A Geometric Model for the Module Category of a Gentle Algebra

In this article, gentle algebras are realised as tiling algebras, which are associated to partial triangulations of unpunctured surfaces with marked points on the boundary. This notion of tiling algebras generalise the notion of Jacobian algebras of triangulations of surfaces and the notion of surface algebras. We use this description to give a geometric model of the module category of any gentle algebra.

[1]  Vincent Pilaud,et al.  Non-kissing complexes and tau-tilting for gentle algebras , 2017, Memoirs of the American Mathematical Society.

[2]  Sibylle Schroll,et al.  On Extensions for Gentle Algebras , 2017, Canadian Journal of Mathematics.

[3]  A. G. Elsener.  Gentle m-Calabi-Yau tilted algebras , 2017, Algebra and Discrete Mathematics.

[4]  T. McConville,et al.  Oriented Flip Graphs and Noncrossing Tree Partitions , 2016, Discrete Mathematics & Theoretical Computer Science.

[5]  Hermund André Torkildsen,et al.  A Geometric Interpretation of Categories of Type Ã$\tilde {A}$ and of Morphisms in the Infinite Radical , 2020, Algebras and Representation Theory.

[6]  T. McConville,et al.  ORIENTED FLIP GRAPHS, NONCROSSING TREE PARTITIONS, AND REPRESENTATION THEORY OF TILING ALGEBRAS , 2019, Glasgow Mathematical Journal.

[7]  A. Polishchuk,et al.  Derived equivalences of gentle algebras via Fukaya categories , 2018, Mathematische Annalen.

[8]  H. Thomas,et al.  On the Combinatorics of Gentle Algebras , 2017, Canadian Journal of Mathematics.

[9]  Sibylle Schroll,et al.  Mapping cones in the bounded derived category of a gentle algebra , 2016, Journal of Algebra.

[10]  Viviana Gubitosi.  m-cluster tilted algebras of type , 2018 .

[11]  Pierre-Guy Plamondon,et al.  A geometric model for the derived category of gentle algebras , 2018, 1801.09659.

[12]  Nathan Broomhead.  Thick subcategories of discrete derived categories , 2016, Advances in Mathematics.

[13]  Viviana Gubitosi.  Derived class of m-cluster tilted algebras of type 𝔸̃ , 2015, Journal of Algebra and Its Applications.

[14]  M. J. Parsons,et al.  Endomorphism algebras for a class of negative Calabi-Yau categories , 2016, 1602.02318.

[15]  Laurent Demonet.  Algebras of partial triangulations , 2016, 1602.01592.

[16]  K. Baur,et al.  A geometric realization of tame categories , 2015, 1502.06489.

[17]  Hermund André Torkildsen.  A Geometric Realization of the m-cluster Category of Affine Type A , 2015 .

[18]  L. Lamberti.  Combinatorial model for the cluster categories of type E , 2014, 1403.0549.

[19]  M. Kontsevich,et al.  Flat surfaces and stability structures , 2014, 1409.8611.

[20]  Sibylle Schroll,et al.  Extensions in Jacobian Algebras and Cluster Categories of Marked Surfaces , 2014, 1408.2074.

[21]  Sibylle Schroll.  Trivial extensions of gentle algebras and Brauer graph algebras , 2014, 1405.6419.

[22]  David Pauksztello,et al.  Torsion pairs in a triangulated category generated by a spherical object , 2014, 1404.4623.

[23]  K. Baur,et al.  Torsion Pairs and Rigid Objects in Tubes , 2011, 1112.6132.

[24]  F. Labourie.  Lectures on Representations of Surface Groups , 2013 .

[25]  K. Baur,et al.  Torsion Pairs and Rigid Objects in Tubes , 2013, Algebras and Representation Theory.

[26]  R. Schiffler,et al.  Algebras from surfaces without punctures , 2011, 1103.4357.

[27]  T. Brustle,et al.  On the cluster category of a marked surface without punctures , 2010, 1005.2422.

[28]  Martin Rubey,et al.  Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type An , 2010, 1010.1184.

[29]  Ibrahim Assem,et al.  Gentle algebras arising from surface triangulations , 2009, 0903.3347.

[30]  Nathan Broomhead.  Dimer Models and Calabi-Yau Algebras , 2009, 0901.4662.

[31]  G. Murphy.  Derived equivalence classification of m-cluster tilted algebras of type An , 2008, 0807.3840.

[32]  Claire Amiot.  Cluster categories for algebras of global dimension 2 and quivers with potential , 2008, 0805.1035.

[33]  D. Labardini-Fragoso,et al.  Quivers with potentials associated to triangulated surfaces , 2008, 0803.1328.

[34]  D. Thurston,et al.  Cluster algebras and triangulated surfaces. Part I: Cluster complexes , 2006, math/0608367.

[35]  K. Baur,et al.  A geometric description of m-cluster categories , 2006, math/0607151.

[36]  C. Geiss,et al.  Classification of discrete derived categories , 2004 .

[37]  R. Schiffler,et al.  Quivers with relations arising from clusters $(A_n$ case) , 2004, math/0401316.

[38]  A. Zimmermann,et al.  Stable endomorphism algebras of modules over special biserial algebras , 2002, math/0208150.

[39]  M. Khovanov,et al.  A Category for the Adjoint Representation , 2000, math/0002060.

[40]  J. Schröer.  Modules without Self-Extensions over Gentle Algebras , 1999 .

[41]  H. Krause.  Maps between tree and band modules , 1991 .

[42]  W. Crawley-Boevey,et al.  Maps between representations of zero-relation algebras , 1989 .

[43]  I. Assem,et al.  Iterated tilted algebras of type $$\tilde {\mathbb{A}}_n $$ , 1987 .

[44]  C. Ringel,et al.  Auslander-reiten sequences with few middle terms and applications to string algebrass , 1987 .

[45]  C. Geiss,et al.  Gentle algebras are Gorenstein , 2022 .