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Cluster algebras and cluster categories associated with triangulated surfaces: an introduction
Claire Amiot1
1 Université Grenoble Alpes, IF, 38000 Grenoble, France
Winter Braids Lecture Notes, Volume 5 (2018), Talk no. 1, 14 p.

These are the notes of a three hours minicourse given at the school Winterbraids VIII, CIRM Luminy in March 2018.

Published online:
DOI: 10.5802/wbln.21
Claire Amiot 1

1 Université Grenoble Alpes, IF, 38000 Grenoble, France 
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Claire Amiot. Cluster algebras and cluster categories associated with triangulated surfaces: an introduction. Winter Braids Lecture Notes, Volume 5 (2018), Talk no. 1, 14 p. doi : 10.5802/wbln.21. https://wbln.centre-mersenne.org/articles/10.5802/wbln.21/

[ABCJP10] Ibrahim Assem; Thomas Brüstle; Gabrielle Charbonneau-Jodoin; Pierre-Guy Plamondon Gentle algebras arising from surface triangulations, Algebra Number Theory, Volume 4 (2010) no. 2, pp. 201-229 | Article | MR: 2592019 | Zbl: 1242.16011

[Ami09] Claire Amiot Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 6, pp. 2525-2590 | Article | Numdam | MR: 2640929 | Zbl: 1239.16011

[Ami11] Claire Amiot On generalized cluster categories, Representation of algebras and related topics (EMS Ser. Congr. Rep.), Eur. Math. Soc., Zürich, 2011, pp. 1-53 | MR: 2764906 | Zbl: 1309.18009

[ASS06] Ibrahim Assem; Daniel Simson; Andrzej Skowroński Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, Volume 65, Cambridge University Press, Cambridge, 2006 (Techniques of representation theory) | MR: 2197389 | Zbl: 1092.16001

[BFZ05] Arkady Berenstein; Sergey Fomin; Andrei Zelevinsky Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J., Volume 126 (2005) no. 1, pp. 1-52 | Article | MR: 2110627 | Zbl: 1135.16013

[BIRS09] A. B. Buan; O. Iyama; I. Reiten; J. Scott Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compos. Math., Volume 145 (2009) no. 4, pp. 1035-1079 | Article | MR: 2521253 | Zbl: 1181.18006

[BMR07] Aslak Bakke Buan; Robert J. Marsh; Idun Reiten Cluster-tilted algebras, Trans. Amer. Math. Soc., Volume 359 (2007) no. 1, pp. 323-332 | Article | MR: 2247893 | Zbl: 1123.16009

[BMR + 06] Aslak Bakke Buan; Robert Marsh; Markus Reineke; Idun Reiten; Gordana Todorov Tilting theory and cluster combinatorics, Adv. Math., Volume 204 (2006) no. 2, pp. 572-618 | Article | MR: 2249625 | Zbl: 1127.16011

[BR87] M. C. R. Butler; Claus Michael Ringel Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra, Volume 15 (1987) no. 1-2, pp. 145-179 | Article | MR: 876976 | Zbl: 0612.16013

[BZ11] Thomas Brüstle; Jie Zhang On the cluster category of a marked surface without punctures, Algebra Number Theory, Volume 5 (2011) no. 4, pp. 529-566 | Article | MR: 2870100 | Zbl: 1250.16013

[BZ13] Thomas Brüstle; Jie Zhang A module-theoretic interpretation of Schiffler’s expansion formula, Comm. Algebra, Volume 41 (2013) no. 1, pp. 260-283 | Article | MR: 3010535 | Zbl: 1271.13043

[CC06] Philippe Caldero; Frédéric Chapoton Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv., Volume 81 (2006) no. 3, pp. 595-616 | Article | MR: 2250855 | Zbl: 1119.16013

[CCS06] Philippe Caldero; Frédéric Chapoton; Ralf Schiffler Quivers with relations and cluster tilted algebras, Algebr. Represent. Theory, Volume 9 (2006) no. 4, pp. 359-376 | Article | MR: 2250652 | Zbl: 1127.16013

[CIKLFP13] Giovanni Cerulli Irelli; Bernhard Keller; Daniel Labardini-Fragoso; Pierre-Guy Plamondon Linear independence of cluster monomials for skew-symmetric cluster algebras, Compos. Math., Volume 149 (2013) no. 10, pp. 1753-1764 | Article | MR: 3123308 | Zbl: 1288.18011

[CS17] Ilke Canakci; Sibylle Schroll Extensions in Jacobian algebras and cluster categories of marked surfaces, Adv. Math., Volume 313 (2017), pp. 1-49 (With an appendix by Claire Amiot) | Article | MR: 3649219 | Zbl: 1401.13064

[FG06] Vladimir Fock; Alexander Goncharov Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. (2006) no. 103, pp. 1-211 | Article | Numdam | Zbl: 1099.14025

[FST08] Sergey Fomin; Michael Shapiro; Dylan Thurston Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math., Volume 201 (2008) no. 1, pp. 83-146 | Article | MR: 2448067 | Zbl: 1263.13023

[FWZ16] Sergey Fomin; Lauren Williams; Andrei Zelevinsky Introduction to Cluster Algebras. Chapters 1-3 (2016) (Preprint https://arxiv.org/abs/1608.05735)

[FWZ17] Sergey Fomin; Lauren Williams; Andrei Zelevinsky Introduction to Cluster Algebras. Chapters 4-5 (2017) (Preprint https://arxiv.org/abs/1707.07190)

[FZ02] Sergey Fomin; Andrei Zelevinsky Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | Article | MR: 1887642 | Zbl: 1021.16017

[Gab72] Peter Gabriel Unzerlegbare Darstellungen. I, Manuscripta Math., Volume 6 (1972), p. 71-103; correction, ibid. 6 (1972), 309 | Article | MR: 332887 | Zbl: 0232.08001

[GHKK18] Mark Gross; Paul Hacking; Sean Keel; Maxim Kontsevich Canonical bases for cluster algebras, J. Amer. Math. Soc., Volume 31 (2018) no. 2, pp. 497-608 | Article | MR: 3758151 | Zbl: 06836101

[GLS06] Christof Geiß; Bernard Leclerc; Jan Schröer Rigid modules over preprojective algebras, Invent. Math., Volume 165 (2006) no. 3, pp. 589-632 | Article | MR: 2242628 | Zbl: 1167.16009

[GLS13] Ch. Geiss; B. Leclerc; J. Schröer Cluster algebras in algebraic Lie theory, Transform. Groups, Volume 18 (2013) no. 1, pp. 149-178 | Article | MR: 3022762 | Zbl: 1278.13027

[Hap88] Dieter Happel Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, Volume 119, Cambridge University Press, Cambridge, 1988, x+208 pages | MR: 935124 | Zbl: 0635.16017

[HL10] David Hernandez; Bernard Leclerc Cluster algebras and quantum affine algebras, Duke Math. J., Volume 154 (2010) no. 2, pp. 265-341 | Article | MR: 2682185 | Zbl: 1284.17010

[IY08] Osamu Iyama; Yuji Yoshino Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math., Volume 172 (2008) no. 1, pp. 117-168 | Article | MR: 2385669 | Zbl: 1140.18007

[Kel] Bernhard Keller https://webusers.imj-prg.fr/ bernhard.keller/quivermutation/

[Kel11a] B. Keller Cluster algebras and cluster categories, Bull. Iranian Math. Soc., Volume 37 (2011) no. 2, pp. 187-234 | MR: 2890585 | Zbl: 1271.13045

[Kel11b] Bernhard Keller Algèbres amassées et applications (d’après Fomin-Zelevinsky, ...), Séminaire Bourbaki. Vol. 2009/2010. Exposés 1012–1026 (Astérisque), Société Mathématique de France, 2011 no. 339 (Exp. No. 1014, vii, 63–90) | MR: 2906350 | Zbl: 1375.13034

[Kel11c] Bernhard Keller Deformed Calabi-Yau completions, J. Reine Angew. Math., Volume 654 (2011), pp. 125-180 (With an appendix by Michel Van den Bergh) | MR: 2795754 | Zbl: 1220.18012

[Kel12] Bernhard Keller Cluster algebras and derived categories, Derived categories in algebraic geometry (EMS Ser. Congr. Rep.), Eur. Math. Soc., Zürich, 2012, pp. 123-183 | Zbl: 1299.13027

[KY11] Bernhard Keller; Dong Yang Derived equivalences from mutations of quivers with potential, Adv. Math., Volume 226 (2011) no. 3, pp. 2118-2168 | Article | MR: 2739775 | Zbl: 1272.13021

[LS15] Kyungyong Lee; Ralf Schiffler Positivity for cluster algebras, Ann. of Math. (2), Volume 182 (2015) no. 1, pp. 73-125 | MR: 3374957 | Zbl: 1350.13024

[MSW11] Gregg Musiker; Ralf Schiffler; Lauren Williams Positivity for cluster algebras from surfaces, Adv. Math., Volume 227 (2011) no. 6, pp. 2241-2308 | Article | MR: 2807089 | Zbl: 1331.13017

[MSW13] Gregg Musiker; Ralf Schiffler; Lauren Williams Bases for cluster algebras from surfaces, Compos. Math., Volume 149 (2013) no. 2, pp. 217-263 | Article | MR: 3020308 | Zbl: 1263.13024

[Mus02] Gregg Musiker Cluster Algebras, Somos sequences and exchange graphs (2002)

[Pal08] Yann Palu Cluster characters for 2-Calabi-Yau triangulated categories, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 6, pp. 2221-2248 | Article | Numdam | MR: 2473635 | Zbl: 1154.16008

[Pla] Pierre-Guy Plamondon Cluster characters, Homological methods, representation theory, and cluster algebras (CRM Short Courses), pp. 101-125 | MR: 3823389 | Zbl: 1419.13043

[Sco06] Joshua S. Scott Grassmannians and cluster algebras, Proc. London Math. Soc. (3), Volume 92 (2006) no. 2, pp. 345-380 | Article | MR: 2205721 | Zbl: 1088.22009

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