Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-06T23:25:59.922Z Has data issue: false hasContentIssue false

3 - From Frieze Patterns to Cluster Categories

Published online by Cambridge University Press:  aN Invalid Date NaN

Edited by
David Jordan ,
Nadia Mazza  and
Sibylle Schroll
David Jordan
Affiliation:
University of Edinburgh
Nadia Mazza
Affiliation:
Lancaster University
Sibylle Schroll
Affiliation:
Universität zu Köln
Get access

Summary

Motivated by Conway and Coxeter’s combinatorial results concerning frieze patterns, we sketch an introduction to the theory of cluster algebras and cluster categories for acyclic quivers. The goal is to show how these more abstract theories provide a conceptual explanation for phenomena concerning friezes, such as integrality and periodicity.

Keywords

cluster algebrascluster categoriesacyclic quiversfrieze patterns
Type
Chapter
Information
Modern Trends in Algebra and Representation Theory , pp. 109 - 145
Publisher: Cambridge University Press
Print publication year: 2023

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adachi, T., Iyama, O. and Reiten, I. τ-tilting theory. Compos. Math., 150(3):415– 452, 2014.CrossRefGoogle Scholar
[2]Amiot, C. Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier, 59(6):2525–2590, 2009.CrossRefGoogle Scholar
[3]Angeleri Hu¨gel, L. 2006. An introduction to Auslander–Reiten theory. http://profs.sci.univr.it/ angeleri/trieste.pdf.Google Scholar
[4]Assem, I., Simson, D., and Skowron´ski, A. 2006. Elements of the Representation Theory of Associative Algebras. Vol. 1. London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge. Techniques of representation theory.CrossRefGoogle Scholar
[5]August, J. 2020. The tilting theory of contraction algebras. Adv. Math., 374, Article 107372.Google Scholar
[6]Auslander, M. and Reiten, I. 1975. Representation theory of Artin algebras. III. Almost split sequences. Comm. Algebra, 3, 239–294.Google Scholar
[7]Auslander, M., Reiten, I. and Smalø, S. O. 1997. Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge. Corrected reprint of the 1995 original.Google Scholar
[8]Baur, K., C¸ anakc¸ı, I˙., Jacobsen, K. M., Kulkarni, M. C. and Todorov, G. 2020. Infinite friezes and triangulations of annuli. Preprint. (arXiv:2007.09411 [math.CO])Google Scholar
[9]Baur, K., Faber, E., Gratz, S., Serhiyenko, K. and Todorov, G. 2021. Friezes satisfying higher SLk -determinants. Algebra Number Theory, 15(1), 29–68.Google Scholar
[10]Baur, K., King, A. D. and Marsh, B. R. 2016. Dimer models and cluster categories of Grassmannians. Proc. Lond. Math. Soc. (3), 113(2), 213–260.Google Scholar
[11]Baur, K., Parsons, M. J. and Tschabold, M. 2016. Infinite friezes. European J. Combin., 54, 220–237.Google Scholar
[12]Bazier-Matte, V., Douville, G., Mousavand, K., Thomas, H. and Yıldırım, E. 2018. ABHY associahedra and Newton polytopes of F-polynomials for finite type cluster algebras.Google Scholar
[13]Bennett-Tennenhaus, R. and Shah, A. 2021. Transport of structure in higher homological algebra. J. Algebra, 574, 514–549.Google Scholar
[14]Berenstein, A., Fomin, S. and Zelevinsky, A. 2005. Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J., 126(1), 1–52.Google Scholar
[15]Bessenrodt, C., Holm, T. and Jørgensen, P. 2014. Generalized frieze pattern determinants and higher angulations of polygons. J. Combin. Theory Ser. A, 123, 30–42.Google Scholar
[16]Bridgeland, T. 2017. Scattering diagrams, Hall algebras and stability conditions. Algebr. Geom., 4(5), 523–561.Google Scholar
[17]Bru¨stle, T., Smith, D. and Treffinger, H. 2019. Wall and chamber structure for finite-dimensional algebras. Adv. Math., 354, Article 106746.Google Scholar
[18]Buan, A. B., Iyama, O., Reiten, I. and Scott, J. 2009. Cluster structures for 2-Calabi–Yau categories and unipotent groups. Compos. Math., 145(4), 1035–1079.Google Scholar
[19]Buan, A. B., Marsh, B. R., Reineke, M., Reiten, I. and Todorov, G. 2006. Tilting theory and cluster combinatorics. Adv. Math., 204(2), 572–618.Google Scholar
[20]Buan, A. B., Marsh, B. R. and Reiten, I. 2007. Cluster-tilted algebras. Trans. Amer. Math. Soc., 359(1), 323–332.Google Scholar
[21]Buan, A. B., Marsh, B. R., Reiten, I. and Todorov, G. 2007. Clusters and seeds in acyclic cluster algebras. Proc. Amer. Math. Soc., 135(10), 3049–3060.Google Scholar
[22]Caldero, P. and Chapoton, F. 2006. Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv., 81(3), 595–616.Google Scholar
[23]Caldero, P., Chapoton, F. and Schiffler, R. 2006. Quivers with relations and cluster tilted algebras. Algebr. Represent. Theory, 9(4), 359–376.Google Scholar
[24]Caldero, P., Chapoton, F. and Schiffler, R. 2006. Quivers with relations arising from clusters (An case). Trans. Amer. Math. Soc., 358(3), 1347–1364.Google Scholar
[25]Caldero, P. and Keller, B. 2006. From triangulated categories to cluster algebras. II. Ann. Sci. E´cole Norm. Sup. (4), 39(6), 983–1009.Google Scholar
[26]C¸ anakc¸ı, I˙. and Jørgensen, P. 2020. Friezes, weak friezes, and T-paths. Adv. in Appl. Math., 131, Paper No. 102253.Google Scholar
[27]Cheung, M. W., Gross, M., Muller, G., Musiker, G., Rupel, D., Stella, S. and Williams, H. 2017. The greedy basis equals the theta basis: a rank two haiku. J. Combin. Theory Ser. A, 145, 150–171.Google Scholar
[28]Conway, J. H. and Coxeter, H. S. M. 1973. Triangulated polygons and frieze patterns. Math. Gaz., 57(400), 87–94.Google Scholar
[29]Conway, J. H. and Coxeter, H. S. M. 1973. Triangulated polygons and frieze patterns. Math. Gaz., 57(401), 175–183.Google Scholar
[30]Cordes, C. M. and Roselle, D. P. 1972. Generalized frieze patterns. Duke Math. J., 39, 637–648.Google Scholar
[31]Coxeter, H. S. M. 1971. Frieze patterns. Acta Arith., 18, 297–310.Google Scholar
[32]Cuntz, M., Holm, T. and Jørgensen, P. 2020. Frieze patterns with coefficients. Forum Math. Sigma, 8, Paper No. e17.Google Scholar
[33]Derksen, H., Weyman, J. and Zelevinsky, A. 2008. Quivers with potentials and their representations. I. Mutations. Selecta Math. (N.S.), 14(1), 59–119.Google Scholar
[34]Fock, V. V. and Goncharov, A. B. 2009. Cluster ensembles, quantization and the dilogarithm. Ann. Sci. E´c. Norm. Supe´r. (4), 42(6), 865–930.Google Scholar
[35]Fomin, S. Cluster algebras portal. www.math.lsa.umich.edu/ fomin/cluster.html.Google Scholar
[36]Fomin, S., Shapiro, M. and Thurston, D. 2008. Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math., 201(1), 83–146.Google Scholar
[37]Fomin, S. and Thurston, D. 2018. Cluster algebras and triangulated surfaces Part II: Lambda lengths. Mem. Amer. Math. Soc., 255(1223).Google Scholar
[38]Fomin, S., Williams, L. and Zelevinsky, A. 2016. Introduction to Cluster Algebras. Chapters 1–3. Preprint. (arXiv:1608.05735 [math.CO])Google Scholar
[39]Fomin, S., Williams, L. and Zelevinsky, A. 2017. Introduction to Cluster Algebras. Chapters 4–5. Preprint. (arXiv:1707.07190 [math.CO])Google Scholar
[40]Fomin, S., Williams, L. and Zelevinsky, A. 2020. Introduction to Cluster Algebras. Chapter 6. Preprint. (arXiv:2008.09189 [math.AC])Google Scholar
[41]Fomin, S. and Zelevinsky, A. 2002. Cluster algebras. I. Foundations. J. Amer. Math. Soc., 15(2), 497–529.Google Scholar
[42]Fomin, S. and Zelevinsky, A. 2003. Cluster algebras. II. Finite type classification. Invent. Math., 154(1), 63–121.Google Scholar
[43]Fomin, S. and Zelevinsky, A. 2007. Cluster algebras. IV. Coefficients. Compos. Math., 143(1), 112–164.Google Scholar
[44]Fontaine, B. and Plamondon, P.-G. 2016. Counting friezes in type Dn. J. Algebraic Combin., 44(2), 433–445.Google Scholar
[45]Fordy, A. P. and Hone, A. 2014. Discrete integrable systems and Poisson algebras from cluster maps. Comm. Math. Phys., 325(2), 527–584.Google Scholar
[46]Franco, S. 2012. Bipartite field theories: from D-brane probes to scattering amplitudes. J. High Energy Phys., JHEP11(2012)141.Google Scholar
[47]Fulton, W. and Harris, J. 1991. Representation Theory: A First Course. Graduate Texts in Mathematics, vol. 129. Springer-Verlag, New York.Google Scholar
[48]Gabriel, P. 1972. Unzerlegbare Darstellungen. I. Manuscripta Math., 6, 71–103.Google Scholar
[49]Geiß, C., Leclerc, B. and Schro¨er, J. 2008. Partial flag varieties and preprojective algebras. Ann. Inst. Fourier (Grenoble), 58(3), 825–876.Google Scholar
[50]Geiß, C., Leclerc, B. and Schro¨er, J. 2010. Cluster algebra structures and semi-canonical bases for unipotent groups. Preprint. (arXiv:math/0703039 [math.RT])Google Scholar
[51]Gekhtman, M., Shapiro, M. and Vainshtein, A. 2010. Cluster algebras and Poisson geometry. Mathematical Surveys and Monographs, no. 167. Providence, RI: American Mathematical Society.Google Scholar
[52]Gelfand, S. I. and Manin, Y. I. 2003. Methods of homological algebra. Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[53]Gross, M., Hacking, P. and Keel, S. 2015. Birational geometry of cluster algebras. Algebr. Geom., 2(2), 137–175.Google Scholar
[54]Gross, M., Hacking, P., Keel, S. and Kontsevich, M. 2018. Canonical bases for cluster algebras. J. Amer. Math. Soc., 31(2), 497–608.Google Scholar
[55]Gunawan, E., Musiker, G. and Vogel, H. 2019. Cluster algebraic interpretation of infinite friezes. European J. Combin., 81, 22–57.Google Scholar
[56]Happel, D. 1987. On the derived category of a finite-dimensional algebra. Comment. Math. Helv., 62(3), 339–389.Google Scholar
[57]Happel, D. 1988. Triangulated categories in the representation theory of finite-dimensional algebras. London Math. Soc. Lecture Note Ser., no. 119. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
[58]Hernandez, D. and Leclerc, B. 2010. Cluster algebras and quantum affine algebras. Duke Math. J., 154(2), 265–341.Google Scholar
[59]Holm, T. and Jørgensen, P. 2015. Generalized friezes and a modified Caldero– Chapoton map depending on a rigid object. Nagoya Math. J., 218, 101–124.Google Scholar
[60]Ingalls, C. and Thomas, H. 2009. Noncrossing partitions and representations of quivers. Compos. Math., 145(6), 1533–1562.Google Scholar
[61]Iyama, O. 2007. Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories. Adv. Math., 210(1), 22–50.Google Scholar
[62]Iyama, O. and Yoshino, Y. 2008. Mutation in triangulated categories and rigid Cohen–Macaulay modules. Invent. Math., 172(1), 117–168.Google Scholar
[63]Jensen, B. T., King, A. D. and Su, X. 2016. A categorification of Grassmannian cluster algebras. Proc. Lond. Math. Soc. (3), 113(2), 185–212.Google Scholar
[64]Kang, S.-J., Kashiwara, M., Kim, M. and Oh, S.-j. 2018. Monoidal categorification of cluster algebras. J. Amer. Math. Soc., 31(2), 349–426.Google Scholar
[65]Keller, B. Quiver mutation in Java. www.math.jussieu.fr/ keller/quiver-mutation/Google Scholar
[66]Keller, B. 1996. Derived categories and their uses. Pages 671–701 of: Handbook of algebra, Vol. 1. Elsevier/North-Holland, Amsterdam.Google Scholar
[67]Keller, B. 2005. On triangulated orbit categories. Doc. Math., 10, 551–581.Google Scholar
[68]Keller, B. 2008. Calabi–Yau triangulated categories. Pages 467–489 of: Trends in representation theory of algebras and related topics. EMS Ser. Congr. Rep. Eur. Math. Soc., Zu¨rich.CrossRefGoogle Scholar
[69]Keller, B. 2010. Cluster algebras, quiver representations and triangulated categories. Pages 76–160 of: Triangulated categories. London Math. Soc. Lecture Note Ser., no. 375. Cambridge: Cambridge Univ. Press.CrossRefGoogle Scholar
[70]Keller, B. 2013. The periodicity conjecture for pairs of Dynkin diagrams. Ann. of Math. (2), 177(1), 111–170.Google Scholar
[71]Labardini-Fragoso, D. 2009. Quivers with potentials associated to triangulated surfaces. Proc. Lond. Math. Soc. (3), 98(3), 797–839.Google Scholar
[72]Marsh, B. R. 2013. Lecture notes on cluster algebras. Zu¨rich Lectures in Advanced Mathematics. Zu¨rich: European Mathematical Society.Google Scholar
[73]Morier-Genoud, S. 2012. Arithmetics of 2-friezes. J. Algebraic Combin., 36(4), 515–539.Google Scholar
[74]Morier-Genoud, S. 2015. Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics. Bull. Lond. Math. Soc., 47(6), 895–938.Google Scholar
[75]Morier-Genoud, S. 2019. Symplectic frieze patterns. SIGMA Symmetry Integrability Geom. Methods Appl., 15, Paper No. 089.Google Scholar
[76]Plamondon, P.-G. 2018. Cluster characters. Pages 101–125 of: Homological methods, representation theory, and cluster algebras. CRM Short Courses. Springer, Cham.CrossRefGoogle Scholar
[77]Pressland, M. 2017. Internally Calabi–Yau algebras and cluster-tilting objects. Math. Z., 287(1-2), 555–585.Google Scholar
[78]Pressland, M. 2022. Calabi–Yau properties of Postnikov diagrams. Forum Math. Sigma, 10, Paper No. e56.Google Scholar
[79]Propp, J. 2020. The combinatorics of frieze patterns and Markoff numbers. Integers, 20, Paper No. A12.Google Scholar
[80]Qin, F. 2019. Bases for upper cluster algebras and tropical points. Preprint. (arXiv:1902.09507 [math.RT])Google Scholar
[81]Schiffler, R. 2014. Quiver representations. CMS Books in Mathematics/Ouvrages de Mathe´matiques de la SMC. Springer, Cham.CrossRefGoogle Scholar
[82]Scott, J. S. 2006. Grassmannians and cluster algebras. Proc. London Math. Soc. (3), 92(2), 345–380.Google Scholar
[83]Shah, A. 2019. Partial cluster-tilted algebras via twin cotorsion pairs, quasiabelian categories and Auslander–Reiten theory. Ph.D. thesis, University of Leeds.Google Scholar
[84]Verdier, J.-L. 1977. Cate´gories de´rive´es: quelques re´sultats (e´tat 0). Pages 262–311 of: Cohomologie e´tale. Lecture Notes in Math., vol. 569. Springer, Berlin.Google Scholar
[85]Verdier, J.-L. 1996. Des cate´gories de´rive´es des cate´gories abe´liennes. Aste´risque. With a preface by Luc Illusie, edited and with a note by Georges Maltsiniotis.Google Scholar
[86]Wemyss, M. 2018. Flops and clusters in the homological minimal model programme. Invent. Math., 211(2), 435–521.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×