\section{Introduction}\label{section_introduction}

One problem in the theory of compact quantum groups is to find examples whose invariants can be calculated. The fusion rules of a compact quantum group are one of these invariants. Fusion rules give a complete description of equivalence classes of irreducible corepresentations and a decomposition of the tensor product of two of them into irreducible corepresentations. One approach to this problem is given by 'free quantum groups' as defined in \cite{banicaspeicher09}. These are orthogonal quantum groups, i.e. subgroups of the free orthogonal Wang algebra, whose intertwiners can be described by non-crossing partitions.\\
Given natural numbers $k$ and $l$ the set $\PART(k,l)$ denotes the set of all partitions on two rows with $k$ and $l$ points, respectively. That is, an element $P \in \PART(k,l)$ is a partition of the disjoint union $\{1,...,k\} \sqcup \{1,...,l\}$. Alternatively it can be described by a diagram
\begin{equation*}
\left\{\begin{xy}
	\POS(0,5) 	*{\cdot}
	\POS(4,5) 	*{\cdot}
	\POS(8,5) 	*{...}
	\POS(12,5)		*{\cdot}
	
	\POS(6,0)		*{P}
	
	\POS(0,-5) 	*{\cdot}
	\POS(4,-5) 	*{\cdot}
	\POS(8,-5) 	*{...}
	\POS(12,-5)	*{\cdot}
\end{xy}\right\}
\end{equation*}
connecting the $k$ points in the upper row and the $l$ points in the lower row according to the partition of $\{1,...,k\} \sqcup \{1,...,l\}$. $P$ is called non-crossing if it can be represented by a diagram with no lines crossing. The set of all non-crossing partitions on $k$ and $l$ points is denoted by $\NC(k,l)$.\\
Let $n, k, l \in \NN$ and let $(e_i)$ be the standard basis of $\CC^n$. Let $i = (i_1,...,i_k) \in \{1,...,n\}^k$ and $j = (j_1,...,j_l) \in \{1,...,n\}^l$ be multi indices and $P \in \PART(k,l)$. We set $P(i,j) = 1$ if and only if the diagram $P$ joins only equal numbers after writing the entries of $i$ in the upper row of the above diagram and those of $j$ in the lower row. If $P$ connects different numbers set $P(i,j) = 0$.\\
Using this notation, a partition $P \in \PART(k,l)$ defines a linear map $T_P$ from $(\CC^n)^{\otimes k}$ to $(\CC^n)^{\otimes l}$ by \[T_P(e_{i_1} \otimes ... \otimes e_{i_k}) = \sum_{j_1, ..., j_l}{P(i_1,...,i_k;j_1,...,j_l) \cdot e_{j_1} \otimes ... \otimes e_{j_l}}.\] A subspace of $\Hom((\CC^n)^{\otimes k}, (\CC^n)^{\otimes l})$ is by definition spanned by partitions if it is linearly generated by a family $(T_P)$ where $P$ runs through some subset of $\PART(k,l)$.\\
In \cite{wang95} the free unitary Wang algebra
\[A_u(n) := \Cstar(u_{ij}, \; 1 \leq i,j \leq n | (u_{ij})_{ij}, (u_{ij}^*)_{ij} \text{ are unitary})\]
and the free orthogonal Wang algebra
\[A_o(n) := \Cstar(u_{ij}, \; 1 \leq i,j \leq n | (u_{ij})_{ij} = (u_{ij}^*)_{ij} \text{ is unitary})\]
were introduced. Moreover in \cite{wang98} the quantum permutation group 
\[A_s(n) := \Cstar\left(u_{ij}, \; 1 \leq i,j \leq n \middle|
\begin{array}{l} (u_{ij}) = (u_{ij}^*) \text{ is unitary and } u_{ij} \text{ are} \\
\text{partial isometries summing up to one} \\
\text{in every row and every column} \end{array}\right)\]
was defined. Note that ``are partial isometries'' can be replaced by ``are projections''. The three last named algebras are compact matrix quantum groups in the sense of Woronowicz \cite{woronowicz91}. \\
The following class of quantum groups will be of interest in this paper.

\begin{definition}\label{definition_free_quantum_groups}
Let $(A,U)$ be a compact matrix quantum group. Then it is called free if
\begin{itemize}
  \item The morphism $(\Aun, U_\textup{u}) \rightarrow (\Asn, U_\textup{s})$ mapping the entries of $U_\textup{u}$ to those of $U_\textup{s}$ factorizes through $(A,U)$.
  \item The intertwiner spaces $\Hom(U^{i_1} \boxtimes \dotsm \boxtimes U^{i_k}, U^{j_1}\boxtimes \dotsm \boxtimes U^{j_l})$, $i_\alpha, j_\beta \in \{1, \overline{\phantom{U}}\}$ are spanned by partitions, where $\overline{U} = (u_{ij}^*)$ is the conjugate corepresentation of $U$ and $\boxtimes$ denotes the tensor product of corepresentations.
\end{itemize}
If the first condition is strengthened by requiring that the morphism $(\Aon, U_\textup{o}) \rightarrow (\Asn, U_\textup{s})$ factors through $(A,U)$, then $A$ it is called orthogonal free.
\end{definition}

In \cite{banicaspeicher09} the following classification was achieved.

\begin{theorem}
  There are exactly six orthogonal free quantum groups. Namely
  \begin{enumerate}
    \item The free orthogonal Wang algebra.
    \item The quantum permutation group.
    \item The hyperoctahedral quantum group
    	\[\Ahn := \Cstar\left(u_{ij}, \; 1 \leq i,j \leq n \middle|
    	\begin{array}{l} (u_{ij}) = (u_{ij}^*) \text{ is unitary and } \\
    	u_{ij} \text{ are partial isometries} \end{array}\right).\]
    \item The bistochastic quantum group 
    	\[\Abn := \Cstar\left(u_{ij}, \; 1 \leq i,j \leq n \middle|
 			\begin{array}{l} (u_{ij}) = (u_{ij}^*) \text{ is unitary and } \\
 			u_{ij} \text{ sum up to one} \\
 			\text{in every row and every column} \end{array}\right).\]
    \item The symmetrized bistochastic quantum group
    	\[\Abprimen := \Cstar\left(u_{ij}, \; 1 \leq i,j \leq n \middle|
 			\begin{array}{l} (u_{ij}) = (u_{ij}^*) \text{ is unitary and } \\
 			u_{ij} \text{ sum up to the same element} \\
 			\text{in every row and every column} \end{array}\right).\]
    \item The symmetrized quantum permutation group
    	\[\Asprimen := \Cstar\left(u_{ij}, \; 1 \leq i,j \leq n \middle|
 			\begin{array}{l} (u_{ij}) = (u_{ij}^*) \text{ is unitary and } \\
 			u_{ij} \text{ are partial isometries} \\
 			\text{summing up to the same element} \\
 			\text{in every row and every column} \end{array}\right).\]
  \end{enumerate}
\end{theorem}

The fusion rules of (1) were calculated in \cite{banica96}, those of (2) in \cite{banica99} and those of (3) in \cite{banicavergnioux09}. We show that the remaining examples are slight modifications of $\Aon$ and $\Asn$. In particular we can derive their fusion rules and find that $\Abprimen$ and $\Asprimen$ are counterexamples to a conjecture by Banica and Vergnioux given in \cite{banicavergnioux09}.\\
In \cite{banica08} the free complexification of orthogonal free quantum groups was considered. If $(A,U)$ is a orthogonal free quantum group, then its free complexification $(\widetilde{A}, \widetilde{U})$ is by definition the sub-C*-algebra of the free product $A * \cont(S^1)$ generated by the entries of $\widetilde{U} := U \cdot \id_{\textup{S}^1} = (u_{ij} \cdot \id_{\textup{S}^1})$. Here $\id_{\textup{S}^1}$ denotes the canonical generator of $\cont(\textup{S}^1)$. As Banica shows in \cite{banica08} the intertwiners between tensor products of the fundamental corepresentation and its conjugate can be described by the intertwiners of the orthogonal free quantum group it comes from. With additional requirements we can calculate the fusion rules of the free complexification from the fusion rules of the original orthogonal free quantum group. These additional requirements are fulfilled by $\Aon$ and $\Ahn$, which gives the fusion rules of $\Akn = \widetilde{\Ahn}$. Those of $\Aun = \widetilde{\Aon}$ are known from \cite{banica97}. \\
From \cite{banica08} we know that $\widetilde{\Abn} = \widetilde{\Abprimen}$ and $\widetilde{\Asn} = \widetilde{\Asprimen}$. We denote $\widetilde{\Abn} =: \Acn$ and $\widetilde{\Asn} =: \Apn$. They can be decomposed and described in terms of $\Aon$ and $\Asn$ again. \\
\textbf{Acknowledgment:} I want to thank both Thomas Timmermann for suggesting to work on fusion rules of free quantum groups and Stefaan Vaes for helpful discussions about this article, especially on the last section. Moreover, I want to thank the referee for helpful comments.
{"config": "arxiv", "file": "1006.2979/Introduction.tex"}