free_bistochastic_quantum_group
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free_bistochastic_quantum_group [2020/01/03 21:45] amang [Basic Properties] |
free_bistochastic_quantum_group [2021/11/23 11:56] (current) |
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| ===== Definition ===== | ===== Definition ===== | ||
| Given $N\in \N$, the **free bistochastic quantum group** $B_N^+$ is the [[compact matrix quantum group]] $(C(B_N^+),u)$ where $u=(u_{i,j})_{i,j=1}^N$ organizes the generators $\{u_{i,j}\}_{i,j=1}^N$ of the (unital) [[wp>Universal_C*-algebra|universal C*-algebra]] | Given $N\in \N$, the **free bistochastic quantum group** $B_N^+$ is the [[compact matrix quantum group]] $(C(B_N^+),u)$ where $u=(u_{i,j})_{i,j=1}^N$ organizes the generators $\{u_{i,j}\}_{i,j=1}^N$ of the (unital) [[wp>Universal_C*-algebra|universal C*-algebra]] | ||
| - | $$C(B_N^+)\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u,\, uu^t=u^tu=I_N\otimes 1, \, {\textstyle\sum_{k=1}^N} u_{i,k}={\textstyle\sum_{l=1}^N} u_{l,j}=1\big\rangle,$$ | + | $$C(B_N^+)\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u,\, uu^t=u^tu=I_N\otimes 1, \, \forall_{i,j=1}^N: {\textstyle\sum_{k=1}^N} u_{i,k}={\textstyle\sum_{l=1}^N} u_{l,j}=1\big\rangle,$$ |
| where $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ is the complex conjugate of $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose, where $I_N$ is the identity $N\!\times \!N$-matrix and where $1$ is the unit of the universal $C^\ast$-algebra. | where $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ is the complex conjugate of $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose, where $I_N$ is the identity $N\!\times \!N$-matrix and where $1$ is the unit of the universal $C^\ast$-algebra. | ||
| - | The definition can also be expressed by saying that the fundamental corpresentation matrix $u$ of $S_N^+$ is **bistochastic** (or **doubly stochastic**), which is to say that $u$ is orthogonal and each of its rows and columns sums to $1$. | + | The definition can also be expressed by saying that the fundamental corpresentation matrix $u$ of $B_N^+$ is **bistochastic** (or **doubly stochastic**), which is to say that $u$ is orthogonal and each of its rows and columns sums to $1$. |
| ===== Basic Properties ===== | ===== Basic Properties ===== | ||
free_bistochastic_quantum_group.1578087905.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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