Higher zigzag algebras

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Given a Koszul algebra of finite global dimension we define its higher zigzag algeba 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 SLd+1 then these relations occur between spherical twists for G-equivariant sheaves on affine (d + 1)-space.
2010 Mathematics Subject Classification: 16. Associative rings and algebras; 18. Category theory, homological algebra; 14. Algebraic geometry
Original languageEnglish
Pages (from-to)749-814
Number of pages66
JournalDocumenta Mathematica
Publication statusPublished - 2019


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

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