category_of_partitions
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| ====== Category of partitions ====== | ====== Category of partitions ====== | ||
| - | This page is about //easy// categories of partitions in the sense of Banica and Speicher. If we equip those with a linear structure, we get [[linear_category_of_partitions|linear categories of partitions]]. | + | This page is about //easy// categories of partitions in the sense of Banica and Speicher [(ref:BS09)]. If we equip those with a linear structure, we get [[linear_category_of_partitions|linear categories of partitions]]. |
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| ==== Non-crossing partitions ==== | ==== Non-crossing partitions ==== | ||
| - | There are the following seven easy [[Non-crossing category of partitions|non-crossing categories]] of partitions. | + | There are the following seven easy [[Non-crossing category of partitions|non-crossing categories]] of partitions [(ref:BS09)],[(ref:Web13)]. |
| $$NC_2=\langle\rangle\subset\Big\{\begin{matrix}\langle\singleton\otimes\singleton\rangle\subset\langle\Labac\rangle\subset\langle\singleton\rangle\\\langle\fourpart\rangle\subset\langle\fourpart,\singleton\otimes\singleton\rangle\end{matrix}\Big\}\subset\langle\fourpart,\singleton\rangle=NC$$ | $$NC_2=\langle\rangle\subset\Big\{\begin{matrix}\langle\singleton\otimes\singleton\rangle\subset\langle\Labac\rangle\subset\langle\singleton\rangle\\\langle\fourpart\rangle\subset\langle\fourpart,\singleton\otimes\singleton\rangle\end{matrix}\Big\}\subset\langle\fourpart,\singleton\rangle=NC$$ | ||
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| A category of partition $\Cscr$ is called a **group category** if it contains the **crossing partition** $\crosspart$. It is called a //group category// because the crossing partition corresponds to commutativity relation. So, a quantum group corresponding to a group category is actually a group. | A category of partition $\Cscr$ is called a **group category** if it contains the **crossing partition** $\crosspart$. It is called a //group category// because the crossing partition corresponds to commutativity relation. So, a quantum group corresponding to a group category is actually a group. | ||
| - | For any group category $\Cscr$ it holds that $\Cscr':=\Cscr\cap NC$ is a non-crossing category of partitions. Conversely, it holds that $\Cscr=\langle\Cscr',\crosspart\rangle$. Thus, all categories with the crossing partition arise by adding the crossing to a non-crossing category. The seven non-crossing categories induce the following six categories with crossing | + | For any group category $\Cscr$ it holds that $\Cscr':=\Cscr\cap NC$ is a non-crossing category of partitions. Conversely, it holds that $\Cscr=\langle\Cscr',\crosspart\rangle$. Thus, all categories with the crossing partition arise by adding the crossing to a non-crossing category. The seven non-crossing categories induce the following six categories with crossing [(ref:BS09)],[(ref:Web13)] |
| $$\Pscr_2=\langle\crosspart\rangle\subset\Big\{\begin{matrix}\langle\crosspart,\singleton\otimes\singleton\rangle\subset\langle\crosspart,\singleton\rangle\\\langle\crosspart,\fourpart\rangle\subset\langle\crosspart,\fourpart,\singleton\otimes\singleton\rangle\end{matrix}\Big\}\subset\langle\crosspart,\fourpart,\singleton\rangle=\Pscr$$ | $$\Pscr_2=\langle\crosspart\rangle\subset\Big\{\begin{matrix}\langle\crosspart,\singleton\otimes\singleton\rangle\subset\langle\crosspart,\singleton\rangle\\\langle\crosspart,\fourpart\rangle\subset\langle\crosspart,\fourpart,\singleton\otimes\singleton\rangle\end{matrix}\Big\}\subset\langle\crosspart,\fourpart,\singleton\rangle=\Pscr$$ | ||
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| ==== The half-liberated categories ==== | ==== The half-liberated categories ==== | ||
| - | A category of partitions $\Cscr$ is called **half-liberated** if $\halflibpart\in\Cscr$, but $\crosspart\not\in\Cscr$. Their classification consists of the categories $\langle\halflibpart\rangle$, $\langle\halflibpart,\singleton\otimes\singleton\rangle$, $\halflibpart,\fourpart\rangle$ and an infinite series $\langle\halflibpart,\fourpart,h_s\rangle$ for $s\ge 3$. Here, $h_s$ is a partition on $2s$ points consisting of two blocks, where all odd points are in one block and all even points are in the second block (in the [[partition#Word representation|word representation]] $h_s=(\mathsf{ab})^s=\mathsf{ab}\,\mathsf{ab}\cdots\mathsf{ab}$). | + | A category of partitions $\Cscr$ is called **half-liberated** if $\halflibpart\in\Cscr$, but $\crosspart\not\in\Cscr$. Their classification [(ref:Web13)] consists of the categories $\langle\halflibpart\rangle$, $\langle\halflibpart,\singleton\otimes\singleton\rangle$, $\halflibpart,\fourpart\rangle$ and an infinite series $\langle\halflibpart,\fourpart,h_s\rangle$ for $s\ge 3$. Here, $h_s$ is a partition on $2s$ points consisting of two blocks, where all odd points are in one block and all even points are in the second block (in the [[partition#Word representation|word representation]] $h_s=(\mathsf{ab})^s=\mathsf{ab}\,\mathsf{ab}\cdots\mathsf{ab}$). |
| ==== The hyperoctahedral categories ==== | ==== The hyperoctahedral categories ==== | ||
| - | A category of partitions $\Cscr$ is called **hyperoctahedral** if $\fourpart\in\Cscr$ but $\singleton\otimes\singleton\not\in\Cscr$. | + | A category of partitions $\Cscr$ is called **hyperoctahedral** if $\fourpart\in\Cscr$ but $\singleton\otimes\singleton\not\in\Cscr$. We have the following classification of hyperoctahedral categories [(ref:RW16)]. |
| + | |||
| + | If $\Paabaab\not\in\Cscr$, then $\Cscr$ is equal to either $\langle \pi_k\rangle$ or $\langle \pi_l\mid l\in\N\rangle$, where $\pi_k\in\Pscr(0,4k)$ is a partition, whose [[partition#word_representation|word representation]] can be written as | ||
| + | $$\pi_k=\mathsf{a}_1\mathsf{a}_2\cdots\mathsf{a}_k\mathsf{a}_k\cdots\mathsf{a}_2\mathsf{a}_1\mathsf{a}_1\mathsf{a}_2\cdots\mathsf{a}_k\mathsf{a}_k\cdots\mathsf{a}_2\mathsf{a}_1.$$ | ||
| + | Note that $\pi_1=\fourpart$ is a non-crossing partition. All the $\pi_k$ for $k>2$ have a crossing and the corresponding categories are pairwise distinct and also different from the above mentioned. | ||
| + | |||
| + | If $\Paabaab\in\Cscr$, then $\Cscr$ is so-called [[Group-theoretical category|group-theoretical category]]. There is a certain normal subgroup $A\subset\Z_2^{*n}$ such that the set of all partitions in $\Cscr$ written in the [[partition#word_representation|word representation]] using the generators of $\Z_2^{*n}$ as the alphabet coincide with $A$ [(ref:RW14)], [(ref:RW15)]. | ||
| + | |||
| + | ===== Further reading ===== | ||
| + | |||
| + | * Moritz Weber, //Introduction to compact (matrix) quantum groups and Banica–Speicher (easy) quantum groups//, [[https://doi.org/10.1007/s12044-017-0362-3|Proceedings – Mathematical Sciences, Vol. 127, No. 5, pp. 881–933, 2017]]. [[https://www.ias.ac.in/article/fulltext/pmsc/127/05/0881-0933|Full text online.]] | ||
| + | |||
| + | ===== References ===== | ||
| + | |||
| + | [(ref:BS09>> | ||
| + | title : Liberation of orthogonal Lie groups | ||
| + | journal : Advances in Mathematics | ||
| + | volume : 222 | ||
| + | number : 4 | ||
| + | pages : 1461--1501 | ||
| + | year : 2009 | ||
| + | url : http://dx.doi.org/10.1016/j.aim.2009.06.009 | ||
| + | author : Teodor Banica and Roland Speicher | ||
| + | )] | ||
| + | |||
| + | [(ref:Web13>> | ||
| + | title : On the classification of easy quantum groups | ||
| + | journal : Advances in Mathematics | ||
| + | volume : 245 | ||
| + | pages : 500--533 | ||
| + | year : 2013 | ||
| + | url : http://dx.doi.org/10.1016/j.aim.2013.06.019 | ||
| + | author : Moritz Weber | ||
| + | )] | ||
| + | |||
| + | [(ref:RW14>> | ||
| + | author : Sven Raum and Moritz Weber | ||
| + | title : The combinatorics of an algebraic class of easy quantum groups | ||
| + | journal : Infinite Dimensional Analysis, Quantum Probability and Related Topics | ||
| + | volume : 17 | ||
| + | number : 03 | ||
| + | pages : 1450016 | ||
| + | year : 2014 | ||
| + | url : http://dx.doi.org/10.1142/S0219025714500167 | ||
| + | )] | ||
| + | |||
| + | [(ref:RW15>> | ||
| + | author : Sven Raum and Moritz Weber | ||
| + | title : Easy quantum groups and quantum subgroups of a semi-direct product quantum group | ||
| + | journal : Journal of Noncommutative Geometry | ||
| + | volume : 9 | ||
| + | number : 4 | ||
| + | pages : 1261--1293 | ||
| + | year : 2015 | ||
| + | url : http://dx.doi.org/10.4171/JNCG/223 | ||
| + | )] | ||
| + | |||
| + | [(ref:RW16>> | ||
| + | author : Sven Raum and Moritz Weber | ||
| + | title : The Full Classification of Orthogonal Easy Quantum Groups | ||
| + | journal : Communications in Mathematical Physics | ||
| + | year : 2016 | ||
| + | volume : 341 | ||
| + | number : 3 | ||
| + | pages : 751--779 | ||
| + | url : http://dx.doi.org/10.1007/s00220-015-2537-z | ||
| + | )] | ||
| + | |||
| + | ~~REFNOTES ref ~~ | ||
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