Higher zigzag algebras

Abstract

Given any Koszul algebra of finite global dimension one can define a new algebra, which we call a higher zigzag algebra, as a twisted trivial extension of the Koszul dual of our original algebra. If our original algebra is the path algebra of a quiver whose underlying graph is a tree, this construction recovers the zigzag algebras of Huerfano and Khovanov. We study examples of higher zigzag algebras coming from Iyama's iterative construction of type A higher representation finite algebras. We give presentations of these algebras by quivers and relations, and describe relations between spherical twists acting on their derived categories. We then make a connection to the McKay correspondence in higher dimensions: if G is a finite abelian subgroup of the special linear group acting on affine space, then the skew group algebra which controls the category of G-equivariant sheaves is Koszul dual to a higher zigzag algebra. Using this, we show that our relations between spherical twists appear naturally in examples from algebraic geometry.