Categorified Jones-Wenzl Projectors and Generalizations

TL;DR

This paper develops a categorification framework for Jones-Wenzl projectors, extending their theory to type B and D, with proofs of key theorems and generalizations in quantum algebra and diagrammatics.

Contribution

It introduces a type B analogue of Jones-Wenzl projectors and proves their properties, including convergence and duality, expanding the categorification landscape.

Findings

Explicit proof of quantum coideal Schur-Weyl duality for non-trivial weight spaces

Generalization of Jones-Wenzl projectors to type B Temperley-Lieb algebra

Convergence of powers of type D full-twist to the Jones-Wenzl projector

Abstract

This preprint comprises the first four out of five chapters of the Master's thesis I wrote 2022 under the supervision of Catharina Stroppel and Paul Wedrich at the University of Bonn titled "Categorified Jones-Wenzl projectors and Generalizations". Chapter 1 can serve as introductory literature for researchers or graduate students familiar with Schur-Weyl duality between the symmetric group and the general linear Lie algebra or their quantum counter parts, and who are seeking to learn a type B/D analogue of the famous type A story. Chapter 1 serves as an introduction to the type B combinatorics and blob-diagrammatics. Chapters 2,3, and 4 contain proofs of three main theorem each - the first one is an explicit/computational proof of the easiest case of quantum coideal Schur-Weyl duality involving non-trivial quantizations of weight spaces, the second one shows the generalization of Jones-Wenzl projectors to the type B Temperley-Lieb algebra and the third one uses generalized Reidemeister moves and a functional analytic argument to prove a version of the fact that powers of the type D full-twist converge towards the type D Jones-Wenzl projector.