Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 5

Special issue on “Subfactors and Related Topics” in memory of Vaughan Jones

Guest Editors: Dietmar Bisch, Arthur Jaffe, Yasuyuki Kawahigashi, and Zhengwei Liu

Planar diagrammatics of self-adjoint functors and recognizable tree series

Pages: 2409 – 2499

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n5.a4

Authors

Mikhail Khovanov (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Robert Laugwitz (School of Mathematical Sciences, University of Nottingham, United Kingdom)

Abstract

A pair of biadjoint functors between two categories produces a collection of elements in the centers of these categories, one for each isotopy class of nested circles in the plane. If the centers are equipped with a trace map into the ground field, then one assigns an element of that field to a diagram of nested circles. We focus on the self-adjoint functor case of this construction and study the reverse problem of recovering such a functor and a category given values associated to diagrams of nested circles.

Keywords

self-adjoint functor, universal construction, monoidal category, Temperley–Lieb category, recognizable tree series

Full Text (PDF format)

M. K. was partially supported via NSF grant DMS-1807425.

R. L. was supported by a Nottingham Research Fellowship.

Received 5 April 2021

Received revised 6 October 2021

Accepted 30 October 2021

Published 30 January 2024