Cluster algebras and cluster categories associated with triangulated surfaces: an introduction

Amiot, Claire ¹

¹ Université Grenoble Alpes, IF, 38000 Grenoble, France

Winter Braids VIII (Marseille, 2018), Winter Braids Lecture Notes (2018), Exposé no. 1, 14 p.

Résumé

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

DOI : 10.5802/wbln.21

Affiliations des auteurs :

Amiot, Claire ¹

¹ Université Grenoble Alpes, IF, 38000 Grenoble, France

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Amiot, Claire. Cluster algebras and cluster categories associated with triangulated surfaces: an introduction, dans Winter Braids VIII (Marseille, 2018), Winter Braids Lecture Notes (2018), Exposé no. 1, 14 p. doi : 10.5802/wbln.21. http://www.numdam.org/articles/10.5802/wbln.21/

Bibliographie
Cité par

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[Kel11b] Keller, Bernhard 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 | Zbl

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

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

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

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

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

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

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

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

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

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

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