Cluster algebras: an introduction

Williams, Lauren

doi:10.1090/S0273-0979-2013-01417-4

Cluster algebras: an introduction

Author: Lauren K. Williams
Journal: Bull. Amer. Math. Soc. 51 (2014), 1-26
MSC (2010): Primary 13F60, 30F60, 82B23, 05E45
DOI: https://doi.org/10.1090/S0273-0979-2013-01417-4
Published electronically: June 10, 2013
MathSciNet review: 3119820
Full-text PDF Free Access

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Abstract: Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced by Fomin and Zelevinsky in 2000 in the context of Lie theory, but have since appeared in many other contexts, from Poisson geometry to triangulations of surfaces and Teichmüller theory. In this expository paper we give an introduction to cluster algebras, and illustrate how this framework naturally arises in Teichmüller theory. We then sketch how the theory of cluster algebras led to a proof of the Zamolodchikov periodicity conjecture in mathematical physics.

[1] Claire Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2525–2590 (English, with English and French summaries). MR 2640929
[2] Philippe Caldero and Markus Reineke, On the quiver Grassmannian in the acyclic case, J. Pure Appl. Algebra 212 (2008), no. 11, 2369–2380. MR 2440252, https://doi.org/10.1016/j.jpaa.2008.03.025
[3] Giovanni Cerulli Irelli, Bernhard Keller, Daniel Labardini-Fragoso, and Pierre-Guy Plamondon, Linear independence of cluster monomials for skew-symmetric cluster algebras, ArXiv Mathematics e-prints (2012), arXiv:1203.1307.
[4] Frédéric Chapoton, Sergey Fomin, and Andrei Zelevinsky, Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), no. 4, 537–566. Dedicated to Robert V. Moody. MR 1941227, https://doi.org/10.4153/CMB-2002-054-1
[5] Philippe Di Francesco and Rinat Kedem, 𝑄-systems, heaps, paths and cluster positivity, Comm. Math. Phys. 293 (2010), no. 3, 727–802. MR 2566162, https://doi.org/10.1007/s00220-009-0947-5
[6] 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
[7] 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
[8] 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, https://doi.org/10.1007/s10240-006-0039-4
[9] Vladimir V. Fock and Alexander B. Goncharov, Dual Teichmüller and lamination spaces, Handbook of Teichmüller theory. Vol. I, IRMA Lect. Math. Theor. Phys., vol. 11, Eur. Math. Soc., Zürich, 2007, pp. 647–684. MR 2349682, https://doi.org/10.4171/029-1/16
[10] 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, https://doi.org/10.24033/asens.2112
[11] 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, https://doi.org/10.1007/s11511-008-0030-7
[12] Sergey Fomin and Dylan Thurston, Cluster algebras and triangulated surfaces. Part II: lambda lengths, ArXiv Mathematics e-prints (2012), arXiv:1210.5569.
[13] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. MR 1887642, https://doi.org/10.1090/S0894-0347-01-00385-X
[14] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63–121. MR 2004457, https://doi.org/10.1007/s00222-003-0302-y
[15] Sergey Fomin and Andrei Zelevinsky, 𝑌-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977–1018. MR 2031858, https://doi.org/10.4007/annals.2003.158.977
[16] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), no. 1, 112–164. MR 2295199, https://doi.org/10.1112/S0010437X06002521
[17] Edward Frenkel and András Szenes, Thermodynamic Bethe ansatz and dilogarithm identities. I, Math. Res. Lett. 2 (1995), no. 6, 677–693. MR 1362962, https://doi.org/10.4310/MRL.1995.v2.n6.a2
[18] Christof Geiss, Bernard Leclerc, and Jan Schröer, Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 3, 825–876 (English, with English and French summaries). MR 2427512
[19] Christof Geiss, Bernard Leclerc, and Jan Schröer, Preprojective algebras and cluster algebras, Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 253–283. MR 2484728, https://doi.org/10.4171/062-1/6
[20] Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster algebras and Weil-Petersson forms, Duke Math. J. 127 (2005), no. 2, 291–311. MR 2130414, https://doi.org/10.1215/S0012-7094-04-12723-X
[21] F. Gliozzi and R. Tateo, Thermodynamic Bethe ansatz and three-fold triangulations, Internat. J. Modern Phys. A 11 (1996), no. 22, 4051–4064. MR 1403679, https://doi.org/10.1142/S0217751X96001905
[22] John L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), no. 1, 157–176. MR 830043, https://doi.org/10.1007/BF01388737
[23] Allen Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991), no. 2, 189–194. MR 1123262, https://doi.org/10.1016/0166-8641(91)90050-V
[24] David Hernandez and Bernard Leclerc, Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2010), no. 2, 265–341. MR 2682185, https://doi.org/10.1215/00127094-2010-040
[25] Rei Inoue, Osama Iyama, Bernhard Keller, Atsuo Kuniba, and Tomoki Nakanishi, Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras I: Type ., ArXiv Mathematics e-prints (2010), arXiv:1001.1880, to appear in Publ. RIMS.
[26] -, Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Types , , and ., ArXiv Mathematics e-prints (2010), arXiv:1001.1881, to appear in Publ. RIMS.
[27] Osamu Iyama and Yuji Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), no. 1, 117–168. MR 2385669, https://doi.org/10.1007/s00222-007-0096-4
[28] M. Kashiwara, On crystal bases of the 𝑄-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, https://doi.org/10.1215/S0012-7094-91-06321-0
[29] Bernhard Keller, Cluster algebras, quiver representations and triangulated categories, Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 76–160. MR 2681708
[30] Bernhard Keller, The periodicity conjecture for pairs of Dynkin diagrams, Ann. of Math. (2) 177 (2013), no. 1, 111–170. MR 2999039, https://doi.org/10.4007/annals.2013.177.1.3
[31] Yoshiyuki Kimura and Fan Qin, Graded quiver varieties, quantum cluster algebras, and dual canonical basis, ArXiv Mathematics e-prints (2012), arXiv:1205.2066.
[32] A. Kuniba and T. Nakanishi, Spectra in conformal field theories from the Rogers dilogarithm, Modern Phys. Lett. A 7 (1992), no. 37, 3487–3494. MR 1192727, https://doi.org/10.1142/S0217732392002895
[33] Atsuo Kuniba, Tomoki Nakanishi, and Junji Suzuki, Functional relations in solvable lattice models. I. Functional relations and representation theory, Internat. J. Modern Phys. A 9 (1994), no. 30, 5215–5266. MR 1304818, https://doi.org/10.1142/S0217751X94002119
[34] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, https://doi.org/10.1090/S0894-0347-1990-1035415-6
[35] Robert Marsh, Markus Reineke, and Andrei Zelevinsky, Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), no. 10, 4171–4186. MR 1990581, https://doi.org/10.1090/S0002-9947-03-03320-8
[36] Lee Mosher, Tiling the projective foliation space of a punctured surface, Trans. Amer. Math. Soc. 306 (1988), no. 1, 1–70. MR 927683, https://doi.org/10.1090/S0002-9947-1988-0927683-0
[37] Gregg Musiker, Ralf Schiffler, and Lauren Williams, Positivity for cluster algebras from surfaces, Adv. Math. 227 (2011), no. 6, 2241–2308. MR 2807089, https://doi.org/10.1016/j.aim.2011.04.018
[38] Hiraku Nakajima, Quiver varieties and cluster algebras, Kyoto J. Math. 51 (2011), no. 1, 71–126. MR 2784748, https://doi.org/10.1215/0023608X-2010-021
[39] R. C. Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987), no. 2, 299–339. MR 919235
[40] R. C. Penner, Decorated Teichmüller theory of bordered surfaces, Comm. Anal. Geom. 12 (2004), no. 4, 793–820. MR 2104076
[41] Pierre-Guy Plamondon, Cluster algebras via cluster categories with infinite-dimensional morphism spaces, Compos. Math. 147 (2011), no. 6, 1921–1954. MR 2862067, https://doi.org/10.1112/S0010437X11005483
[42] F. Ravanini, A. Valleriani, and R. Tateo, Dynkin TBAs, Internat. J. Modern Phys. A 8 (1993), no. 10, 1707–1727. MR 1216231, https://doi.org/10.1142/S0217751X93000709
[43] Joshua S. Scott, Grassmannians and cluster algebras, Proc. London Math. Soc. (3) 92 (2006), no. 2, 345–380. MR 2205721, https://doi.org/10.1112/S0024611505015571
[44] András Szenes, Periodicity of Y-systems and flat connections, Lett. Math. Phys. 89 (2009), no. 3, 217–230. MR 2551180, https://doi.org/10.1007/s11005-009-0332-5
[45] William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, https://doi.org/10.1090/S0273-0979-1988-15685-6
[46] William P. Thurston, The geometry and topology of three-manifolds, Princeton University notes, 1980.
[47] Alexandre Yu. Volkov, On the periodicity conjecture for 𝑌-systems, Comm. Math. Phys. 276 (2007), no. 2, 509–517. MR 2346398, https://doi.org/10.1007/s00220-007-0343-y
[48] Al. B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless 𝐴𝐷𝐸 scattering theories, Phys. Lett. B 253 (1991), no. 3-4, 391–394. MR 1092210, https://doi.org/10.1016/0370-2693(91)91737-G