An introduction to diagrammatic algebra and categorified quantum sl(2)

TL;DR

This paper introduces diagrammatic methods in the categorification of quantum sl(2), detailing new structures, algebraic isomorphisms, and interpretations relevant to categorified quantum groups.

Contribution

It provides a comprehensive exposition of diagrammatic algebra in categorified quantum sl(2), including new structural insights and algebraic isomorphisms.

Findings

Rescaling isomorphisms for categorified quantum sl(2)

Cyclotomic quotients of nilHecke algebra are isomorphic to cohomology rings of Grassmannians

Fake bubbles are interpreted using symmetric functions

Abstract

This expository article explains how planar diagrammatics naturally arise in the study of categorified quantum groups with a focus on the categorification of quantum sl2. We derive the definition of categorified quantum sl2 and highlight some of the new structure that arises in categorified quantum groups. The expert will find a discussion of rescalling isomorphisms for categorified quantum sl2, a proof that cyclotomic quotients of the nilHecke algebra are isomorphic to matrix rings over the cohomology ring of Grassmannians, and an interpretation of `fake bubbles' using symmetric functions.