Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910436234887168 |
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| author | Rollier, Lukas |
| author_facet | Rollier, Lukas |
| contents | We define quantum automorphism groups of a wide range of discrete structures. The central tool for their construction is a generalisation of the Tannaka-Krein reconstruction theorem. For any direct sum of matrix algebras $M$, and any concrete unitary 2-category of finite type Hilbert-$M$-bimodules $\mathcal{C}$, under reasonable conditions, we construct an algebraic quantum group $\mathbb{G}$ which acts on $M$ by $α$, such that the category of $α$-equivariant corepresentations of $\mathbb{G}$ on finite type Hilbert-$M$-bimodules is equivalent to $\mathcal{C}$. Moreover, we explicitly describe how to get such categories from connected locally finite discrete structures. As an example, we define the quantum automorphism group of a quantum Cayley graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures Rollier, Lukas Operator Algebras Quantum Algebra Representation Theory We define quantum automorphism groups of a wide range of discrete structures. The central tool for their construction is a generalisation of the Tannaka-Krein reconstruction theorem. For any direct sum of matrix algebras $M$, and any concrete unitary 2-category of finite type Hilbert-$M$-bimodules $\mathcal{C}$, under reasonable conditions, we construct an algebraic quantum group $\mathbb{G}$ which acts on $M$ by $α$, such that the category of $α$-equivariant corepresentations of $\mathbb{G}$ on finite type Hilbert-$M$-bimodules is equivalent to $\mathcal{C}$. Moreover, we explicitly describe how to get such categories from connected locally finite discrete structures. As an example, we define the quantum automorphism group of a quantum Cayley graph. |
| title | Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures |
| topic | Operator Algebras Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2405.03364 |