Affine zigzag algebras and imaginary strata for KLR algebras

TL;DR

This paper introduces affine zigzag algebras and demonstrates their Morita equivalence to imaginary strata of affine ADE KLR algebras, advancing understanding of their stratification structure in suitable characteristics.

Contribution

It defines affine zigzag algebras and proves their Morita equivalence to all imaginary strata of affine ADE KLR algebras in sufficiently large characteristic.

Findings

Affine zigzag algebras are Morita equivalent to imaginary strata.

Imaginary strata of affine ADE KLR algebras are classified via affine zigzag algebras.

The results depend on the characteristic of the ground field being above a certain bound.

Abstract

KLR algebras of affine ADE types are known to be properly stratified if the characteristic of the ground field is greater than some explicit bound. Understanding the strata of this stratification reduces to semicuspidal cases, which split into real and imaginary subcases. Real semicuspidal strata are well-understood. We show that the smallest imaginary stratum is Morita equivalent to Huerfano-Khovanov's zigzag algebra tensored with a polynomial algebra in one variable. We introduce affine zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above.