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2018 | OriginalPaper | Buchkapitel

Cluster Algebras from Surfaces

Lecture Notes for the CIMPA School Mar del Plata, March 2016

verfasst von : Ralf Schiffler

Erschienen in: Homological Methods, Representation Theory, and Cluster Algebras

Verlag: Springer International Publishing

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Abstract

Cluster algebras were introduced by Fomin and Zelevinsky [17] in 2002. Their original motivation was coming from canonical bases in Lie Theory. Today cluster algebras are connected to various fields of mathematics, including

Combinatorics (polyhedra, frieze patterns, green sequences, snake graphs, T-paths, dimer models, triangulations of surfaces)
Representation theory of finite dimensional algebras (cluster categories, cluster-tilted algebras, preprojective algebras, tilting theory, 2-Calabi–Yau categories, invariant theory)
Poisson geometry and algebraic geometry (cluster varieties, Grassmannians, stability conditions, scattering diagrams, Poisson structures on \({{\mathrm{SL}}}(n)\))
Teichmüller theory (lambda-lengths, Penner coordinates, cluster varieties)
Knot theory (Chern–Simons invariants, volume conjecture, Legendrian knots)
Dynamical systems (frieze patterns, pentagram map, integrable systems, T-systems, sine-Gordon Y-systems)
Mathematical Physics (statistical mechanics, Donaldson–Thomas invariants, quantum dilogarithm identities, BPS particles).

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Vorheriges Kapitel Auslander–Reiten Theory for Finite-Dimensional Algebras

Nächstes Kapitel Cluster Characters

Nur mit Berechtigung zugänglich

This means that \(\oplus \) is commutative, associative, and distributive with respect to the multiplication in \(\mathbb {P}\).

A homotopy between two continuous maps \(f,g:X\rightarrow Y\) is a continuous map \(h:[0, 1]\times X\rightarrow Y\) such that \(h(0,x) =f(x)\) and \(h(1,x)=g(x)\). An isotopy is a homotopy \(h\) such that for all \(t\in [0,1]\) the map \(h(t,\text {--}):X\rightarrow h(t,X)\) is a homeomorphism. In particular an isotopy of curves cannot create self-crossings.

Ouroboros: a snake devouring its tail.

There is a choice involved here which of the two is \(P_-\) and this will make a difference later when we consider expansion formulas for cluster algebras with nontrivial coefficients. One can determine \(P_-\) as follows. If a tile \(G_j\) in the snake graph has the same orientation as the corresponding quadrilateral in the surface \(S\), then \(P_-\) contains the south and the north edges of \(G_j\) if they are boundary edges, and \(P_-\) does not contain the east or the west edge of \(G_j\).

So far, restricting to the case without punctures has been for the sake of simplicity. But now we really need to make this restriction.

Amiot, C.: Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier (Grenoble) 59(6), 2525–2590 (2009). DOI https://doi.org/10.5802/aif.2499

Assem, I., Brüstle, T., Schiffler, R.: Cluster-tilted algebras as trivial extensions. Bull. Lond. Math. Soc. 40(1), 151–162 (2008). DOI https://doi.org/10.1112/blms/bdm107

Buan, A.B., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006). DOI https://doi.org/10.1016/j.aim.2005.06.003

Buan, A.B., Marsh, R.J., Reiten, I.: Cluster-tilted algebras. Trans. Amer. Math. Soc. 359(1), 323–332 (2007). DOI https://doi.org/10.1090/S0002-9947-06-03879-7

Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (\(A_n\) case). Trans. Amer. Math. Soc. 358(3), 1347–1364 (2006). DOI https://doi.org/10.1090/S0002-9947-05-03753-0

Çanakçı, İ., Lee, K., Schiffler, R.: On cluster algebras from unpunctured surfaces with one marked point. Proc. Amer. Math. Soc. Ser. B 2, 35–49 (2015). DOI https://doi.org/10.1090/bproc/21

Çanakçı, İ., Schiffler, R.: Snake graph calculus and cluster algebras from surfaces. J. Algebra 382, 240–281 (2013). DOI https://doi.org/10.1016/j.jalgebra.2013.02.018

Çanakçı, İ., Schiffler, R.: Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs. Math. Z. 281(1–2), 55–102 (2015). DOI https://doi.org/10.1007/s00209-015-1475-y

Çanakçı, İ., Schiffler, R.: Cluster algebras and continued fractions. Compos. Math. 154(3) (2018). DOI https://doi.org/10.1112/S0010437X17007631

10.

Çanakçı, İ., Schiffler, R.: Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings. Int. Math. Res. Not. rnx157, 1–82 (2017). DOI https://doi.org/10.1093/imrn/rnx157

11.

Felikson, A., Shapiro, M., Tumarkin, P.: Bases for cluster algebras from orbifolds. arXiv:1511.08023

12.

Felikson, A., Shapiro, M., Tumarkin, P.: Cluster algebras and triangulated orbifolds. Adv. Math. 231(5), 2953–3002 (2012). DOI https://doi.org/10.1016/j.aim.2012.07.032

13.

Felikson, A., Shapiro, M., Tumarkin, P.: Skew-symmetric cluster algebras of finite mutation type. J. Eur. Math. Soc. (JEMS) 14(4), 1135–1180 (2012). DOI https://doi.org/10.4171/JEMS/329

14.

Fock, V., Goncharov, A.: Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. 103, 1–211 (2006). DOI https://doi.org/10.1007/s10240-006-0039-4

15.

Fock, V.V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. Ann. Sci. Éc. Norm. Supér. (4) 42(6), 865–930 (2009). DOI https://doi.org/10.1007/978-0-8176-4745-2_15

16.

Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math. 201(1), 83–146 (2008). DOI https://doi.org/10.1007/s11511-008-0030-7

17.

Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15(2), 497–529 (2002). DOI https://doi.org/10.1090/S0894-0347-01-00385-X

18.

Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003). DOI https://doi.org/10.1007/s00222-003-0302-y

19.

Fomin, S., Zelevinsky, A.: Cluster algebras. IV. Coefficients. Compos. Math. 143(1), 112–164 (2007). DOI https://doi.org/10.1112/S0010437X06002521

20.

Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Weil–Petersson forms. Duke Math. J. 127(2), 291–311 (2005). DOI https://doi.org/10.1215/S0012-7094-04-12723-X

21.

Labardini-Fragoso, D.: Quivers with potentials associated to triangulated surfaces. Proc. Lond. Math. Soc. (3) 98(3), 797–839 (2009). DOI https://doi.org/10.1112/plms/pdn051

22.

Musiker, G., Schiffler, R.: Cluster expansion formulas and perfect matchings. J. Algebraic Combin. 32(2), 187–209 (2010). DOI https://doi.org/10.1007/s10801-009-0210-3

23.

Musiker, G., Schiffler, R., Williams, L.: Positivity for cluster algebras from surfaces. Adv. Math. 227(6), 2241–2308 (2011). DOI https://doi.org/10.1016/j.aim.2011.04.018

24.

Musiker, G., Schiffler, R., Williams, L.: Bases for cluster algebras from surfaces. Compos. Math. 149(2), 217–263 (2013). DOI https://doi.org/10.1112/S0010437X12000450

25.

Musiker, G., Williams, L.: Matrix formulae and skein relations for cluster algebras from surfaces. Int. Math. Res. Not. IMRN 2013(13), 2891–2944 (2013). DOI https://doi.org/10.1093/imrn/rns118

26.

Propp, J.: The combinatorics of frieze patterns and Markoff numbers. arXiv:math/0511633

27.

Thurston, D.P.: Positive basis for surface skein algebras. Proc. Natl. Acad. Sci. USA 111(27), 9725–9732 (2014). DOI https://doi.org/10.1073/pnas.1313070111

Titel: Cluster Algebras from Surfaces
verfasst von: Ralf Schiffler
Verlag: Springer International Publishing
Buch: Homological Methods, Representation Theory, and Cluster Algebras
Print ISBN: 978-3-319-74584-8

Electronic ISBN: 978-3-319-74585-5

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-74585-5_3

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