Rewriting modulo isotopies in Khovanov-Lauda-Rouquier's categorification of quantum groups

Abstract

We study a presentation of Khovanov -Lauda -Rouquier's candidate 2-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to compute a family of linear bases for all the vector spaces of 2-cells in this 2-category. We show that these bases correspond to Khovanov and Lauda's conjectured generating sets, proving the non-degeneracy of their diagrammatic calculus. This implies that this 2-category is a categorification of Lusztig's idempotent and integral quantum group U q (g) associated to a symmetrizable simply-laced Kac-Moody algebra g.

Super rewriting theory and nondegeneracy of odd categorified sl(2)

“…Remark 2.5. Throughout this paper, we read our compositions cells as is common in higher category theory, just as the first author does in [14,13]. This composition is read backwards from the more prevalent way of reading composition used by Brundan and Ellis [6, Definition 2.1].…”

“…It provides a set of tools for determining when a presentation of a 2-category will be coherent and allows for a determination of a normal form for a given 2-morphism within a given rewriting class, constructively providing bases from a specific presentation of a 2-category. The techniques of higher-dimensional rewriting have been effectively applied in a number of important examples in higher-representation theory [4,3,14,13] including cases where a determination of these bases have eluded experts for some time [13].…”

A diamond lemma for Hecke-type algebras

“…Addendum. Diamond lemmas for monoidal categories have appeared in the literature, see the references in [6].…”

“…Addendum. The original plan was to post this paper when [8] was complete, but I'm posting it now in response to recent work of Dupont [6]. Dupont's work has made it clear how much of the literature on Bergman-style arguments in 2-categories I was unaware of when I wrote this, and the reader should read [6] for a more detailed account.…”

“…The original plan was to post this paper when [8] was complete, but I'm posting it now in response to recent work of Dupont [6]. Dupont's work has made it clear how much of the literature on Bergman-style arguments in 2-categories I was unaware of when I wrote this, and the reader should read [6] for a more detailed account. Dupont applies a Bergman-style argument to the 2-category which categorifies the quantum group, rather than to quiver Hecke algebras which categorify only the positive half of the quantum group, and thus his work is more impressive.…”