glued_products
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| - | ====== Glued product of compact matrix quantum groups ====== | ||
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| - | Glued products form a less standard product construction, which is defined only for matrix quantum groups. It was formally defined in [(ref:TW17)] to interpret some coloured categories of partitions in terms of compact matrix quantum groups. | ||
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| - | ===== Definition ===== | ||
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| - | ==== Glued tensor product ==== | ||
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| - | 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 ==== | ||
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| - | 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 $u_{ij}v_{kl}$. | ||
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| - | ==== Main application: Complexifications ==== | ||
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| - | 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 [(ref:Ban99)], [(ref:Ban08)]. | ||
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| - | ==== Remark on the distinction with ordinary products ==== | ||
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| - | 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. | ||
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| - | ===== Examples ===== | ||
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| - | ==== Tensor complexification for classical groups ==== | ||
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| - | 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. | ||
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| - | 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 $\pm 1$. | ||
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| - | 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 $\hat\Z=\T\subset\C$ | ||
| - | $$G\tiltimes\T=\{zA\mid z\in\T;\;A\in G\}.$$ | ||
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| - | ==== Unitary quantum groups as free complexifications ==== | ||
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| - | It holds that $U_N^+=O_N^+\hatstar\haz\Z$ [(ref:Ban97)] | ||
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| - | ===== References ===== | ||
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| - | [(ref:TW17>> | ||
| - | author: Pierre Tarrago and Moritz Weber | ||
| - | title: Unitary Easy Quantum Groups: The Free Case and the Group Case | ||
| - | journal: International Mathematics Research Notices | ||
| - | volume: 2017 | ||
| - | number: 18 | ||
| - | pages: 5710–5750 | ||
| - | year: 2017 | ||
| - | url: https://dx.doi.org/10.1093/imrn/rnw185 | ||
| - | )] | ||
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| - | [(ref:Ban99>> | ||
| - | author: Teodor Banica | ||
| - | title: Representations of compact quantum groups and subfactors | ||
| - | journal: Journal für die reine und angewandte Mathematik | ||
| - | year: 1999 | ||
| - | volume: 509 | ||
| - | pages: 167–198 | ||
| - | url: http://dx.doi.org/10.1515/crll.1999.509.167 | ||
| - | )] | ||
| - | |||
| - | [(ref:Ban08>> | ||
| - | author: Teodor Banica | ||
| - | title: A Note on Free Quantum Groups | ||
| - | journal: Annales Mathématiques Blaise Pascal | ||
| - | volume: 15 | ||
| - | number: 2 | ||
| - | year: 2008 | ||
| - | pages: 135–146 | ||
| - | url: http://dx.doi.org/10.5802/ambp.243 | ||
| - | )] | ||
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| - | |||
| - | [(ref:Ban97>> | ||
| - | author: Teodor Banica | ||
| - | title: Le Groupe Quantique Compact Libre U(n) | ||
| - | journal: Communications in Mathematical Physics | ||
| - | year: 1997 | ||
| - | volume: 190 | ||
| - | number: 1 | ||
| - | pages: 143–172 | ||
| - | url: http://dx.doi.org/10.1007/s002200050237 | ||
| - | )] | ||
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