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Subfactors from Graphs Induced by Association Schemes

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Abstract

We characterize anyonic systems algebraically by identifying the mathematical structures that support duality and fusion, Reidemeister moves, that are invariants of knots, braids, and modular data. The characterization is based on the connection between fusion algebras relevant in conformal field theories and character algebras related to association schemes. To make this abstract connection concrete, we provide the example of Hamming association schemes and relate them to representations of quantum groups \(SU_q(2)\) that are closely connected to \(SU(2)_k\) algebras whose fusion rules describe well known anyons. Our primary object of interest is the interacting Fock space which is deeply connected to an association scheme and the corresponding Bose-Mesner algebra, a combinatorial gadget with built-in duality and fusion rules, that leads to matrices (invariant under Reidemeister II and III moves in knots) which aid construction of subfactors with projections that braid. This way we set up a subfactor, a 3D topological quantum field theory, and a 2D rational conformal field theory and relate them to anyon systems described by fusion algebras. We discuss in detail a large family of graphs of self-dual association schemes that can be treated with this algebraic framework.

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Acknowledgements

The author is grateful to Paul Terwilliger for suggesting, in a private communication, the conditions under which the Leonard pairs of q-Krawtchauk polynomials have \(u_q(2, C)\) modules. The author is grateful to the anonymos rereferee whose detailed feedback improved the presentation of the manuscript.

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  1. Army Research Office, RTP, NC, 21005-5069, USA

    Radhakrishnan Balu

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Balu, R. Subfactors from Graphs Induced by Association Schemes. Int J Theor Phys 62, 260 (2023). https://doi.org/10.1007/s10773-023-05510-w

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