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Book Review

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MathSciNet review: 3497798
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Book Information:

Author: Robert J. Marsh
Title: Lecture notes on cluster algebras
Additional book information: Zürich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, ii+117 pp., ISBN 978-3-03719-130-9

[1] Takahide Adachi, Osamu Iyama, and Idun Reiten, 𝜏-tilting theory, Compos. Math. 150 (2014), no. 3, 415–452. MR 3187626, https://doi.org/10.1112/S0010437X13007422
[2] 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
[3] I. Assem, T. Brüstle, and R. Schiffler, Cluster-tilted algebras as trivial extensions, Bull. Lond. Math. Soc. 40 (2008), no. 1, 151–162. MR 2409188, https://doi.org/10.1112/blms/bdm107
[4] Arkady Berenstein and Andrei Zelevinsky, Triangular bases in quantum cluster algebras, Int. Math. Res. Not. IMRN 6 (2014), 1651–1688. MR 3180605, https://doi.org/10.1093/imrn/rns268
[5] Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), no. 2, 572–618. MR 2249625, https://doi.org/10.1016/j.aim.2005.06.003
[6] Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), no. 1, 323–332. MR 2247893, https://doi.org/10.1090/S0002-9947-06-03879-7
[7] Philippe Caldero and Frédéric Chapoton, Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv. 81 (2006), no. 3, 595–616. MR 2250855, https://doi.org/10.4171/CMH/65
[8] P. Caldero, F. Chapoton, and R. Schiffler, Quivers with relations arising from clusters (𝐴_{𝑛} case), Trans. Amer. Math. Soc. 358 (2006), no. 3, 1347–1364. MR 2187656, https://doi.org/10.1090/S0002-9947-05-03753-0
[9] Philippe Caldero and Bernhard Keller, From triangulated categories to cluster algebras, Invent. Math. 172 (2008), no. 1, 169–211. MR 2385670, https://doi.org/10.1007/s00222-008-0111-4
[10] I. Canakci, K. Lee and R. Schiffler, On cluster algebras for surfaces without punctures and one marked point, preprint, arXiv:1407.5060.
[11] I. Canakci and R. Schiffler, Snake graph calculus and cluster algebras from surfaces, J. Algebra, 382, (2013) 240-281.
[12] I. Canakci and R. Schiffler, Snake graph calculus and cluster algebras from surfaces II: Self-crossing snake graphs, Math. Z. 281 (2015), no. 1, 55-102.
[13] I. Canakci and R. Schiffler, Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings, preprint, arXiv:1506.01742.
[14] Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky, Quivers with potentials and their representations II: applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), no. 3, 749–790. MR 2629987, https://doi.org/10.1090/S0894-0347-10-00662-4
[15] 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
[16] 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
[17] 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
[18] S. Fomin and D. Thurston, Cluster algebras and triangulated surfaces. Part II: Lambda lengths, preprint, arXiv:1210.5569.
[19] 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
[20] 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
[21] Christof Geiß, Bernard Leclerc, and Jan Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589–632. MR 2242628, https://doi.org/10.1007/s00222-006-0507-y
[22] Christof Geiss, Bernard Leclerc, and Jan Schröer, Generic bases for cluster algebras and the Chamber ansatz, J. Amer. Math. Soc. 25 (2012), no. 1, 21–76. MR 2833478, https://doi.org/10.1090/S0894-0347-2011-00715-7
[23] 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
[24] Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster algebras and Poisson geometry, Mathematical Surveys and Monographs, vol. 167, American Mathematical Society, Providence, RI, 2010. MR 2683456
[25] M. Gross, P. Hacking, S. Keel and M. Kontsevich, Canonical bases for cluster algebras, preprint arXiv:1411.1394.
[26] 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
[27] Bernhard Keller, Calabi-Yau triangulated categories, Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 467–489. MR 2484733, https://doi.org/10.4171/062-1/11
[28] 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
[29] Kyungyong Lee, Li Li, and Andrei Zelevinsky, Greedy elements in rank 2 cluster algebras, Selecta Math. (N.S.) 20 (2014), no. 1, 57–82. MR 3147413, https://doi.org/10.1007/s00029-012-0115-1
[30] K. Lee and R. Schiffler, Positivity for cluster algebras, Annals of Math. 182 (1), (2015) 73-125.
[31] 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
[32] 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
[33] 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
[34] 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, https://doi.org/10.1093/imrn/rns118
[35] 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
[36] 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
[37] 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

Review Information:

Reviewer: Ralf Schiffler
Affiliation: Department of Mathematics, University of Connecticut,
Email: schiffler@math.uconn.edu

Journal: Bull. Amer. Math. Soc. 53 (2016), 325-330
MSC (2010): Primary 13F60
DOI: https://doi.org/10.1090/bull/1514
Published electronically: August 26, 2015
Additional Notes: The reviewer was supported by NSF-CAREER grant DMS-1254567.
Review copyright: © Copyright 2015 American Mathematical Society