Representations of Quiver Hecke Algebras via Lyndon Bases

TL;DR

This paper uses Lyndon word combinatorics to classify irreducible representations of quiver Hecke algebras associated with finite type Cartan data, completing prior classification efforts.

Contribution

It introduces a Lyndon basis approach to explicitly construct and classify simple modules for quiver Hecke algebras of finite type.

Findings

Complete classification of simple modules achieved

Construction of irreducible representations via Lyndon words

Advances understanding of categorification in quantum groups

Abstract

A new class of algebras have been introduced by Khovanov and Lauda and independently by Rouquier. These algebras categorify one-half of the Quantum group associated to arbitrary Cartan data. In this paper, we use the combinatorics of Lyndon words to construct the irreducible representations of those algebras associated to Cartan data of finite type. This completes the classification of simple modules for the quiver Hecke algebra initiated by Kleshchev and Ram.