Homological properties of finite type Khovanov-Lauda-Rouquier algebras

TL;DR

This paper constructs standard modules for finite type Khovanov-Lauda-Rouquier algebras and proves they have affine quasi-hereditary properties, providing elementary algebraic proofs and Koszul-like resolutions.

Contribution

It offers an algebraic construction of standard modules and demonstrates their homological properties, extending known results to all finite types without geometric methods.

Findings

Algebraic construction of standard modules categorifying PBW basis

Proof that these algebras are affine quasi-hereditary in all finite types

Construction of Koszul-like projective resolutions for certain modules

Abstract

We give an algebraic construction of standard modules (infinite dimensional modules categorifying the PBW basis of the underlying quantized enveloping algebra) for Khovanov-Lauda-Rouquier algebras in all finite types. This allows us to prove in an elementary way that these algebras satisfy the homological properties of an `affine quasi-hereditary algebra.' In simply-laced types these properties were established originally by Kato via a geometric approach. We also construct some Koszul-like projective resolutions of standard modules corresponding to multiplicity-free positive roots.