Derived categories of gentle algebras and surface dissections

AbstractThe purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

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q-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras

Pacific Journal of Mathematics ◽

10.2140/pjm.2007.229.25 ◽

2007 ◽

Vol 229 (1) ◽

pp. 25-47 ◽

Cited By ~ 5

Author(s):

Christine Bessenrodt ◽

Thorsten Holm

Keyword(s):

Derived Categories ◽

Gentle Algebras ◽

Cartan Matrices

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Curved Rickard complexes and link homologies

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2019-0044 ◽

2020 ◽

Vol 2020 (769) ◽

pp. 87-119

Author(s):

Sabin Cautis ◽

Aaron D. Lauda ◽

Joshua Sussan

Keyword(s):

Spectral Sequence ◽

Quantum Groups ◽

Braid Group ◽

Derived Categories ◽

Duke Math ◽

Khovanov Homology ◽

Hilbert Schemes ◽

Coherent Sheaves ◽

Knot Homology ◽

Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

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