Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras

TL;DR

This paper proves a conjecture that certain algebraic structures called cyclotomic Khovanov-Lauda-Rouquier algebras categorify all irreducible highest weight modules of symmetrizable Kac-Moody algebras, advancing the understanding of quantum group representations.

Contribution

It establishes the cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras, linking Khovanov-Lauda-Rouquier algebras to highest weight modules.

Findings

Proves the cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras.

Shows that the algebra $R^{ ext{Lambda}}$ categorifies the irreducible highest weight module $V( ext{Lambda})$.

Extends previous results to a broader class of algebras.

Abstract

In this paper, we prove Khovanov-Lauda's cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras. Let $U_q(g)$ be the quantum group associated with a symmetrizable Cartan datum and let $V(\Lambda)$ be the irreducible highest weight $U_q(g)$-module with a dominant integral highest weight $\Lambda$. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra $R^{\Lambda}$ gives a categorification of $V(\Lambda)$.