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Linear category of partitions

Partition categories also known as linear categories of partitions have been recently heavily studied by researchers from different fields of mathematics and physics such as group theory, compact quantum groups, operator algebras, tensor categories or statistical physics. In the theory of (compact quantum) groups they are used to model the representation theory of a given quantum group.

Note that in the theory of compact matrix quantum groups, the term category of partitions often means an easy category of partitions in the sense of Banica and Speicher.

Definition

A linear category of partitions is a rigid monoidal $*$ -subcategory of the linear category of all partitions. That is, any collection $\mathscr{K}$ of linear subspaces $\mathscr{K}(k,l)\subset\Partlin_\delta(k,l)$ containing the identity partition $\idpart\in\mathscr{K}(1,1)$ and the pair partition $\pairpart\in\mathscr{K}(0,2)$ , which is closed under the tensor product, composition and involution is called a linear category of partitions.

For given $p_1,\dots,p_n\in\Partlin_\delta$ , we denote by $\langle p_1,\dots,p_n\rangle_\delta$ the smallest linear category of partitions containing $p_1,\dots,p_n$ . We say that $p_1,\dots,p_n$ generate $\langle p_1,\dots,p_n\rangle_\delta$ . Note that the pair partition is contained in the category by definition and hence will not be explicitly listed as a generator. Note that any element in $\langle p_1,\dots,p_n\rangle_\delta$ can be obtained from the generators $p_1,\dots,p_n$ and the pair partition $\pairpart$ by performing a finite amount of category operations and linear combinations.