Category of non-crossing partitions of even size with small blocks and even distances between legs

The category of non-crossing partitions of even size with small blocks and even distances between legs is a Banica-Speicher category of partitions inducing the corepresentation category of the freely modified bistochastic quantum groups.

Definition

By the category of non-crossing partitions of even size with small blocks and even distances between legs one denotes the subcategory of the category of all partitions $\Pscr$ whose morphism class is the set of non-crossing partitions of even size with small blocks and even distances between legs. It was introduced by Banica and Speicher in [BanSp09].

A partition $p\in \Pscr$ belongs to this set if the following conditions are met:

$p$ has small blocks, meaning that every block in $p$ is of size $1$ or $2$ .
$p$ is of even size, i.e., if $k,l\in \{0\}\cup \N$ are such that $p\in \Pscr(k,l)$ , then $k+l$ is an even number, which is to say that $p$ has evenly many points. Given that $p$ has small blocks, $p$ is of even size if and only if it has an even number of singleton blocks.
$p$ is non-crossing, which means that there exist no blocks $B$ and $B'$ of $p$ with $B\neq B'$ and no legs $i,j\in B$ and $i',j'\in B'$ such that $i\prec i'\prec j$ and $i'\prec j\prec j'$ with respect to the cyclic order of $p$ . See also category of all non-crossing partitions.
$p$ has even distances between legs. This property has been expressed in three different but equivalent ways:
- For any given block $B$ of $p$ only evenly many blocks $B'$ of $p$ with $B\neq B'$ exist which cross $B$ , i.e., such that one can find $i,j\in B$ and $i',j'\in B'$ with $i\prec i'\prec j$ and $i'\prec j\prec j'$ (where $\cdot\!\prec\!\cdot\!\prec\!\cdot$ is the cyclic order of $p$ ).
- For any block $B$ of $p$ and any two legs $i,j\in B$ there is an even number of points located between $i$ and $j$ , i.e. in the interval $]i,j[_p$ given by the set $\{ k\,\vert\, i\prec k\prec j\}$ .
- If one labels the points of $p$ in alternating fashion with one of two symbols $\oplus$ and $\ominus$ along the cyclic order of $p$ , then blocks of $p$ may only join points with unequal labels.
The name set of all non-crossing partitions of even size with small blocks and even distances between legs is to be taken literally.

Canonical Generator

The category of non-crossing partitions of even size with small blocks and even distances between legs is the subcategory of $\Pscr$ generated by the partition $\singleton\otimes \singleton$ .

Associated easy quantum group

Via Tannaka-Krein duality for compact quantum groups, the category of non-crossing partitions of even size with small blocks and even distances between legs corresponds to the family $(B^{\#+}_N)_{N\in \N}$ of freely modified bistochastic quantum groups.

References

[BanSp09] Banica, Teodor and Speicher, Roland, 2009. Liberation of orthogonal Lie groups. Advances in Mathematics, 222, pp.1461–150.

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Category of non-crossing partitions of even size with small blocks and even distances between legs

Definition

Canonical Generator

Associated easy quantum group

References

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