A Coboundary Temperely-Lieb Category for $\mathfrak{sl}_2$-Crystals

TL;DR

This paper explores a specialized Temperley--Lieb category at q=0, establishing its structure, bases, and coboundary properties, and relating it to $rak{sl}_2$-crystals and cactus group actions.

Contribution

It introduces a coboundary structure on the q=0 Temperley--Lieb category and links it to $rak{sl}_2$-crystals via idempotent completion and diagrammatic methods.

Findings

Provides a closed formula for Jones--Wenzl projectors at q=0

Establishes a coboundary monoidal equivalence with $rak{sl}_2$-crystals

Describes the action of the cactus group on the category

Abstract

By considering a suitable renormalization of the Temperley--Lieb category, we study its specialization to the case $q=0$. Unlike the $q\neq 0$ case, the obtained monoidal category, $\mathcal{TL}_0(\Bbbk)$, is not rigid or braided. We provide a closed formula for the Jones--Wenzl projectors in $\mathcal{TL}_0(\Bbbk)$ and give semisimple bases for its endomorphism algebras. We explain how to obtain the same basis using the representation theory of finite inverse monoids, via the associated M\"obius inversion. We then describe a coboundary structure on $\mathcal{TL}_0(\Bbbk)$ and show that its idempotent completion is coboundary monoidally equivalent to the category of $\mathfrak{sl}_{2}$-crystals. This gives a diagrammatic description of the commutor for $\mathfrak{sl}_{2}$-crystals defined by Henriques and Kamnitzer and of the resulting action of the cactus group. We also study fiber functors of $\mathcal{TL}_0(\Bbbk)$ and discuss how they differ from the $q\neq 0$ case.