Category of non-crossing partitions of even size with small blocks

Category of non-crossing partitions of even size with small blocks

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

Definition

By the category of non-crossing partitions of even size with small blocks one denotes the subcategory of the category of all partitions $\Pscr$ whose morphism set is the set of all non-crossing partitions of even size with small blocks. It was introduced by Weber in [Web12].

For all $k,l\in \{0\}\cup \N$ , a partition $p\in \Pscr(k,l)$ is said to be of even size if $k+l$ is an even number, i.e., if $p$ has evenly many points.
We say that $p$ has small blocks if every block of $p$ has size $1$ or $2$ .
The partition $p$ is said to be non-crossing if 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.
The name set of all non-crossing partitions of even size with small blocks is to be taken literally.

Canonical generator

The category of all non-crossing partitions of even size with small blocks is the subcategory of $\Pscr$ generated by the partition $\Labac$ .

Associated easy quantum group

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

References

[Web12] Weber, Moritz, 2013. On the classification of easy quantum groups. Advances in Mathematics, 245, pp.500–533.

Table of Contents

Category of non-crossing partitions of even size with small blocks

Definition

Canonical generator

Associated easy quantum group

References