Hall categories and KLR categorification

TL;DR

This paper initiates the categorification of the bialgebra structure of half of quantum groups using geometric and Hall algebra methods, connecting to KLR categorification.

Contribution

It introduces a new geometric framework for categorifying the bialgebra structure of quantum groups via D-modules and Hall algebras, extending Waldhausen's construction.

Findings

Establishes a category of D-modules as an algebra object in stable $mbda$-categories.

Connects the construction to Khovanov-Lauda-Rouquier categorification.

Provides a foundation for future categorification of the entire quantum group structure.

Abstract

This paper is the first step in the project of categorifying the bialgebra structure on the half of quantum group $U_{q}(\mathfrak{g})$ by using geometry and Hall algebras. We equip the category of D-modules on the moduli stack of objects of the category $Rep_{\mathbb{C}}(Q)$ of representations of a quiver with the structure of an algebra object in the category of stable $\infty$-categories. The data for this construction is provided by an extension of the Waldhausen construction for the category $Rep_{\mathbb{C}}(Q)$. We discuss the connection to the Khovanov-Lauda-Rouquier categorification of half of the quantum group $U_{q}(\mathfrak{g})$ associated to the quiver $Q$ and outline our approach to the categorification of the bialgebra structure.