Categories of the higher hyperoctahedral series

The categories of the higher hyperoctahedral series are a family of Banica-Speicher categories of partitions, indexed by $\{3,4,\ldots,\infty\}$ introduced by Raum and Weber in [RaWe15].

Definition

Let $\Z_2^{\ast\infty}$ be the free product group of $\aleph_0$ many copies of the cyclic group $\Z_2\equiv \Z/2\Z$ of order $2$ and for every $k\in\N$ let $a_k$ be the image of $1\equiv 1+2\Z$ under the embedding of $\Z_2$ as the $k$ -th free factor in $\Z_2^{\ast\infty}$ . Then, $\{a_k\,\vert\,k\in\N\}$ generates $\Z_2^{\ast\infty}$ and any group endomorphism of $\Z_2^{\ast\infty}$ is uniquely determined by its restriction to $\{a_k\,\vert\, k\in\N\}$ .

The strong symmetric semigroup $\mathrm{sS}_\infty$ is the subsemigroup of the semigroup $\mathrm{End}(\Z_2^{\ast\infty})$ of group endomorphisms of $\Z_2^{\ast\infty}$ generated by the endomorphisms defined by $a_k\mapsto a_{i(k)}$ for all mappings $i:\N\to\N$ such that $|\N\backslash i(\N)|<\infty$ . In other words, $\mathrm{sS}_\infty$ is given by all identifications of finitely many letters in words in an alphabet of countably many letters from $\Z_2$ .

A set $A\subseteq \Z_2^\infty$ is said to be $\mathrm{sS}_\infty$ -invariant if $\varphi(w)\in A$ for all $w\in A$ and $\varphi\in \mathrm{sS}_\infty$ .

For every $s\in \N\cup \{\infty\}$ with $3\leq s$ by the category of the higher hyperoctahedral series with parameter $s$ one denotes the subcategory of the category of all partitions $\Pscr$ whose morphism class is given by the set of all (uncolored) partitions $p\in\Pscr$ with the property that the word representation of $p$ , if interpreted as a product in $\Z_2^{\ast\infty}$ (generally the word representation is not a fully reduced word), is an element of,

if $s=\infty$ , the trivial subgroup of $\Z^{\ast\infty}_2$ ,
if $s<\infty$ , the smallest $\mathrm{sS}_\infty$ -invariant normal subgroup of $\Z^{\ast\infty}_2$ which contains $(a_1a_2)^s$ .

Had one allowed $s=2$ in the above definition one would have obtained the category of partitions with blocks of even size. Permitting $s=1$ yields the category of partitions of even size.

The categories of the higher hyperoctahedral series are special cases of the class of group-theoretical hyperoctahedral categories of partitions.

Canonical Generator

The category of the higher hyperoctahedral series with parameter $s\in N\cup\{\infty\}$ with $3\leq s$ is the smallest subcategory of $\Pscr$ containing,

if $s=\infty$ , the partition $\Paabaab$ ,
if $s<\infty$ , the partition $h_s$ [RaWe14], the partition whose word representation is given by $(\mathsf{ab})^s$ .

Associated easy quantum group

Via Tannaka-Krein duality for compact quantum groups, the categories of the higher hyperoctahedral series induce the corepresentation categories of the quantum groups belonging to, as the name suggests, the higher hyperoctahedral series $(H^{[s]}_N)_{N\in \N,s\in \{3,\ldots,\infty\}}$ .

References

[RaWe15] Raum, Sven and Weber, Moritz, 2015. Easy quantum groups and quantum subgroups of a semi-direct product quantum group. Journal of Noncommutative Geometry, 9, pp.1261–1293.

[RaWe14] Raum, Sven and Weber, Moritz, 2014. The combinatorics of an algebraic class of easy quantum groups. Infinite Dimensional Analysis, Quantum Probability and related topics, 17.

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Categories of the higher hyperoctahedral series

Definition

Canonical Generator

Associated easy quantum group

References

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