====== Easy unitary quantum group ======

**Easy unitary quantum groups** are a particular class of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Freslon and Weber in [(:ref:FreWeb16)], generalizing the definition of [[easy quantum group|easy orthogonal quantum groups]] given by Banica and Speicher in [(:ref:BanSp09)]. Every easy unitary quantum group is by definition a compact quantum subgroup of a [[free unitary quantum group]]. 

===== Definition =====

Informally, a [[compact matrix quantum group]] is called **easy unitary** if it is a compact quantum subgroup of the corresponding free unitary quantum group and if its corepresentation category is generated by a category of partitions. Formally, for every $N\in \N$, any compact $N\times N$-matrix quantum group $G\cong (C(G),u)$ is called an **easy unitary quantum group** if the [[corepresentation category]] $\FundRep(G)$ of $G$ has as objects the set $\bigcup_{k\in\N\cup \{0\}}\{\circ,\bullet\}^{\times k}$ of tuples of arbitrary lengths with two distinct possible entries $\circ$ and $\bullet$ and if there exists some [[categories of two-colored partitions|category of two-colored partitions]] $\Cscr\subseteq \Pscr^{\circ\bullet}$ such that for all $k,\ell\in\N\cup \{0\}$ and all $c^1,\ldots,c^k,c_1,\ldots,c_\ell\in \{\circ,\bullet\}$ the morphism set $(c^1,\ldots,c^k)\to(c_1,\ldots,c_\ell)$ of $\FundRep(G)$ is given by 
$$\mathrm{Hom}((c^1,\ldots,c^k),(c_1,\ldots,c_\ell))=\spanlin_\C(\{ T_p\,\vert\, p\in \Cscr(c^1,\ldots,c^k,c_1,\ldots,c_\ell)\}),$$
where for all $p\in \Cscr(k,l)$ the linear map $T_p:\,(\C^N)^{\otimes k}\to (\C^N)^{\otimes \ell}$ satisfies for all $j_1,\ldots,j_k\in N$,
$$T_p(e_{j_1}\otimes\cdots\otimes e_{j_k})=\sum_{i_1,\dots,i_\ell=1}^N\delta_p(j_1,\ldots,j_k,i_1,\ldots,i_\ell)(e_{i_1}\otimes\cdots\otimes e_{i_\ell}),$$
where $(e_i)_{i=1}^N$ is the standard basis of $\C^N$ and where for all $i_1,\ldots,i_\ell\in N$ the symbol $\delta_p(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ is $1$ if the kernel, i.e., the induced partition with $k$ upper and $\ell$ lower points, of $(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ refines $p$ and is $0$ otherwise.


===== References =====

[( :ref:FreWeb16 >>
author:  Freslon, Amaury and Weber, Moritz
title:   On the representation theory of partition (easy) quantum groups
year:    2016
journal: Journal für die reine und angewandte Mathematik [Crelle's Journal]
volume:  2016
issue:   720
url:     https://doi.org/10.1515/crelle-2014-0049
archivePrefix: arXiv
eprint   :1308.6390v2
)]

[( :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
)]