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Easy unitary quantum group

Easy unitary quantum groups are a particular class of compact matrix quantum groups introduced by Freslon and Weber in [FreWeb16], generalizing the definition of easy orthogonal quantum groups given by Banica and Speicher in [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 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

[FreWeb16] Freslon, Amaury and Weber, Moritz, 2016. On the representation theory of partition (easy) quantum groups. Journal für die reine und angewandte Mathematik [Crelle's Journal], 2016.
[BanSp09] Banica, Teodor and Speicher, Roland, 2009. Liberation of orthogonal Lie groups. Advances in Mathematics, 222, pp.1461–150.