Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Moaaz Alqady, Mateusz Stroiński

TL;DR

This work constructs a diagrammatic framework for $\mathfrak{sl}_2$-crystals by renormalizing the Temperley--Lieb category at $q=0$, producing $\mathcal{TL}_0(\Bbbk)$ which is non-rigid and non-braided but admits a semisimple endomorphism structure and an explicit basis from Temperley--Lieb diagrams. It gives a closed formula for Jones--Wenzl projectors in this setting, and connects endomorphism algebras to contracted monoid algebras via Möbius inversion on finite inverse monoids, providing a rich combinatorial and representation-theoretic description. The main result is a monoidal equivalence between the Cauchy completion $\mathbf{CrysTL}$ and the category of $\mathfrak{sl}_2$-crystals, together with a diagrammatic coboundary structure that yields Henriques–Kamnitzer style commutors and cactus-group actions in a purely diagrammatic setting. The paper also analyzes fiber functors in this non-generic regime, showing non-uniqueness and revealing affine-GIT parameter spaces for these functors, thereby contrasting with the $q\neq 0$ case and highlighting the richer fiber-functor landscape at $q=0$.

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 Möbius 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.

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

TL;DR

This work constructs a diagrammatic framework for -crystals by renormalizing the Temperley--Lieb category at , producing which is non-rigid and non-braided but admits a semisimple endomorphism structure and an explicit basis from Temperley--Lieb diagrams. It gives a closed formula for Jones--Wenzl projectors in this setting, and connects endomorphism algebras to contracted monoid algebras via Möbius inversion on finite inverse monoids, providing a rich combinatorial and representation-theoretic description. The main result is a monoidal equivalence between the Cauchy completion and the category of -crystals, together with a diagrammatic coboundary structure that yields Henriques–Kamnitzer style commutors and cactus-group actions in a purely diagrammatic setting. The paper also analyzes fiber functors in this non-generic regime, showing non-uniqueness and revealing affine-GIT parameter spaces for these functors, thereby contrasting with the case and highlighting the richer fiber-functor landscape at .

Abstract

By considering a suitable renormalization of the Temperley--Lieb category, we study its specialization to the case . Unlike the case, the obtained monoidal category, , is not rigid or braided. We provide a closed formula for the Jones--Wenzl projectors in 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 Möbius inversion. We then describe a coboundary structure on and show that its idempotent completion is coboundary monoidally equivalent to the category of -crystals. This gives a diagrammatic description of the commutor for -crystals defined by Henriques and Kamnitzer and of the resulting action of the cactus group. We also study fiber functors of and discuss how they differ from the case.

Paper Structure

This paper contains 16 sections, 56 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathfrak{g}$-Crystals
  4. Coboundary Categories and Cactus Actions
  5. $\mathfrak{g}\mathbf{-Crys}$ as a Coboundary Category
  6. The Temperley--Lieb Category
  7. The Crystal Temperley--Lieb Category
  8. Renormalizing the Temperley--Lieb Category
  9. Jones--Wenzl Projectors
  10. Properties of TL0(k)
  11. Pointed Sets, Monoid Algebras, Möbius Inversion and Cells
  12. Equivalence of $\mathfrak{sl}_2\mathbf{-Crys}$ and $\mathbf{CrysTL}$
  13. Equivalence as Monoidal Categories
  14. A Commutor for the Crystal Temperley--Lieb Category
  15. Equivalence as Coboundary Categories
  16. ...and 1 more sections

Key Result

Theorem 2.1

HKam Let $\mathcal{C}$ be a coboundary category, and let $A_1,\dots, A_n\in \mathcal{C}$. If $\rho\in J_n$, the natural isomorphisms $\tau(\rho; A_1, \dots, A_n)$ defined above satisfy Moreover, these natural isomorphisms are exactly the ones which can be generated by using the commutor of $\mathcal{C}$.

Figures (1)

  • Figure :

Theorems & Definitions (133)

  • Definition 1
  • Definition 2
  • Remark 3
  • Definition 4
  • Example 5
  • Example 6
  • Definition 7
  • Definition 8: D
  • Definition 9
  • Theorem 2.1
  • ...and 123 more