Higher zigzag algebras

TL;DR

This paper introduces higher zigzag algebras derived from Koszul algebras, explores their properties, presentations, and relations to spherical twists, and connects them to higher-dimensional McKay correspondence in algebraic geometry.

Contribution

It defines higher zigzag algebras for Koszul algebras, studies their structure, and links them to geometric and representation-theoretic contexts.

Findings

Higher zigzag algebras generalize classical zigzag algebras.

Explicit quiver and relations for these algebras are provided.

Relations between spherical twists are characterized and connected to algebraic geometry.

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.