unitary_easy_quantum_group
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unitary_easy_quantum_group [2020/02/14 07:39] amang created |
unitary_easy_quantum_group [2021/11/23 11:56] (current) |
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| ===== Definition ===== | ===== Definition ===== | ||
| - | Informally, a [[compact matrix quantum group]] is called **easy unitary** if it is a compact quantum subgroup of the corresponding free unitary quantum group and if its corepresentation category is generated by a category of partitions. Formally, for every $N\in \N$, any compact $N\times N$-matrix quantum group $G\cong (C(G),u)$ is called an **easy unitary quantum group** if the [[corepresentation category]] $\FundRep(G)$ of $G$ has as objects the set $\bigcup_{k\in\N\cup \{0\}}\{\circ,\bullet\}^{\times k}$ of tuples of arbitrary lengths with two distinct possible entries $\circ$ and $\bullet$ and if there exists some [[category of two-colored partitions]] $\Cscr\subseteq \Pscr^{\circ\bullet}$ such that for all $k,\ell\in\N\cup \{0\}$ and all $c^1,\ldots,c^k,c_1,\ldots,c_\ell\in \{\circ,\bullet\}$ the morphism set $(c^1,\ldots,c^k)\to(c_1,\ldots,c_\ell)$ of $\FundRep(G)$ is given by | + | Informally, a [[compact matrix quantum group]] is called **easy unitary** if it is a compact quantum subgroup of the corresponding free unitary quantum group and if its corepresentation category is generated by a category of partitions. Formally, for every $N\in \N$, any compact $N\times N$-matrix quantum group $G\cong (C(G),u)$ is called an **easy unitary quantum group** if the [[corepresentation category]] $\FundRep(G)$ of $G$ has as objects the set $\bigcup_{k\in\N\cup \{0\}}\{\circ,\bullet\}^{\times k}$ of tuples of arbitrary lengths with two distinct possible entries $\circ$ and $\bullet$ and if there exists some [[categories of two-colored partitions|category of two-colored partitions]] $\Cscr\subseteq \Pscr^{\circ\bullet}$ such that for all $k,\ell\in\N\cup \{0\}$ and all $c^1,\ldots,c^k,c_1,\ldots,c_\ell\in \{\circ,\bullet\}$ the morphism set $(c^1,\ldots,c^k)\to(c_1,\ldots,c_\ell)$ of $\FundRep(G)$ is given by |
| $$\mathrm{Hom}((c^1,\ldots,c^k),(c_1,\ldots,c_\ell))=\spanlin_\C(\{ T_p\,\vert\, p\in \Cscr(c^1,\ldots,c^k,c_1,\ldots,c_\ell)\}),$$ | $$\mathrm{Hom}((c^1,\ldots,c^k),(c_1,\ldots,c_\ell))=\spanlin_\C(\{ T_p\,\vert\, p\in \Cscr(c^1,\ldots,c^k,c_1,\ldots,c_\ell)\}),$$ | ||
| where for all $p\in \Cscr(k,l)$ the linear map $T_p:\,(\C^N)^{\otimes k}\to (\C^N)^{\otimes \ell}$ satisfies for all $j_1,\ldots,j_k\in N$, | where for all $p\in \Cscr(k,l)$ the linear map $T_p:\,(\C^N)^{\otimes k}\to (\C^N)^{\otimes \ell}$ satisfies for all $j_1,\ldots,j_k\in N$, | ||
unitary_easy_quantum_group.1581665952.txt.gz ยท Last modified: 2021/11/23 11:56 (external edit)
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