Affine Cellularity of Khovanov-Lauda-Rouquier algebras in type A

TL;DR

This paper proves that Khovanov-Lauda-Rouquier algebras of type A are affine cellular with idempotent-generated ideals, leading to finite global dimension and a new module theory.

Contribution

It establishes the affine cellularity of these algebras and shows that their affine cell ideals are generated by idempotents, which was not previously known.

Findings

Affine cellularity of $R_eta$ in type A proven

Affine cell ideals are generated by idempotents

Finite global dimension and new module theory developed

Abstract

We prove that the Khovanov-Lauda-Rouquier algebras $R_\al$ of type $A_\infty$ are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\al$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\al$ is finite, and yields a theory of standard and reduced standard modules for $R_\al$.