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arXiv · 2405.03364

Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures

Lukas Rollier
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Abstract

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.

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Lukas Rollier. 2024-05-06. Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures. https://arxiv.org/abs/2405.03364

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