✎ linear_category_of_partitions [CMQG Wiki]

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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 group|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_group|compact matrix quantum groups]], the term **category of partitions** often means an [[Category of partitions|easy category of partitions]] in the sense of Banica and Speicher.

===== Definition =====

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

For given $p_1,\dots,p_n\in\Part_\delta$, 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. 

===== Additional operations =====

Thanks to the fact that any linear category of partitions $\Kscr$ contains the pair partition, we can introduce additional operations on $\Part_\delta$ that stabilize any category $\Kscr\subset\Part_\delta$.

==== Rotations ====

For $p\in\Pscr(k,l)$, $k>0$, we define its **left rotation** as a partition $\Lrot p\in\Part(k-1,l+1)$ obtained by moving the leftmost point of the upper row on the beginning of the lower row. Similarly, for $p\in\Pscr(k,l)$, $l>0$, we can define its **right rotation** $\Rrot p\in\Part(k+1,l-1)$ by moving the last point of the lower row to the end of the upper row. Both operations are obviously invertible. We extend this operation linearly on $\Part_\delta$.

**Proposition.** Every category $\Kscr\subset\Part_\delta$ is closed under taking left and right rotations and their inverses.

==== One line operations ====



===== Connection with CMQG =====

===== Examples and classification =====

The most important examples of partition categories are the [[category_of_all_partitions|Category of all partitions]], [[Category of all non-crossing partitions|Category of all non-crossing partitions]], [[Category of all pair partitions|Category of all pair partitions]] and [[Category of all non-crossing pair partitions|Category of all non-crossing pair partitions]] forming a square of inclusions
\begin{eqnarray*}
\Pair   & \subset & \Part\\
\cup    &         & \cup\\
\NCPair & \subset & \NCPart
\end{eqnarray*}

Classification of partition categories is available in the case of [[category_of_partitions|easy categories of partitions]]. Another source of examples are the [[group-theoretical categories of partitions|group-theoretical categories of partitions]]. Outside those two classes, several additional examples are known, but no systematical classification result is available.


===== Further reading =====

===== References =====
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