# Extremal weight projectors

## TL;DR

This paper introduces a quotient of the affine Temperley-Lieb category with diagrammatic idempotents that categorify Chebyshev polynomials, providing a new approach to categorifying the Kauffman bracket skein algebra of the annulus and torus.

## Contribution

It defines extremal weight projectors within a quotient of the affine Temperley-Lieb category, linking diagrammatic idempotents to Chebyshev polynomials and skein algebra categorification.

## Key findings

- Idempotents satisfy properties similar to Jones-Wenzl projectors
- Categorifies Chebyshev polynomials of the first kind
- Provides a categorification of the Kauffman bracket skein algebra of the annulus

## Abstract

We introduce a quotient of the affine Temperley-Lieb category that encodes all weight-preserving linear maps between finite-dimensional 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.