Title: 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.