628 | ||
| )] | )] | ||
easy_quantum_group.1580979003.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
