Mathematical Research Letters

Volume 25 (2018)

Number 6

Extremal weight projectors

Pages: 1911 – 1936



Hoel Queffelec (IMAG, CNRS, Université de Montpellier, France)

Paul Wedrich (Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia)


We introduce a quotient of the affine Temperley–Lieb category that encodes all weight-preserving linear maps between finite-dimensional $\mathfrak{sl}_2$-representations.We study the diagrammatic idempotents that correspond to projections onto extremal weight spaces and find that they satisfy similar properties as Jones–Wenzl projectors, and that they categorify the Chebyshev polynomials of the first kind. This gives a categorification of the Kauffman bracket skein algebra of the annulus, which is well adapted to the task of categorifying the multiplication on the Kauffman bracket skein module of the torus.

Received 6 March 2017

Accepted 15 January 2018

Published 25 March 2019