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--- category_of_partitions [2019/03/04 14:37]
d.gromada
+++ category_of_partitions [2021/11/23 11:56] (current)
@@ Line 1: / Line 1: @@
 ====== 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]].
 ===== Definition =====
+==== Partitions ====
+Let $k,l\in\mathbb{N}_0$, by a [[partition|partition]] of $k$ upper and $l$ lower points we mean herre a partition of the set $\{1,\dots,k\}\sqcup\{1,\dots,l\}\approx\{1,\dots,k+l\}$, that is, a decomposition of the set of $k+l$ points into non-empty disjoint subsets, called **blocks**. The first $k$ points are called **upper** and the last $l$ points are called **lower**. The set of all partitions on $k$ upper and $l$ lower points is denoted $\Pscr(k,l)$. We denote the union $\Pscr:=\bigcup_{k,l\in\N_0}\Pscr(k,l)$.
+We illustrate partitions graphically by putting $k$ points in one row and $l$ points on another row below and connecting by lines those points that are grouped in one block. All lines are drawn between those two rows.
+Below, we give an example of two partitions $p\in\Pscr(3,4)$ and $q\in\Pscr(4,4)$ defined by their graphical representation. The first set of points is decomposed into three blocks, whereas the second one is into five blocks. In addition, the first one is an example of a **non-crossing** partition, i.e. a partition that can be drawn in a way that lines connecting different blocks do not intersect (following the rule that all lines are between the two rows of points). On the other hand, the second partition has one crossing.
+$$
+p=
+\BigPartition{
+\Pblock 0 to 0.25:2,3
+\Pblock 1 to 0.75:1,2,3
+\Psingletons 0 to 0.25:1,4
+\Pline (2.5,0.25) (2.5,0.75)
+}
+\qquad
+q=
+\BigPartition{
+\Psingletons 0 to 0.25:1,4
+\Psingletons 1 to 0.75:1,4
+\Pline (2,0) (3,1)
+\Pline (3,0) (2,1)
+\Pline (2.75,0.25) (4,0.25)
+}
+$$
+A block containing a single point is called a **singleton**. In particular, the partitions containing only one point are called singletons and for clarity denoted by an arrow $\singleton\in\Pscr(0,1)$ and $\upsingleton\in\Pscr(1,0)$.
+==== The category structure ====
 The set of all [[Partition|partitions]] $\Pscr$ can be given the structure of a monoidal involutive category by introducing the following operations.
@@ Line 112: / Line 142: @@
 ==== Non-crossing partitions ====
-==== Categories containing the crossing partition ====
+There are the following seven easy [[Non-crossing category of partitions|non-crossing categories]] of partitions [(ref:BS09)],[(ref:Web13)].
+$$NC_2=\langle\rangle\subset\Big\{\begin{matrix}\langle\singleton\otimes\singleton\rangle\subset\langle\Labac\rangle\subset\langle\singleton\rangle\\\langle\fourpart\rangle\subset\langle\fourpart,\singleton\otimes\singleton\rangle\end{matrix}\Big\}\subset\langle\fourpart,\singleton\rangle=NC$$
+Here, we denote by $NC$ the [[Category of all non-crossing partitions|category of all non-crossing partitions]]. The smallest category $\langle\rangle$ is  $NC_2$, the [[Temperley–Lieb category|category of non-crossing pairings]].
+==== The group categories ====
+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 [(ref:BS09)],[(ref:Web13)]
+$$\Pscr_2=\langle\crosspart\rangle\subset\Big\{\begin{matrix}\langle\crosspart,\singleton\otimes\singleton\rangle\subset\langle\crosspart,\singleton\rangle\\\langle\crosspart,\fourpart\rangle\subset\langle\crosspart,\fourpart,\singleton\otimes\singleton\rangle\end{matrix}\Big\}\subset\langle\crosspart,\fourpart,\singleton\rangle=\Pscr$$
+Here, we denote by $\Pscr_2$ the [[Brauer category|category of all pairings]]. The largest category is of course the [[category_of_all_partitions|category of all partitions]] $\Pscr$.
+==== The half-liberated categories ====
+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 ====
+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
+)]
-==== Categories containing the half-liberating partition ====
+[(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
+)]
-==== The hyperoctahedral case ====
+[(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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