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 [Chi15], [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 $H_1,H_2\subset U^+(F)$ .

References

[Chi15] Alexandru Chirvasitu, 2015. Residually finite quantum group algebras. Journal of Functional Analysis, 268(11), pp.3508–3533.

[BCV17] Michael Brannan and Benoît Collins and Roland Vergnioux, 2017. The Connes embedding property for quantum group von Neumann algebras. Transactions of the American Mathematical Society, 369, pp.3799–3819.

Table of Contents

Topological generation

Definition

Uniqueness, existence, and characterization

Further reading

References

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Table of Contents

Topological generation

Definition

Uniqueness, existence, and characterization

Further reading

References

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