intersection_of_quantum_groups
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intersection_of_quantum_groups [2020/06/09 11:51] d.gromada |
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| ====== Intersection of quantum groups ====== | ====== 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. | + | 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 ===== | ===== 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 | 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 $k=1,2$. | + | - $G\subset H_1,H_2$, 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~$G=\tilde G$. | + | - 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 ===== | ===== Uniqueness, existence, and characterization ===== | ||
intersection_of_quantum_groups.1591703474.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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