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

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

The category of 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 half-liberated bistochastic quantum groups.

Definition

By the category of 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 partitions of even size with small blocks and even distances between legs. It was introduced by Weber in [Web12].

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.
$p$ 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.

In particular, the category of partitions of even size with small blocks and even distances between legs is a subcategory of the categories of partitions of even size and of partitions with small blocks.

Canonical Generator

The category of partitions of even size with small blocks and even distances between legs is the subcategory of $\Pscr$ generated by the partitions $\{\Pabcabc,\singleton\otimes \singleton\}$ . The partition $\Pabcabc$ embodies the half-commutation relations $acb=bca$ .

Associated easy quantum group

Via Tannaka-Krein duality for compact quantum groups, the category of partitions of even size with small blocks and even distances between legs corresponds to the family $(B^{\#\ast}_N)_{N\in \N}$ of half-liberated 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 partitions of even size with small blocks and even distances between legs

Definition

Canonical Generator

Associated easy quantum group

References