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arXiv · 2409.01951

Quantum graphs, subfactors and tensor categories I

Michael Brannan·Roberto Hernández Palomares
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Abstract

We develop an equivariant theory of graphs with respect to quantum symmetries and present a detailed exposition of various examples. We portray unitary tensor categories as a unifying framework encompassing all finite classical simple graphs, (quantum) Cayley graphs of finite (quantum) groupoids, and all finite-dimensional quantum graphs. We model a quantum set by a finite-index inclusion of C*-algebras and use the quantum Fourier transform to obtain all possible adjacency operators. In particular, we show every finite-index subfactor can be regarded as a complete quantum graph and describe how to find all its subgraphs. As applications, we prove a version of Frucht's Theorem for finite quantum groupoids, and introduce a version of path spaces for quantum graphs.

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Michael Brannan, Roberto Hernández Palomares. 2024-09-03. Quantum graphs, subfactors and tensor categories I. https://doi.org/10.1142/s0129167x25500399

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