====== Glued product of compact matrix quantum groups ======

Glued products form a less standard product construction, which is defined only for matrix quantum groups. It was formally defined in [(ref:TW17)] to interpret some coloured categories of partitions in terms of compact matrix quantum groups.

===== Definition =====

==== Glued tensor product ====

Let $G=(C(G),u)$ and $H=(C(H),v)$ be compact matrix quantum groups. We define the **glued tensor product**
$$G\tiltimes H:=(C(G\tiltimes H),u\otimes v),$$
where $C(G\tiltimes H)$ is the C*-subalgebra of $C(G)\otimes_{\rm max} C(H)$ generated by $u_{ij}v_{kl}$ -- the elements of the tensor product $u\otimes v$.

==== Glued free product ====

Similarly, we define the **glued free product**
$$G\tilstar H:=(C(G\tilstar H),u\otimes v),$$
where $C(G\tilstar H)$ is the C*-subalgebra of $C(G)*_\C C(H)$ generated by $u_{ij}v_{kl}$.

==== Main application: Complexifications ====

Given a compact matrix quantum group $G$, we call $G\tiltimes\hat\Z$ the **tensor complexification** of $G$, $G\tiltimes\hat\Z_k$ is the **tensor //k//-complexification**, $G\tilstar\hat\Z$ is the **free complexification** and $G\tilstar\hat\Z_k$ is the **free //k//-complexification** of $G$. The free complexification was studied already by Banica in [(ref:Ban99)], [(ref:Ban08)].

==== Remark on the distinction with ordinary products ====

The glued versions of the tensor and free products $G\tiltimes H$ and $G\tilstar H$ are by definition quotient quantum groups of the standard constructions $G\times H$ and $G\hatstar H$. Often it happens that the elements $u_{ij}v_{kl}$ already generate the whole C*-algebra, so actually $G\tiltimes H\simeq G\times H$ or $G\tilstar H\simeq H\hatstar H$. Even in this case, however, we should not put the equality sign here. Although the quantum groups can have the same underlying C*-algebra and hence be isomorphic, they are never identical as compact matrix quantum groups since their fundamental representations are always different -- $u\oplus v$ in the standard case and $u\otimes v$ in the glued case.

===== Examples =====

==== Tensor complexification for classical groups ====

Let us have a look on how the definition of glued tensor product looks like for groups. Let $G$ and $H$ be two matrix groups, then we have
$$G\tiltimes H=\{A\otimes B\mid A\in G,B\in H\},$$
where $\otimes$ denotes the Kronecker product.

As a concrete example, consider the symmetric group $S_N$ represented by the permutation matrices and consider the cyclic group of order two $\hat\Z_2=\Z_2$ represented by a single complex number $\pm 1$. Then $S_N\tiltimes\Z_2$ consists of $N\times N$ permutation matrices multiplied by a global sign. Thus, $S_N\tiltimes\Z_2$ is actually isomorphic to $S_N\times\Z_2$. Nevertheless, by $S_N\times\Z_2$ we mean a different matrix realization. The ordinary product $S_N\times\Z_2$ consists of $(N+1)\times(N+1)$ matrices with block diagonal structure, where one block is formed by an $N\times N$ permutation matrix and the second block is the single number $\pm 1$.

In general, take any cyclic group $\hat\Z_k=\Z_k$ with $k\in\N$ represented by the $k$-th roots of unity. Then for any matrix group $G$, we have
$$G\tiltimes\Z_k=\{{\rm e}^{2\pi ij/k}A\mid j=0,\dots,k-1;\;A\in G\}.$$
We can do the same for the whole unit disk $\hat\Z=\T\subset\C$
$$G\tiltimes\T=\{zA\mid z\in\T;\;A\in G\}.$$

==== Unitary quantum groups as free complexifications ====

It holds that $U_N^+=O_N^+\hatstar\hat\Z$ [(ref:Ban97)]

===== References =====


[(ref:TW17>>
author:  Pierre Tarrago and Moritz Weber
title:   Unitary Easy Quantum Groups: The Free Case and the Group Case
journal: International Mathematics Research Notices
volume:  2017
number:  18
pages:   5710–5750
year:    2017
url:     https://dx.doi.org/10.1093/imrn/rnw185
)]

[(ref:Ban99>>
author:  Teodor Banica
title:   Representations of compact quantum groups and subfactors
journal: Journal für die reine und angewandte Mathematik
year:    1999
volume:  509
pages:   167–198
url:     http://dx.doi.org/10.1515/crll.1999.509.167
)]

[(ref:Ban08>>
author:  Teodor Banica
title:   A Note on Free Quantum Groups
journal: Annales Mathématiques Blaise Pascal
volume:  15
number:  2
year:    2008
pages:   135–146
url:     http://dx.doi.org/10.5802/ambp.243
)]


[(ref:Ban97>>
author:  Teodor Banica
title:   Le Groupe Quantique Compact Libre U(n)
journal: Communications in Mathematical Physics
year:    1997
volume:  190
number:  1
pages:   143–172
url:     http://dx.doi.org/10.1007/s002200050237
)]