Speaker: Junichiro Matsuda (Kyoto Univ.)

Title: Classification of quantum graphs on M₂ and their quantum automorphism groups

Time/Date: 4:45-6:15pm, December 7 (Tue.), 2021

Room: This will be an online seminar on Zoom. Please ask Kawahigashi for the Zoom link.

Abstract: The notion of quantum graphs is introduced in the early 2010s and has been developed by interactions with quantum information theory, operator algebra theory, quantum group theory, etc. Musto, Reutter, Verdon (2018) formulated the entrywise product of matrices (Schur product of operators on a commutative algebra Cⁿ) in terms of string diagrams and applied it to noncommutative algebras. Since the adjacency matrix of a classical graph is a Schur idempotent matrix, they introduced quantum adjacency matrices as Schur idempotent operators on a noncommutative algebra with a tracial state. Brannan, Chirvasitu, Eifler, Harris, Paulsen, Su, Wasilewski (2019) generalized them to nontracial states, and I applied the string diagrammatic approach to such nontracial cases.

In order to capture a concrete image of quantum graphs, I classified the undirected reflexive quantum graphs on M₂ and their quantum automorphism groups both in tracial and nontracial settings. As a corollary, we reprove the monoidal equivalences between SO(3) and S₄⁺, and O(2) and H₂⁺.

This talk is based on my preprint, arXiv:2110.09085.