kazhdan_property
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| ====== Kazhdan property (T) ====== | ====== Kazhdan property (T) ====== | ||
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| + | This property was originally formulated by David Kazhdan for locally compact groups (see [[wp>Kazhdan's_property_(T)|Kazhdan's property (T)]]). This article is about its generalization to the quantum group setting. | ||
| ===== Definition ===== | ===== Definition ===== | ||
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| The definition of Kazhdan property (T) for discrete quantum groups was formulated in [(ref:Fim10)]. | The definition of Kazhdan property (T) for discrete quantum groups was formulated in [(ref:Fim10)]. | ||
| - | Let $G$ be a compact quantum group, $X\subset\Irr(G)$, and $\pi\colon C(G)\to B(H)$ a $*$-representation on a Hilbert space $H$. For $x\in\Irr(G)$, denote $u^x$ its representative acting on $H_x$ and put $U^x:= (\id\otimes\pi)u^x\in B(H_x)\otimes B(H)$. | + | Let $G$ be a compact quantum group, $X\subset\Irr(G)$, and $\pi\colon C(G)\to B(H)$ a $*$-representation on a Hilbert space $H$. For $x\in\Irr(G)$, denote $u^x$ its representative acting on $H_x$ and put $U^x:=(\id\otimes\pi)u^x\in B(H_x)\otimes B(H)$. |
| For $\epsilon>0$ we say that the unit vector $v\in H$ is $(X,\epsilon)$**-invariant** if for all $x\in X$ and all non-zero $\eta\in H_x$ we have | For $\epsilon>0$ we say that the unit vector $v\in H$ is $(X,\epsilon)$**-invariant** if for all $x\in X$ and all non-zero $\eta\in H_x$ we have | ||
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| ===== Results ===== | ===== Results ===== | ||
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| + | ==== Stability results ==== | ||
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| + | * Suppose $H\subset G$. If $\hat G$ has (T), then also $\hat H$ has (T). [(ref:Fim10)] | ||
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| + | ==== Examples ==== | ||
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| + | The following quantum groups have the property (T). | ||
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| + | The following quantum groups do not have the property (T). | ||
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| + | * $\hat O_N^+$ for $N\ge 2$ (since $\Z_2^{*N}$ does not have (T)) [(ref:Fim10)] | ||
| + | * $\hat U_N^+$ for $N\ge 2$ (since the free group $\mathbb{F}_N$ does not have (T)) [(ref:Fim10)] | ||
| + | * $\hat S_N^+$ for $N\ge 2$ [(ref:Fim10)] | ||
| + | * Any free group | ||
| + | * Any infinite Abelian discrete quantum group (since it is [[amenability|amenable]]) | ||
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| ===== Relation with other properties ===== | ===== Relation with other properties ===== | ||
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| * $\Gamma$ is finitely generated [(ref:Fim10)] | * $\Gamma$ is finitely generated [(ref:Fim10)] | ||
| - | * $\Gamma$ is of Kac type [(ref:Fim10)] | + | * $\Gamma$ is [[Unimodularity|unimodular]] [(ref:Fim10)] |
| - | + | ||
| - | Discrete quantum group $\Gamma=\hat G$ has (T) if | + | |
| + | Discrete quantum group $\Gamma=\hat G$ has (T) if... | ||
| + | Discrete quantum group $\Gamma$ has the [[haagerup_property|Haagerup property]] and property (T) if and only if $\Gamma$ is [[compact_quantum_group#Finite quantum groups|finite]] [(ref:DFSW13)]. | ||
| ===== References ===== | ===== References ===== | ||
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| author : Angshuman Bhattacharya, Michael Brannan, Alexandru Chirvasitu, Shuzhou Wang | author : Angshuman Bhattacharya, Michael Brannan, Alexandru Chirvasitu, Shuzhou Wang | ||
| url : https://arxiv.org/abs/1712.08682 | url : https://arxiv.org/abs/1712.08682 | ||
| + | )] | ||
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| + | [(ref:DFSW13>> | ||
| + | author : Matthew Daws, Pierre Fima, Adam Skalski, Stuart White | ||
| + | title : The Haagerup property for locally compact quantum groups | ||
| + | journal : Journal für die reine und angewandte Mathematik | ||
| + | volume : 2016 | ||
| + | number : 711 | ||
| + | pages : 189–229 | ||
| + | year : 2013 | ||
| + | doi : 10.1515/crelle-2013-0113 | ||
| + | url : https://doi.org/10.1515/crelle-2013-0113 | ||
| )] | )] | ||
kazhdan_property.1567774373.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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