Quiver Hecke algebras for Borcherds-Cartan datum II

TL;DR

This paper categorifies the crystal structure of quantum Borcherds algebras using quiver Hecke algebras, extending previous work to arbitrary Borcherds-Cartan data and exploring cyclotomic categorification of highest weight modules.

Contribution

It provides the crystal structure of the Grothendieck group for quiver Hecke algebras associated with Borcherds-Cartan data, enabling categorification of quantum Borcherds algebras.

Findings

Categorification of the crystal $B()$ for quantum Borcherds algebras.

Construction of the crystal structure on $G_0(R)$ for arbitrary Borcherds-Cartan data.

Analysis of cyclotomic categorification of highest weight modules.

Abstract

We give the crystal structure of the Grothendieck group $G_0(R)$ of irreducible modules over the quiver Hecke algebra $R$ constructed in \cite{TW2023}. This leads to the categorification of the crystal $B(\infty)$ of the quantum Borcherds algebra $U_q(\mathscr g)$ and its irreducible highest weight crystal $B(\lambda)$ for arbitrary Borcherds-Cartan data. Additionally, we study the cyclotomic categorification of irreducible highest weight $U_q(\mathscr g)$-modules.