Extended graphical calculus for categorified quantum sl(2)

TL;DR

This paper extends the graphical calculus for categorified quantum sl(2), providing explicit formulas for decompositions and demonstrating that key algebraic structures hold over the integers.

Contribution

It enhances the graphical calculus to include two-morphisms between divided powers and proves integral formulas for decompositions, strengthening the categorification framework.

Findings

Explicit diagrammatic formulas for product decompositions

Identification of the Grothendieck ring over integers

Main results hold over the ring of integers

Abstract

A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. We obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements. These formulas have integral coefficients and imply that one of the main results of Lauda's paper---identification of the Grothendieck ring of his 2-category with the idempotented quantum sl(2)---also holds when the 2-category is defined over the ring of integers rather than over a field.