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====== Topological generation ======

Given two compact matrix groups represented by matrices of the same size $H_1,H_2\subset\GL_N$, we may ask, what compact matrix group they generate. That is, find the smallest compact subgroup $\langle H_1,H_2\rangle\subset\GL_N$ containing both $H_1$ and $H_2$. We may ask the same question also for compact matrix quantum groups. The idea goes back to [(:ref:Chi15)], [(:ref:BCV17)].

===== Definition =====

Let $H_1$ and $H_2$ be compact matrix quantum groups with fundamental representations $v_1$ and $v_2$ of the same size. We define $G:=\langle H_1,H_2\rangle$ to be the smallest compact matrix quantum group containing $H_1$ and $H_2$. We say that $G$ is **topologically generated** by $H_1$ and $H_2$. That is $G=(C(G),u)$ is a quantum group satisfying the following.

  - $H_1,H_2\subset G$, so the size of $u$ coincides with the size of $v_1$ and $v_2$ and there are surjective $*$-homomorphisms $\phi_k\colon C(G)\to C(H_k)$ mapping $u_{ij}\mapsto [v_k]_{ij}$ for $k=1,2$.
  - For every compact quantum group $\tilde G$ such that $H_1,H_2\subset\tilde G\subset G$, we have~$G=\tilde G$.

===== Uniqueness, existence, and characterization =====

Note that already in the case of groups it may happen that for two compact matrix groups $H_1$, $H_2$, the group they generate $\langle H_1,H_2\rangle$ is not compact. We can fix this issue assuming that the compact groups are unitary $H_1,H_2\subset U_N$. Then surely $\langle H_1,H_2\rangle\subset U_N$ and hence it is compact.

Thus, also for compact matrix quantum groups $H_1$ and $H_2$, the quantum group $\langle H_1,H_2\rangle$ may not exist unless we assume $H_1,H_2\subset U^+(F)$ for some common $F\in\GL_N$. This can be formulated as an equivalence.

Suppose $G$, $H_1$, and $H_2$ are compact matrix quantum groups with unitary fundamental representations of the same size. The following are equivalent.
  - $G=\langle H_1,H_2\rangle$,
  - $I_G=I_{H_1}\cap I_{H_2}$,
  - $\FundRep_G(w_1,w_2)=\FundRep_{H_1}(w_1,w_2)\cap\FundRep_{H_2}(w_1,w_2)$ for all $w_1,w_2$.
In particular, $\langle H_1,H_2\rangle$ exists if and only if there is a~matrix~$F$ such that $H_1,H_2\subset U^+(F)$.

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

  * Teodor Banica, //Free quantum groups and related topics// [[https://banica.u-cergy.fr/a3.pdf|Online notes]]
  * Amaury Freslon, //Applications of non-crossing partitions to quantum groups//, lecture notes, 2019. [[https://www.imo.universite-paris-saclay.fr/~freslon/Documents/Talks/Copenhaguen_2019.pdf|Available online]]
  * Daniel Gromada, //Compact matrix quantum groups and their representation categories//, PhD thesis, Saarland University, 2020.



===== References =====


[( :ref:BCV17 >>
title:   The Connes embedding property for quantum group von Neumann algebras
journal: Transactions of the American Mathematical Society
volume:  369
pages:   3799–3819
year:    2017
url:     https://dx.doi.org/10.1090/tran/6752
author:  Michael Brannan and Benoît Collins and Roland Vergnioux
)]


[( :ref:Chi15 >>
title:   Residually finite quantum group algebras
journal: Journal of Functional Analysis
volume:  268
number:  11
pages:   3508--3533
year:    2015
url:     https://dx.doi.org/10.1016/j.jfa.2015.01.013
author:  Alexandru Chirvasitu
)]
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