compact_matrix_quantum_group
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compact_matrix_quantum_group [2020/01/02 11:59] amang [Original version by Woronowicz] |
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| ===== Definition ===== | ===== Definition ===== | ||
| - | The term **compact matrix quantum group** only makes sense with reference to a certain dimension $N\in \N$. Two definitions appear in the literature, the orginal one by Woronowicz from [(:ref:Wor87)] and an equivalent alternative formulation. | + | The term **compact matrix quantum group** (or **CMQG** for short) only makes sense with reference to a certain dimension $N\in \N$. Two definitions appear in the literature, the orginal one by Woronowicz from [(:ref:Wor87)] and an equivalent alternative formulation. |
| Both define a compact matrix quantum group $G$ as a pair $(A,u)$ of a $C^\ast$-algebra $A$ and a matrix $u$ with entries in $A$. In keeping with the general paradigm of non-commutative topology, $A$ is usually referred to as the //algebra of continuous functions// $C(G)$ //on// $G$ even if $A$ is non-commutative. | Both define a compact matrix quantum group $G$ as a pair $(A,u)$ of a $C^\ast$-algebra $A$ and a matrix $u$ with entries in $A$. In keeping with the general paradigm of non-commutative topology, $A$ is usually referred to as the //algebra of continuous functions// $C(G)$ //on// $G$ even if $A$ is non-commutative. | ||
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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)//. |
| - | ==== 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 | ||
| * $A$ is a unital $C^\ast$-algebra, | * $A$ is a unital $C^\ast$-algebra, | ||
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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 ===== | ||
| Line 51: | Line 51: | ||
| )] | )] | ||
| + | [( :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 | ||
| + | )] | ||
compact_matrix_quantum_group.1577966397.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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