Cluster categories for marked surfaces: punctured case

We study cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of $\operatorname{Ext}^{1}$ as intersection numbers of tagged curves and Auslander–Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.

[1]  Y. Qiu,et al.  Decorated marked surfaces II: Intersection numbers and dimensions of Homs , 2014, Transactions of the American Mathematical Society.

[2]  Matthew R. Mills.  Maximal green sequences for quivers of finite mutation type , 2016, 1606.03799.

[3]  C. Geiss,et al.  The representation type of Jacobian algebras , 2013, 1308.0478.

[4]  R. Schiffler,et al.  Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs , 2015, Mathematische Zeitschrift.

[5]  Y. Qiu,et al.  Tagged mapping class groups: Auslander–Reiten translation , 2012, 1212.0007.

[6]  A. King,et al.  Exchange graphs and Ext quivers , 2011, 1109.2924.

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

[8]  Y. Qiu.  Decorated marked surfaces: spherical twists versus braid twists , 2014, 1407.0806.

[9]  Ralf Schiffler,et al.  Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs , 2014, 1407.0500.

[10]  Yann Palu,et al.  Coloured quivers for rigid objects and partial triangulations: the unpunctured case , 2010, 1012.5790.

[11]  D. Labardini-Fragoso.  On triangulations, quivers with potentials and mutations , 2013, 1302.1936.

[12]  I. Reiten,et al.  $\tau $-tilting theory , 2012, Compositio Mathematica.

[13]  R. Schiffler,et al.  Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings , 2012, 1506.01742.

[14]  Yu Zhou,et al.  Cotorsion pairs in the cluster category of a marked surface , 2012, 1205.1504.

[15]  B. Keller.  Cluster algebras and derived categories , 2012, 1202.4161.

[16]  L. Williams,et al.  Bases for cluster algebras from surfaces , 2011, Compositio Mathematica.

[17]  Giovanni Cerulli Irelli,et al.  Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials , 2011, Compositio Mathematica.

[18]  A. King,et al.  EXCHANGE GRAPHS OF ACYCLIC CALABI-YAU CATEGORIES , 2012 .

[19]  C. Vafa,et al.  BPS Quivers and Spectra of Complete N=2 Quantum Field Theories , 2011 .

[20]  C. Vafa,et al.  BPS Quivers and Spectra of Complete $${\mathcal{N} = 2}$$N=2 Quantum Field Theories , 2011, 1109.4941.

[21]  L. Williams,et al.  Matrix formulae and skein relations for cluster algebras from surfaces , 2011, 1108.3382.

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

[23]  Pierre-Guy Plamondon.  Cluster algebras via cluster categories with infinite-dimensional morphism spaces , 2010, Compositio Mathematica.

[24]  H. Thomas,et al.  From m-clusters to m-noncrossing partitions via exceptional sequences , 2010, 1007.0928.

[25]  J. Lott,et al.  The index of a transverse Dirac-type operator: the case of abelian Molino sheaf , 2010, 1005.0161.

[26]  D. Labardini-Fragoso.  Quivers with potentials associated to triangulated surfaces, Part II: Arc representations , 2009, 0909.4100.

[27]  B. Keller,et al.  Deformed Calabi–Yau completions , 2009, 0908.3499.

[28]  B. Keller,et al.  Derived equivalences from mutations of quivers with potential , 2009, 0906.0761.

[29]  L. Williams,et al.  Positivity for cluster algebras from surfaces , 2009, 0906.0748.

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

[31]  M. Shapiro,et al.  Skew-symmetric cluster algebras of finite mutation type , 2008, 0811.1703.

[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]  Y. Yoshino,et al.  Mutation in triangulated categories and rigid Cohen–Macaulay modules , 2006, math/0607736.

[35]  Yann Palu.  Cluster characters for 2-Calabi–Yau triangulated categories , 2008 .

[36]  J. Weyman,et al.  Quivers with potentials and their representations I: Mutations , 2007, 0704.0649.

[37]  Yann Palu.  Cluster characters for triangulated 2-Calabi--Yau categories , 2007, math/0703540.

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

[39]  R. Schiffler.  A geometric model for cluster categories of type Dn , 2006, math/0608264.

[40]  S. Koenig,et al.  From triangulated categories to abelian categories: cluster tilting in a general framework , 2006, math/0605100.

[41]  V. Nicholson.  Twisted Surfaces , 2006 .

[42]  O. Iyama.  Mutations in triangulated categories and rigid Cohen-Macaulay modules , 2006 .

[43]  I. Reiten,et al.  Cluster-tilted algebras are Gorenstein and stably Calabi–Yau , 2005, math/0512471.

[44]  I. Reiten,et al.  Tilting theory and cluster combinatorics , 2004, math/0402054.

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

[46]  A. Goncharov,et al.  Cluster ensembles, quantization and the dilogarithm , 2003, math/0311245.

[47]  A. Goncharov,et al.  Moduli spaces of local systems and higher Teichmüller theory , 2003, math/0311149.

[48]  S. Fomin,et al.  Cluster algebras I: Foundations , 2001, math/0104151.

[49]  B. Deng.  On a problem of Nazarova and Roiter , 2000 .

[50]  C. Geiss.  Maps Between Representations of Clans , 1999 .

[51]  J. A. Peña,et al.  Auslander-Reiten components for clans , 1999 .

[52]  W. Crawley-Boevey.  Functorial Filtrations II: Clans and the Gelfand Problem , 1989 .