Higher Zigzag Algebras

Abstract

Given a Koszul algebra of finite global dimension we define its higher zigzag algebra as a twisted trivial extension of the Koszul dual. If our original algebra is the path algebra of a tree-type quiver, this construction recovers the zigzag algebras of Huerfano-Khovanov. We study examples of higher zigzag algebras coming from Iyama's type A higher representation finite algebras, give their presentations by quivers and relations, and describe relations between spherical twists acting on their derived categories. We connect this to the McKay correspondence in higher dimensions: if $G$ is a finite abelian subgroup of $S{L}_{d+1}$ then these relations occur between spherical twists for $G$-equivariant sheaves on affine $(d+1)$-space.