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compact_matrix_quantum_group [2020/01/02 12:17] amang [Alternative version] |
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 | ||
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Here also, of course, $\Delta$ and //u// are referred to as the //comultiplication// and //fundamental corepresentation (matrix)//, respectively. | Here also, of course, $\Delta$ and //u// are referred to as the //comultiplication// and //fundamental corepresentation (matrix)//, respectively. | ||
===== Compact matrix quantum groups as compact quantum groups ===== | ===== Compact matrix quantum groups as compact quantum groups ===== | ||
- | Given a compact matrix quantum group $(A,u)$ with comultiplication $\Delta$ the pair $(A,\Delta)$ is a [[compact quantum group]]. | + | Given a compact matrix quantum group $(A,u)$ with comultiplication $\Delta$ the pair $(A,\Delta)$ is a [[compact quantum group]] by [(:ref:Wor98)]. |
===== Comparing compact matrix quantum groups ===== | ===== Comparing compact matrix quantum groups ===== | ||
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)] | )] | ||
+ | [( :ref:Wor98 >> | ||
+ | author :Stanisław L. Woronowicz | ||
+ | title :Compact quantum groups | ||
+ | booktitle: Quantum symmetries/Symétries quantiques. Proceedings of the Les Houches summer school, Session LXIV, Les Houches, France, August 1st -- September 8th, 1995 | ||
+ | editor :Connes, Alain | ||
+ | pages :845--884 | ||
+ | year :1998 | ||
+ | )] |