Intersection of quantum groups

Given two groups $H_1$ , $H_2$ embedded into a larger one, we can compute their intersection $H_1\cap H_2$ , which is again a group. In particular, we can take compact matrix groups $H_1$ , $H_2$ represented by matrices of the same size and ask, what is their intersection $H_1\cap H_2$ – the largest subgroup of both. This concept can be generalized to the case of compact matrix quantum groups.

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 denote by $H_1\cap H_2$ the intersection of $H_1$ and $H_2$ defined as the largest quantum subgroup of both $H_1$ and $H_2$ . That is $H_1\cap H_2=:G=(C(G),u)$ is defined by the fact that

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

Uniqueness, existence, and characterization

The quantum group $H_1\cap H_2$ is unique and always exists as follows from the following characterization.

Suppose $G$ , $H_1$ , and $H_2$ are compact matrix quantum groups with unitary fundamental representations of the same size. The following are equivalent.