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On cluster algebras from unpunctured surfaces with one marked point


Authors: Ilke Canakci, Kyungyong Lee and Ralf Schiffler
Journal: Proc. Amer. Math. Soc. Ser. B 2 (2015), 35-49
MSC (2010): Primary 13F60
DOI: https://doi.org/10.1090/bproc/21
Published electronically: November 13, 2015
MathSciNet review: 3422667
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Abstract: We extend the construction of canonical bases for cluster algebras from unpunctured surfaces to the case where the number of marked points on the boundary is one. We show that the cluster algebra is equal to the upper cluster algebra in this case.


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  • [ACCERV] Murad Alim, Sergio Cecotti, Clay Córdova, Sam Espahbodi, Ashwin Rastogi, and Cumrun Vafa, BPS quivers and spectra of complete $ \mathcal {N}=2$ quantum field theories, Comm. Math. Phys. 323 (2013), no. 3, 1185-1227. MR 3106506, https://doi.org/10.1007/s00220-013-1789-8
  • [BDP] Thomas Brüstle, Grégoire Dupont, and Matthieu Pérotin, On maximal green sequences, Int. Math. Res. Not. IMRN 16 (2014), 4547-4586. MR 3250044
  • [BFZ] Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no. 1, 1-52. MR 2110627 (2005i:16065), https://doi.org/10.1215/S0012-7094-04-12611-9
  • [CS] Ilke Canakci and Ralf Schiffler, Snake graph calculus and cluster algebras from surfaces, J. Algebra 382 (2013), 240-281. MR 3034481, https://doi.org/10.1016/j.jalgebra.2013.02.018
  • [CS2] Ilke Canakci and Ralf Schiffler, Snake graph calculus and cluster algebras from surfaces II: Self-crossing snake graphs, Math. Z. 281 (2015), no. 1-2, 55-102. MR 3384863, https://doi.org/10.1007/s00209-015-1475-y
  • [CS3] Ilke Canakci and Ralf Schiffler, Snake graph calculus and cluster algebras from surfaces III, preprint, arxiv:1506.01742.
  • [FeShTu] Anna Felikson, Michael Shapiro, and Pavel Tumarkin, Skew-symmetric cluster algebras of finite mutation type, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 4, 1135-1180. MR 2928847, https://doi.org/10.4171/JEMS/329
  • [FeShTu2] Anna Felikson, Michael Shapiro, and Pavel Tumarkin, Cluster algebras of finite mutation type via unfoldings, Int. Math. Res. Not. IMRN 8 (2012), 1768-1804. MR 2920830, https://doi.org/10.1093/imrn/rnr072
  • [FeShTu3] Anna Felikson, Michael Shapiro, and Pavel Tumarkin, Cluster algebras and triangulated orbifolds, Adv. Math. 231 (2012), no. 5, 2953-3002. MR 2970470, https://doi.org/10.1016/j.aim.2012.07.032
  • [GLS] C. Geiss, D. Labardini-Fragoso, and J. Schröer, The representation type of Jacobian algebras, arXiv:1308.0478.
  • [FG1] Vladimir Fock and Alexander Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1-211. MR 2233852 (2009k:32011), https://doi.org/10.1007/s10240-006-0039-4
  • [FG2] Vladimir V. Fock and Alexander B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 6, 865-930 (English, with English and French summaries). MR 2567745 (2011f:53202), https://doi.org/10.1007/978-0-8176-4745-2_15
  • [FST] Sergey Fomin, Michael Shapiro, and Dylan Thurston, Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (2008), no. 1, 83-146. MR 2448067 (2010b:57032), https://doi.org/10.1007/s11511-008-0030-7
  • [FT] Sergey Fomin and Dylan Thurston, Cluster algebras and triangulated surfaces. Part II: Lambda lengths, preprint (2008), http://www.math.lsa.umich.edu/ $ \sim $fomin/Papers/cats2.ps
  • [FZ1] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529 (electronic). MR 1887642 (2003f:16050), https://doi.org/10.1090/S0894-0347-01-00385-X
  • [FZ2] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63-121. MR 2004457 (2004m:17011), https://doi.org/10.1007/s00222-003-0302-y
  • [FZ4] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), no. 1, 112-164. MR 2295199 (2008d:16049), https://doi.org/10.1112/S0010437X06002521
  • [GSV] Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster algebras and Weil-Petersson forms, Duke Math. J. 127 (2005), no. 2, 291-311. MR 2130414 (2006d:53103), https://doi.org/10.1215/S0012-7094-04-12723-X
  • [GY] Kenneth R. Goodearl and Milen T. Yakimov, Quantum cluster algebras and quantum nilpotent algebras, Proc. Natl. Acad. Sci. USA 111 (2014), no. 27, 9696-9703. MR 3263301, https://doi.org/10.1073/pnas.1313071111
  • [K] Bernhard Keller, On cluster theory and quantum dilogarithm identities, Representations of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 85-116. MR 2931896, https://doi.org/10.4171/101-1/3
  • [Lad] S. Ladkani, On cluster algebras from once punctured closed surfaces, preprint, arXiv:1310.4454.
  • [LS] Kyungyong Lee and Ralf Schiffler, Positivity for cluster algebras, Ann. of Math. (2) 182 (2015), no. 1, 73-125. MR 3374957, https://doi.org/10.4007/annals.2015.182.1.2
  • [L1] George Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447-498. MR 1035415 (90m:17023), https://doi.org/10.2307/1990961
  • [L2] George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098 (94m:17016)
  • [MM] J. Matherne and G. Muller, Computing upper cluster algebras, to appear in Int. Math. Res. Not., arxiv:1307.0579.
  • [M] Greg Muller, Skein algebras and cluster algebras of marked surfaces, preprint, arXiv:1204.0020.
  • [M2] Greg Muller, Locally acyclic cluster algebras, Adv. Math. 233 (2013), 207-247. MR 2995670, https://doi.org/10.1016/j.aim.2012.10.002
  • [M3] Greg Muller, $ \mathcal {A}=\mathcal {U}$ for locally acyclic cluster algebras, SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 094, 8. MR 3261850, https://doi.org/10.3842/SIGMA.2014.094
  • [M4] Greg Muller, private communication.
  • [MSp] Greg Muller and D. Speyer, Cluster algebras of grassmannians are locally acyclic, preprint, arXiv:1401.5137.
  • [MS] Gregg Musiker and Ralf Schiffler, Cluster expansion formulas and perfect matchings, J. Algebraic Combin. 32 (2010), no. 2, 187-209. MR 2661414 (2011m:13047), https://doi.org/10.1007/s10801-009-0210-3
  • [MSW] Gregg Musiker, Ralf Schiffler, and Lauren Williams, Positivity for cluster algebras from surfaces, Adv. Math. 227 (2011), no. 6, 2241-2308. MR 2807089 (2012f:13052), https://doi.org/10.1016/j.aim.2011.04.018
  • [MSW2] Gregg Musiker, Ralf Schiffler, and Lauren Williams, Bases for cluster algebras from surfaces, Compos. Math. 149 (2013), no. 2, 217-263. MR 3020308, https://doi.org/10.1112/S0010437X12000450
  • [MW] Gregg Musiker and Lauren Williams, Matrix formulae and skein relations for cluster algebras from surfaces, Int. Math. Res. Not. IMRN 13 (2013), 2891-2944. MR 3072996
  • [S2] Ralf Schiffler, A cluster expansion formula ($ A_n$ case), Electron. J. Combin. 15 (2008), no. 1, Research paper 64, 9. MR 2398856 (2009d:13029)
  • [S3] Ralf Schiffler, On cluster algebras arising from unpunctured surfaces. II, Adv. Math. 223 (2010), no. 6, 1885-1923. MR 2601004 (2011c:13048), https://doi.org/10.1016/j.aim.2009.10.015
  • [ST] Ralf Schiffler and Hugh Thomas, On cluster algebras arising from unpunctured surfaces, Int. Math. Res. Not. IMRN 17 (2009), 3160-3189. MR 2534994 (2010h:13040), https://doi.org/10.1093/imrn/rnp047
  • [Se] Ahmet I. Seven, Maximal green sequences of exceptional finite mutation type quivers, SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 089, 5. MR 3261855, https://doi.org/10.3842/SIGMA.2014.089
  • [T] Dylan Paul Thurston, Positive basis for surface skein algebras, Proc. Natl. Acad. Sci. USA 111 (2014), no. 27, 9725-9732. MR 3263305, https://doi.org/10.1073/pnas.1313070111

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Additional Information

Ilke Canakci
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom
Address at time of publication: Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Road, Durham DH1 3LE, United Kingdom
Email: ilke.canakci@durham.ac.uk

Kyungyong Lee
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202 — and — Center for Mathematical Challenges, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea
Address at time of publication: Department of Mathematics, University of Nebraska-Lincoln, 210 Avery Hall, Lincoln, NE 68588-0130, USA
Email: klee24@unl.edu, klee1@kias.re.kr

Ralf Schiffler
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: schiffler@math.uconn.edu

DOI: https://doi.org/10.1090/bproc/21
Keywords: Cluster algebra, upper cluster algebra, unpunctured surface, basis
Received by editor(s): August 23, 2014
Published electronically: November 13, 2015
Additional Notes: The first author was supported by EPSRC grant number EP/K026364/1, UK and the University of Leicester
The second author was supported by Wayne State University, the Korea Institute for Advanced Study, AMS Centennial Fellowship and NSA grant H98230-14-1-0323
The third author was supported by NSF grants DMS-1254567, DMS-1101377 and by the University of Connecticut
Communicated by: Harm Derksen
Article copyright: © Copyright 2015 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0)

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