✎ intersection_of_quantum_groups [CMQG Wiki]

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====== 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.
  - $G=H_1\cap H_2$,
  - $I_G=I_{H_1}+I_{H_2}$,
  - $\FundRep_G$ is generated by $\FundRep_{H_1}\cup\FundRep_{H_2}$.

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

  * Teodor Banica, //Free quantum groups and related topics// [[https://banica.u-cergy.fr/a3.pdf|Online notes]]
  * Daniel Gromada, //Compact matrix quantum groups and their representation categories//, PhD thesis, Saarland University, 2020.
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