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Geometric construction of cluster algebras and cluster categories

Profile image of Karin BaurKarin Baur

2008, Trends in Mathematics

Abstract
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This paper surveys geometric approaches to the construction of cluster algebras and m-cluster categories, founded on two significant problems in the field: the study of the canonical basis in quantized enveloping algebras and the concept of total positivity in matrix theory. It discusses the connections of cluster algebras to various mathematical domains such as Poisson geometry, Teichmüller spaces, and thermodynamic Bethe Ansatz, while providing examples and definitions crucial to understanding the geometric foundations of cluster theories.

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References (34)

  1. K. Baur, R. Marsh, A geometric description of the m-cluster categories of type Dn International Mathematical Research Notices (2007), Vol. 2007 doi:10.1093/imrn/rnm011.
  2. K. Baur, R. Marsh, A geometric construction of m-cluster categories, to appear in Trans. AMS
  3. K. Baur, R. Marsh, Ptolemy relations for punctured discs, preprint, arXiv:0711.1443v1 [math.CO].
  4. A. Berenstein, S. Fomin, A. Zelevinsky, Cluster algebras III: Upper bounds and double Bruhat cells. Duke Math. J. 126, 2006, no.1, 1-52.
  5. D. Broline, D. W. Crowe and I. M. Isaacs. The geometry of frieze patterns. Geom. Ded. 3 (1974), 171-176.
  6. A. B. Buan, R. J. Marsh, M. Reineke, I. Reiten, G. Todorov. Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), no. 2, 572-618.
  7. P. Caldero, F. Chapoton, R. Schiffler, Quivers with relations arising from clusters (An case), Transactions of the AMS 358 (2006), no. 3, 1347-1364.
  8. F. Chapoton, S. Fomin, A. Zelevinsky, Polytopal realizations of generalized associa- hedra, Canad. Math. Bull. 45 (2002), no. 4, 537-566.
  9. J. H. Conway and H. S. M. Coxeter, Triangulated discs and frieze patterns. Math. Gaz. 57 (1973), 87-94.
  10. J. H. Conway and H. S. M. Coxeter, Triangulated discs and frieze patterns. Math. Gaz. 57 (1973), 175-186.
  11. S. Fock, A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes tudes Sci. No. 103 (2006), 1-211.
  12. S. Fock, A.B. Goncharov, Dual Teichmüller and lamination spaces, Handbook of Teichmüller theory. Vol. I, 647-684, IRMA Lect. Math. Theor. Phys., 11, Eur. Math. Soc., Zrich, 2007.
  13. S. Fomin, M. Shapiro and D. Thurston, Cluster algebras and triangulated discs. Part I: Cluster complexes. Preprint arxiv:math.RA/0608367v3, August 2006.
  14. S. Fomin and A. Zelevinsky. Cluster algebras I: Foundations. J. Amer. Math. Soc. 15 (2002), no. 2, 497-529.
  15. S. Fomin and A. Zelevinsky. The Laurent phenomenon. Advances in Appl. Math. 28 (2002), no. 2 119 -144.
  16. S. Fomin and A. Zelevinsky. Y -systems and generalized associahedra. Ann. Math. 158 (2003), no. 3, 977-1018.
  17. S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification. Invent. Math. 154 (2003), no. 1, 63-121.
  18. S. Fomin and A. Zelevinsky, Cluster algebras: Notes for the DCM-03 Conference. CDM 203: Current Developments in Mathematics. International Press, 2004.
  19. Thermodynamic Bethe ansatz and dilogarithm identities. I, Math. Res. Lett. 2 (1995), no. 6, 677-693.
  20. Ch. Geiss, B. Leclerc, J. Schröer, Semicanonical bases and preprojective algebras, Annales Scientifiques de l'Ecole Normale Superieure 38 (2005), 193-253.
  21. Ch. Geiss, B. Leclerc, J. Schröer, Semicanonical bases and preprojective algebras II: A multiplication formula, Compositio Mathematica 143 (2007), 1313-1334.
  22. M. Gekhtman, M. Shapiro, A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003), no. 3, 899-934, 1199. Cluster algebras and Weil-Petersson forms
  23. M. Gekhtman, M. Shapiro, A. Vainshtein, Cluster algebras and Weil-Petersson forms, Duke Math. J. 127 (2005), no. 2, 291-311.
  24. F. Gliozzi, R. Tateo, Thermodynamic Bethe ansatz and three-fold triangulations, Internat. J. Modern Phys. A 11 (1996), no. 22, 4051-4064.
  25. A. Hatcher, On triangulations of surfaces, Toplogy Appl. 40 (1991), 189-194.
  26. P. Jorgensen, Quotients of cluster categories, Preprint, arXiv:0705.1117v1.
  27. B. Keller, On triangulated orbit categories, Documenta Mathematica, Vol. 10 (2005), 551-581.
  28. G. Lusztig, Total positivity in reductive groups. In G.I. Lehrer, editor, Lie theory and geometry: in honor of Bertram Kostant, volume 123 of Progress in Mathematics, 531-568. Birkhäuser, Boston, 1994.
  29. D. Panyushev, ad-nilpotent ideals of a Borel subalgebra: generators and duality, J. Algebra 274 (2004), no. 2, 822-846.
  30. Chr. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comm. Math. Helv. 55, no. 2 (1990), 199-224.
  31. R. Schiffler, A geometric model for cluster categories of type Dn. Preprint arxiv:math.RT/0608264, 2006.
  32. J. Scott, Grassmannians and cluster algebras. Proc of the LMS (2006), 92: 345-380
  33. A. Szenes, Periodicity of Y-systems and flat connections, Preprint arxiv:math.RT/0606377.
  34. A.Y. Volkov, On Zamolodchikov's Periodicity Conjecture, Preprint arxiv:hep-th/0606094.

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