Grant, Joseph

Higher Zigzag Algebras

Doc. Math. 24, 749-814 (2019)
DOI: 10.25537/dm.2019v24.749-814
Communicated by Henning Krause

Summary

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 \(SL_{d+1}\) then these relations occur between spherical twists for \(G\)-equivariant sheaves on affine \((d+1)\)-space.

Mathematics Subject Classification

16D50, 16G20, 18E30, 16E35, 16W55, 14F05

Keywords/Phrases

trivial extension, braid group action, spherical twist, quiver, derived category, Koszul algebra, cluster tilting, equivariant sheaves

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Affiliation

Grant, Joseph
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK

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