A diagrammatic approach to categorification of quantum groups I

TL;DR

This paper introduces a diagrammatic method to categorify quantum groups associated with Kac-Moody Lie algebras using rings derived from graphs, advancing the understanding of algebraic structures through categorification.

Contribution

It presents a novel diagrammatic approach to categorify quantum groups via rings linked to graphs, providing new tools for algebraic and categorical analysis.

Findings

Categorification of $U^-_q(rak{g})$ using rings from graphs

Development of a diagrammatic framework for quantum groups

Establishment of categories of projective modules as categorifications

Abstract

To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify $U^-_q(\mathfrak{g})$, where $\mathfrak{g}$ is the Kac-Moody Lie algebra associated with the graph.