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 ${\mathbf{U}}_{\mathbf{q}}(\mathfrak{g})$ associated to a symmetrizable simply-laced Kac-Moody algebra $\mathfrak{g}$.