Geodesic
Higher Zigzag Algebras
Documenta mathematica, Tome 24 (2019), pp. 749-814
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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 SLd+1​ then these relations occur between spherical twists for G-equivariant sheaves on affine (d+1)-space.
DOI : 10.4171/dm/693
Classification : 16D50, 16E35, 16G20, 16W55
Mots-clés : derived category, quiver, Koszul algebra, cluster tilting, trivial extension, braid group action, spherical twist, equivariant sheaves 
@article{10_4171_dm_693,
     author = {Joseph Grant},
     title = {Higher {Zigzag} {Algebras}},
     journal = {Documenta mathematica},
     pages = {749--814},
     year = {2019},
     volume = {24},
     doi = {10.4171/dm/693},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/693/}
}
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Joseph Grant. Higher Zigzag Algebras. Documenta mathematica, Tome 24 (2019), pp. 749-814. doi: 10.4171/dm/693

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