Abstract
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 treetype quiver, this construction recovers the zigzag algebras of HuerfanoKhovanov. 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 Gequivariant sheaves on affine (d + 1)space.
2010 Mathematics Subject Classification: 16. Associative rings and algebras; 18. Category theory, homological algebra; 14. Algebraic geometry
2010 Mathematics Subject Classification: 16. Associative rings and algebras; 18. Category theory, homological algebra; 14. Algebraic geometry
Original language  English 

Pages (fromto)  749814 
Number of pages  66 
Journal  Documenta Mathematica 
Volume  24 
DOIs  
Publication status  Published  2019 
Keywords
 trivial extension
 braid group action
 spherical twist
 quiver
 derived category
 Koszul algebra
 cluster tilting
 equivariant sheaves
Profiles

Joseph Grant
 School of Mathematics  Lecturer in Pure Mathematics
 Algebra and Combinatorics  Member
Person: Research Group Member, Academic, Teaching & Research