====== Free product of quantum groups ======

The following construction was defined by Wang in [(ref:Wan95)]

===== Definition =====

==== Motivation for the compact case: Direct product of groups ====

Given two (compact) groups $G$ and $H$, we can construct their direct product. Such a construction is generalized to the quantum case as the [[tensor_product|tensor product of quantum groups]]. Here, we have $C(G\times H)=C(G)\otimes C(H)$. In particular, the two factors $C(G)$ and $C(H)$ commute.

If we start with some pair of //free// compact quantum groups (such as the [[free orthogonal quantum group]] $O_N^+$), we would like to generalize the direct product in a way, that the resulting C*-algebra will again be //as free as possible//. To achieve that, we replace the tensor product by the C*-algebraic free product.

==== Definition for compact quantum groups ====

Let $G$ and $H$ be compact quantum groups. We define [(ref:Wan95)] their **dual free product** $G\hatstar H$ to be the quantum group with underlying C*-algebra $C(G\hatstar H):=C(G)*_\C C(H)$ and comultiplication the unique unital $*$-homomorphism satisfying
$$\Delta_{G\hatstar H}(a)=\Delta_G(a),\quad\Delta_{G\hatstar H}(b)=\Delta_H(b)\qquad\hbox{for all $a\in C(G)$, $b\in C(H)$.}$$
Formally, we should rather write $\Delta_{G\hatstar H}(\iota_G(a))=(\iota_G\otimes\iota_G)(\Delta_G(a))$, $\Delta_{G\hatstar H}(\iota_H(b))=(\iota_H\otimes\iota_H)(\Delta_H(b))$, where $\iota_G$ and $\iota_H$ are the canonical inclusions into $C(G)*_\C C(H)$.

==== Definition for compact matrix quantum groups ====

Let $G=(C(G),u)$ and $H=(C(H),v)$ be compact matrix quantum groups. Then
$$G\hatstar H=(C(G)*_\C C(H),u\oplus v)$$
is also a compact matrix quantum group. It is a matrix realization of the dual free product of $G$ and $H$ as defined above.

==== Motivation in the discrete case: Free product of groups ====

Given two discrete groups $\Gamma_1$, $\Gamma_2$, we can construct their [[wp>free product]] $\Gamma_1*\Gamma_2$. Note that the above definition of the //dual free product// should //not// be interpreted this way, because the freeness in case of the dual free product appears in the C*-algebra multiplication, not in the comultiplication. This is the reason for the word //dual// in the name. (And indeed, free product of the compact groups is a group that is never compact; in contrast, dual free product of two compact groups is a compact quantum group, but not a group).

However, dualizing the definition above, we may indeed generalize the definition of a free product to the case of discrete quantum groups.

==== Definition ====

Let $\Gamma_1$ and $\Gamma_2$ be discrete quantum groups. We define their **free product** to be the quantum group $\Gamma_1*\Gamma_2:=\widehat{\hat\Gamma_1\hatstar\hat\Gamma_2}$.

In case when $\Gamma_1$ and $\Gamma_2$ are discrete groups, the above defined direct and free product exactly corresponds to the classical construction for groups.

==== Remark on the notation ====

The notation using the hat and calling the product //dual free product// is rather new. It was first used probably in [(ref:DFSW13)]. Some authors omit the word //dual// and denote the product just by asterisk $*$. In most cases, it should be clear, which product is meant – whether the dual free product or the free product – since the former is defined only in the compact case, whereas the latter only in the discrete case. It may, however, cause some confusion if one works with finite (quantum) groups.

===== Properties =====

==== Irreducible representations ====

Let $G$ and $H$ be compact quantum groups. Let $\{u^\alpha\}_{\alpha\in\Irr G}$ and $\{v^\beta\}_{\beta\in\Irr H}$ be complete sets of irreducible representation of $G$ and $H$. Denote by $\iota_G$ and $\iota_H$ the embeddings of $C(G)$ and $C(H)$ into $C(G\hatstar H)$, respectively, and denote $w^\alpha_{ij}:=\iota_G(u^\alpha_{ij})$ and $w^\beta_{ij}:=\iota_H(v^\beta_{ij})$. Then a complete set of irreducible representations of $G\hatstar H$ is formed by the trivial representation together with
$$w^{\gamma_1}\otimes w^{\gamma_2}\otimes\cdots\otimes w^{\gamma_n},$$
where $\gamma_i\in\Irr G\cup\Irr H$ are non-trivial representations such that the sets $\Irr G$ and $\Irr H$ alternate, so if $\gamma_i\in\Irr G$, then $\gamma_{i+1}\in\Irr H$ and vice versa. [(ref:Wan95)]

==== Haar state ====

Let $G$ and $H$ be compact quantum groups and let $h_G$, $h_H$, respectively be the corresponding Haar states. Then the Haar state on $G*H$ is given by the free product $h=h_G*h_H$. [(ref:Wan95)]


===== References =====


[(ref:Wan95>>
author:  Shuzhou Wang
title:   Free products of compact quantum groups
journal: Communications in Mathematical Physics
year:    1995
volume:  167
number:  3
pages:   671–692
url:     http://dx.doi.org/10.1007/BF02101540
)]


[(ref:DFSW13>>
author    : Matthew Daws, Pierre Fima, Adam Skalski, Stuart White
title     : The Haagerup property for locally compact quantum groups
journal   : Journal für die reine und angewandte Mathematik
volume    : 2016
number    : 711
pages     : 189–229
year      : 2013
doi       : 10.1515/crelle-2013-0113
url       : https://doi.org/10.1515/crelle-2013-0113
)]