A KLR-like presentation for the bt-algebra

TL;DR

This work constructs KLR-like presentations for the bt-algebras ${\mathcal{E}}_n(q)$ and ${\mathcal{E}}^{\rm{ord}}_n(q)$ by leveraging Yokonuma–Hecke algebras and cyclotomic quiver Hecke algebras. It develops idempotent theory via Möbius inversion and introduces Jucys–Murphy elements to obtain graded, seminormal structures, including diagrammatic calculi for both standard and ordered variants. The central contributions are two isomorphisms: ${\mathcal{E}}_n(q) \cong {\mathcal{E}}_n(\Gamma_e)$ (when $q^{1/2}$ exists in the base field) and ${\mathcal{E}}^{\rm{ord}}_n(q) \cong {\mathcal{E}}^{\rm{ord}}_n(\Gamma_e)$ (for all $q\neq 1$), which transfer the natural ${\mathbb{Z}}$-gradings from KLR algebras to the bt-algebras. These results illuminate the representation theory of bt-algebras at roots of unity and connect knot-theoretic algebras to categorification frameworks.

Abstract

We consider the bt-algebra ${ \mathcal E}_n(q)$ of knot theory, defined over an arbitrary field $ \Bbbk$. We find a KLR-like presentation for $ {\mathcal E}_n(q) $ showing that it is a $ \mathbb Z$-graded algebra if $ q \in \Bbbk^{\times} \setminus \{1 \} $ admits a square root in $ \Bbbk $. We introduce the ordered bt-algebra $ {\mathcal E}^{\rm{ord}}_n(q)$ and show that it also has a KLR-like presentation, without restriction on $ q \in \Bbbk^{\times} \setminus \{1 \} $. In particular, $ {\mathcal E}^{\rm{ord}}_n(q)$ is a $ \mathbb Z$-graded algebra for all $ q \in \Bbbk^{\times} \setminus \{1 \} $.

Theorems & Definitions (40)

• Definition 1
• Lemma 2
• Corollary 3
• Theorem 4
• Lemma 5
• Definition 6
• Definition 7
• Lemma 8
• Lemma 9
• Lemma 10