category_of_partitions_with_blocks_of_even_size_and_even_distances_between_legs
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| - | ====== Category of partitions with blocks of even size and parity-balanced legs ====== | ||
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| - | The **category of partitions with blocks of even size and parity-balanced legs** is a [[category_of_partitions|Banica-Speicher category of partitions]] inducing the corepresentation category of the [[half-liberated hyperoctahedral quantum group|half-liberated hyperoctahedral quantum groups]]. | ||
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| - | ===== Definition ===== | ||
| - | By the **category of partitions with blocks of even size and even distances between legs** one denotes the subcategory of the [[category of all partitions]] $\Pscr$ whose morphism class is the //set of partitions with blocks of even size and even distances between legs//. It was introduced by Banica, Curran and Speicher in [(:ref:BanCuSp10)]. | ||
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| - | A partition $p\in \Pscr$ belongs to this set if the following conditions are met: | ||
| - | * $p$ has **blocks of even size**, i.e., every block of $p$ has an even number of legs. | ||
| - | * $p$ has **parity-balanced legs**, i.e., for any block of $p$ when counting from an arbitrary point $i$ of the partition, the number of legs of $B$ at even distances from $i$ is equal to the number of legs of $B$ at odd distances from $i$. | ||
| - | * The name **set of partitions with blocks of even size and parity-balanced legs** is to be taken literally. | ||
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| - | A partition with blocks of even size is in particular of even size itself. | ||
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| - | ===== Canonical Generator ===== | ||
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| - | The category of partitions with blocks of even size and parity-balanced legs is the subcategory of $\Pscr$ generated by the set $\{\Pabcabc,\fourpart\}$ of partitions. | ||
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| - | ===== Associated easy quantum group ===== | ||
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| - | Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, the category of partitions with blocks of even size and parity-balanced legs corresponds to the family $(H^{\ast}_N)_{N\in \N}$ of [[half-liberated hyperoctahedral quantum group|half-liberated hyperoctahedral quantum groups]]. | ||
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| - | ===== References ===== | ||
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| - | [( :ref:BanSp09 >> | ||
| - | author: Banica, Teodor and Speicher, Roland | ||
| - | title: Liberation of orthogonal Lie groups | ||
| - | year: 2009 | ||
| - | journal: Advances in Mathematics | ||
| - | volume: 222 | ||
| - | issue: 4 | ||
| - | pages: 1461--150 | ||
| - | url: https://doi.org/10.1016/j.aim.2009.06.009 | ||
| - | archivePrefix: arXiv | ||
| - | eprint :0808.2628 | ||
| - | )] | ||
| - | |||
| - | |||
| - | [( :ref:Web12 >> | ||
| - | author: Weber, Moritz | ||
| - | title: On the classification of easy quantum groups | ||
| - | year: 2013 | ||
| - | journal: Advances in Mathematics | ||
| - | volume: 245 | ||
| - | pages: 500--533 | ||
| - | url: https://doi.org/10.1016/j.aim.2013.06.019 | ||
| - | archivePrefix: arXiv | ||
| - | eprint :1201.4723v2 | ||
| - | )] | ||
| - | |||
| - | [( :ref:BanCuSp10 >> | ||
| - | author: Banica, Teodor and Curran, Stephen and Speicher, Roland | ||
| - | title: Classification results for easy quantum groups | ||
| - | year: 2010 | ||
| - | journal: Pacific Journal of Mathematics | ||
| - | volume: 247 | ||
| - | issue: 1 | ||
| - | pages: 1-26 | ||
| - | url: https://doi.org/10.2140/pjm.2010.247.1 | ||
| - | archivePrefix: arXiv | ||
| - | eprint :0906.3890v1 | ||
| - | )] | ||
category_of_partitions_with_blocks_of_even_size_and_even_distances_between_legs.txt ยท Last modified: 2021/11/23 11:56 (external edit)
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