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

Abstract We study a presentation of the 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 of 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 idempotented and integral quantum group U q ( g ) associated with a symmetrizable simply-laced Kac-Moody algebra g.