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| - | ====== CMQG Wiki ====== | ||
| - | This website was created to summarize all known properties of [[compact_quantum_group|compact quantum groups]], in particular [[Compact matrix quantum group|compact matrix quantum groups]] and associated [[Linear category of partitions|partition categories]]. | ||
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| - | All researchers working on compact quantum groups and related fields are invited to create and edit pages to push this project closer to its goal being the summary of all the current knowledge about compact matrix quantum groups. Before creating or editing pages, you are advised to read the [[Instructions for authors|instructions for authors]]. | ||
| - | |||
| - | ===== Content ===== | ||
| - | |||
| - | In this section, all the central content of this wiki is listed. | ||
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| - | ==== Compact matrix quantum groups ==== | ||
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| - | * [[Easy quantum group|Orthogonal easy quantum groups]] | ||
| - | * [[Free orthogonal easy quantum groups]] | ||
| - | * [[free_orthogonal_quantum_group|Free orthogonal quantum group]] ($O^+$) | ||
| - | * [[Free bistochastic quantum group]] ($B^+$) | ||
| - | * [[Free modified bistochastic quantum group]] ($B^{\prime +}$) | ||
| - | * [[Freely modified bistochastic quantum group]] ($B^{\# +}$) | ||
| - | * [[Free symmetric quantum group]] ($S^+$) | ||
| - | * [[Free modified symmetric quantum group]] ($S^{\prime +}$) | ||
| - | * [[Free hyperoctahedral quantum group]] ($H^+$) | ||
| - | * [[Half-liberated orthogonal easy quantum groups]] | ||
| - | * [[Half-liberated orthogonal quantum group]] ($O^\ast$) | ||
| - | * [[Half-liberated bistochastic quantum group]] ($B^{\# \ast}$) | ||
| - | * [[Half-liberated hyperoctahedral quantum group]] ($H^\ast$) | ||
| - | * [[Hyperoctahedral series]] ($H^{(\,\cdot\,)}$) | ||
| - | * [[Group-theoretical hyperoctahedral easy orthogonal quantum groups]] | ||
| - | * [[Higher hyperoctahedral series]] ($H^{[\,\cdot\,]}$) | ||
| - | * [[Non-group-theoretical hyperoctahedral easy orthogonal quantum groups]] ($H^{[\pi_{(\,\cdot\,)}]}$) | ||
| - | * [[Square commuting hyperoctahedral quantum group]] ($H^\diamond$) | ||
| - | * [[Classical orthogonal matrix groups]] | ||
| - | * [[Orthogonal group]] ($O$) | ||
| - | * [[Bistochastic group]] ($B$) | ||
| - | * [[Modified bistochastic group]] ($B'$) | ||
| - | * [[Symmetric group]] ($S$) | ||
| - | * [[Modified symmetric group]] ($S'$) | ||
| - | * [[Hyperoctahedral group]] ($H$) | ||
| - | * [[Unitary easy quantum group|Unitary easy quantum groups]] | ||
| - | * [[Free unitary easy quantum groups]] | ||
| - | * [[Free unitary quantum group]] ($U^+$) | ||
| - | * [[Half-liberated unitary quantum groups]] | ||
| - | * [[Classical unitary matrix groups]] | ||
| - | * [[Unitary group]] ($U$) | ||
| - | * [[Homogeneous compact matrix quantum group|Homogeneous compact matrix quantum groups]] | ||
| - | * [[Group-theoretical quantum group|Group-theoretical quantum groups]] | ||
| - | * [[Graph-theoretical quantum group|Graph-theoretical quantum groups]] | ||
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| - | ==== Partition categories ==== | ||
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| - | * [[category_of_partitions|Banica–Speicher easy categories (of one-colored partitions)]] | ||
| - | * [[Category of all partitions]] ($\mathrm{S}$) | ||
| - | * [[Category of partitions of even size]] ($\mathrm{S}'$) | ||
| - | * [[Category of all non-crossing partitions]] ($\mathrm{S}^+$) | ||
| - | * [[Category of non-crossing partitions of even size]] ($\mathrm{S}^{\prime+}$) | ||
| - | * [[brauer category|Brauer category (of all pair partitions)]] ($\mathrm{O}$) | ||
| - | * [[Category of pair partitions with even distances between legs]] ($\mathrm{O}^\ast$) | ||
| - | * [[temperley_lieb_category|Temperley–Lieb category (of all non-crossing pair partitions)]] ($\mathrm{O}^+$) | ||
| - | * [[Category of all partitions with small blocks]] ($\mathrm{B}$) | ||
| - | * [[Category of partitions of even size with small blocks]] ($\mathrm{B}'$) | ||
| - | * [[Category of partitions of even size with small blocks and even distances between legs]] ($\mathrm{B}^{\#\ast}$) | ||
| - | * [[Category of non-crossing partitions with small blocks]] ($\mathrm{B}^+$) | ||
| - | * [[Category of non-crossing partitions of even size with small blocks]] ($\mathrm{B}^{\prime+}$) | ||
| - | * [[Category of non-crossing partitions of even size with small blocks and even distances between legs]] ($\mathrm{B}^{\#+}$) | ||
| - | * [[Category of partitions with blocks of even size]] ($\mathrm{H}$) | ||
| - | * [[Category of partitions with blocks of even size and even distances between legs]] ($\mathrm{H}^\ast$) | ||
| - | * [[Category of non-crossing partitions with blocks of even size]] ($\mathrm{H}^+$) | ||
| - | * [[Group-theoretical hyperoctahedral categories of partitions]] | ||
| - | * [[Categories of the hyperoctahedral series]] ($\mathrm{H}^{(\,\cdot\,)}$) | ||
| - | * [[Categories of the higher hyperoctahedral series]] ($\mathrm{H}^{[\,\cdot\,]}$) | ||
| - | * [[Non-group-theoretical hyperoctahedral categories of partitions]] ($\mathrm{H}^{[\pi_{(\,\cdot\,)}]}$) | ||
| - | * [[Categories of two-colored partitions]] | ||
| - | * [[Category of non-crossing two-colored pair partitions with neutral blocks]] | ||
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| - | ==== Properties of CMQGs and associated algebras ==== | ||
| - | |||
| - | * [[unimodularity|Unimodularity]] | ||
| - | * (Co)homology of quantum groups | ||
| - | * Approximation properties ([[approximation properties summary|summary]]) | ||
| - | * [[Connes embedding property]] | ||
| - | * [[Kirchberg property|Kirchberg factorization property (F)]] | ||
| - | * [[Residual finiteness|Residual finiteness]] | ||
| - | * [[Kazhdan property|Kazhdan property (T)]] | ||
| - | * [[Amenability]] | ||
| - | * [[Weak amenability]] | ||
| - | * [[Haagerup property]] | ||
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| - | ==== Quantum group constructions ==== | ||
| - | |||
| - | * Direct products (tensor product, free product) | ||
| - | * Glued products | ||
| - | * Semidirect products | ||
| - | * Wreath products | ||
| - | * Topological generation | ||
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| - | ===== Literature ===== | ||
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| - | Here, we recommend some monographies and surveys related to the topic. | ||
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| - | * Sergey Neshveyev, Lars Tuset, //Compact Quantum Groups and Their Representation Categories//, Société Mathématique de France, 2013. [[https://www.sciencesmaths-paris.fr/upload/Contenu/Fichiers_chaire/livre%20Sergey%20Neyshveyev.pdf|Available online]] | ||
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