PBW bases and KLR algebras

TL;DR

This paper extends Lusztig's geometric construction of PBW bases for finite quantum groups of type ADE, linking them to KLR algebras, and proves positivity and finiteness conjectures in this setting.

Contribution

It generalizes Lusztig's construction within the Varagnolo-Vasserot framework, establishing semi-orthogonal collections and proving key conjectures for ADE type KLR algebras.

Findings

PBW bases induce semi-orthogonal collections in KLR module categories

Proves Lusztig's positivity conjecture for ADE quantum groups

Verifies Kashiwara's finiteness conjecture for ADE KLR algebras

Abstract

We generalize Lusztig's geometric construction of the PBW bases of finite quantum groups of type $\mathsf{ADE}$ under the framework of [Varagnolo-Vasserot, J. reine angew. Math. 659 (2011)]. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the KLR-algebras. This enables us to prove Lusztig's conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases in the $\mathsf{ADE}$ case. In addition, we verify Kashiwara's problem on the finiteness of the global dimensions of the KLR-algebras of type $\mathsf{ADE}$.