compact_matrix_quantum_group
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compact_matrix_quantum_group [2020/01/02 12:31] amang [References] |
compact_matrix_quantum_group [2021/11/23 11:56] (current) |
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| * there exists a linear antimultiplicative mapping $S:\mathscr{A}\to \mathscr{A}$ with $S(S(a^\ast)^\ast)=a$ for all $a\in \mathscr{A}$ and with $\sum_{k=1}^N S(u_{i,k})u_{k,j}=\delta_{i,j}I$ and $\sum_{k=1}^N u_{i,k}S(u_{k,j})=\delta_{i,j}I$ for all $i,j=1,\ldots,N$, where $I$ is the unit of $A$. | * there exists a linear antimultiplicative mapping $S:\mathscr{A}\to \mathscr{A}$ with $S(S(a^\ast)^\ast)=a$ for all $a\in \mathscr{A}$ and with $\sum_{k=1}^N S(u_{i,k})u_{k,j}=\delta_{i,j}I$ and $\sum_{k=1}^N u_{i,k}S(u_{k,j})=\delta_{i,j}I$ for all $i,j=1,\ldots,N$, where $I$ is the unit of $A$. | ||
| - | If so, then $\Delta$ and $S$ are uniquely determined. They are called the //comultiplication// and the //antipode// (or //coinverse//), respectively. And $u$ is called the //fundamental corepresentation (matrix)//. | + | If so, then $\Delta$ and $S$ are uniquely determined by [(:ref:Wor87)]. They are called the //comultiplication// and the //antipode// (or //coinverse//), respectively. And $u$ is called the //fundamental corepresentation (matrix)//. |
| ==== Equivalent alternative version ==== | ==== Equivalent alternative version ==== | ||
| A **compact** $N\!\times\! N$**-matrix quantum group** is a pair $(A,u)$ such that | A **compact** $N\!\times\! N$**-matrix quantum group** is a pair $(A,u)$ such that | ||
compact_matrix_quantum_group.1577968283.txt.gz ยท Last modified: 2021/11/23 11:56 (external edit)
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