Category of pair partitions with even distances between legs

The category of pair partitions with even distances between legs is a Banica-Speicher category of partitions inducing the corepresentation category of the half-liberated orthogonal quantum groups. It is a proper subcategory of the Brauer category. Its unique proper subcategory is the Temperley-Lieb category.

By the category of pair partitions with even distances between legs one denotes the subcategory of the category of all partitions whose morphism class is the set of all pair partitions with even distances between legs. It was introduced by Banica and Speicher in [BanSp09].

What it means for a partition to belong to this set has been said in three different but equivalent ways:

• For any given block of only evenly many blocks of with exist which cross , i.e., such that one can find and with and (where is the cyclic order of ).
• For any block of and any two legs there is an even number of points located between and , i.e. in the interval given by the set .
• If one labels the points of in alternating fashion with one of two symbols and along the cyclic order of , then blocks of may only join points with unequal labels.

The set of all pair partitions with even distances between legs is denoted by in [BanSp09].

The category of pair partitions with even distances between legs is the subcategory of generated by the partition . This canonical generator embodies the half-commutation relations .

Via Tannaka-Krein duality for compact quantum groups, the category of all pair partitions with even distances between legs corresponds to the family of half-liberated orthogonal quantum groups.