Glued product of compact matrix quantum groups

Glued products form a less standard product construction, which is defined only for matrix quantum groups. It was formally defined in [TW17] to interpret some coloured categories of partitions in terms of compact matrix quantum groups.

Definition

Glued tensor product

Let $G=(C(G),u)$ and $H=(C(H),v)$ be compact matrix quantum groups. We define the glued tensor product

$G\tiltimes H:=(C(G\tiltimes H),u\otimes v),$

where $C(G\tiltimes H)$ is the C*-subalgebra of $C(G)\otimes_{\rm max} C(H)$ generated by $u_{ij}v_{kl}$ – the elements of the tensor product $u\otimes v$ .

Glued free product

Similarly, we define the glued free product

$G\tilstar H:=(C(G\tilstar H),u\otimes v),$

where $C(G\tilstar H)$ is the C*-subalgebra of $C(G)*_\C C(H)$ generated by $u_{ij}v_{kl}$ .

Main application: Complexifications

Given a compact matrix quantum group $G$ , we call $G\tiltimes\hat\Z$ the tensor complexification of $G$ , $G\tiltimes\hat\Z_k$ is the tensor k-complexification, $G\tilstar\hat\Z$ is the free complexification and $G\tilstar\hat\Z_k$ is the free k-complexification of $G$ . The free complexification was studied already by Banica in [Ban99], [Ban08].

Remark on the distinction with ordinary products

The glued versions of the tensor and free products $G\tiltimes H$ and $G\tilstar H$ are by definition quotient quantum groups of the standard constructions $G\times H$ and $G\hatstar H$ . Often it happens that the elements $u_{ij}v_{kl}$ already generate the whole C*-algebra, so actually $G\tiltimes H\simeq G\times H$ or $G\tilstar H\simeq H\hatstar H$ . Even in this case, however, we should not put the equality sign here. Although the quantum groups can have the same underlying C*-algebra and hence be isomorphic, they are never identical as compact matrix quantum groups since their fundamental representations are always different – $u\oplus v$ in the standard case and $u\otimes v$ in the glued case.

Examples

Tensor complexification for classical groups

Let us have a look on how the definition of glued tensor product looks like for groups. Let $G$ and $H$ be two matrix groups, then we have

$G\tiltimes H=\{A\otimes B\mid A\in G,B\in H\},$

where $\otimes$ denotes the Kronecker product.

As a concrete example, consider the symmetric group $S_N$ represented by the permutation matrices and consider the cyclic group of order two $\hat\Z_2=\Z_2$ represented by a single complex number $\pm 1$ . Then $S_N\tiltimes\Z_2$ consists of $N\times N$ permutation matrices multiplied by a global sign. Thus, $S_N\tiltimes\Z_2$ is actually isomorphic to $S_N\times\Z_2$ . Nevertheless, by $S_N\times\Z_2$ we mean a different matrix realization. The ordinary product $S_N\times\Z_2$ consists of $(N+1)\times(N+1)$ matrices with block diagonal structure, where one block is formed by an $N\times N$ permutation matrix and the second block is the single number $\pm 1$ .

In general, take any cyclic group $\hat\Z_k=\Z_k$ with $k\in\N$ represented by the $k$ -th roots of unity. Then for any matrix group $G$ , we have

$G\tiltimes\Z_k=\{{\rm e}^{2\pi ij/k}A\mid j=0,\dots,k-1;\;A\in G\}.$

We can do the same for the whole unit disk $\hat\Z=\T\subset\C$

$G\tiltimes\T=\{zA\mid z\in\T;\;A\in G\}.$

Unitary quantum groups as free complexifications

It holds that $U_N^+=O_N^+\hatstar\hat\Z$ [Ban97]

References

[TW17] Pierre Tarrago and Moritz Weber, 2017. Unitary Easy Quantum Groups: The Free Case and the Group Case. International Mathematics Research Notices, 2017(18), pp.5710–5750.

[Ban99] Teodor Banica, 1999. Representations of compact quantum groups and subfactors. Journal für die reine und angewandte Mathematik, 509, pp.167–198.

[Ban08] Teodor Banica, 2008. A Note on Free Quantum Groups. Annales Mathématiques Blaise Pascal, 15(2), pp.135–146.

[Ban97] Teodor Banica, 1997. Le Groupe Quantique Compact Libre U(n). Communications in Mathematical Physics, 190(1), pp.143–172.

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