The periodicity conjecture for pairs of Dynkin diagrams

Abstract

We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorification via triangulated categories.

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    @misc{InoueIyamaKellerKunibaNakanishi10a,
      author={Inoue, R. and Iyama, O. and Keller, B. and Kuniba, A. and Nakanishi, T.},
      TITLE={Periodicities of {$T$} and {$Y$}-systems, dilogarithm identities, and cluster algebras {I}: type {$B_r$}},
      NOTE={to appear in \emph{Publ. RIMS}},
      ARXIV={1001.1880},
     }
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    @misc{InoueIyamaKellerKunibaNakanishi10b,
      author={Inoue, R. and Iyama, O. and Keller, B. and Kuniba, A. and Nakanishi, T.},
      TITLE={Periodicities of {$T$} and {$Y$}-systems, dilogarithm identities, and cluster algebras {II}: types {$C_r$, $F_4$, and $G_2$}},
      ARXIV={1001.1881},
      NOTE={to appear in {\it Publ. RIMS}},
      SORTYEAR={2013},
     }
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      AUTHOR = {Inoue, Rei and Iyama, Osamu and Kuniba, Atsuo and Nakanishi, Tomoki and Suzuki, Junji},
      TITLE = {Periodicities of {$T$}-systems and {$Y$}-systems},
      JOURNAL = {Nagoya Math. J.},
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      ISSN = {0020-9910},
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    @incollection{Keller09b, MRKEY = {2906350},
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      BOOKTITLE = {S{é}minaire Bourbaki. Vol. 2009/2010. Expos{é}s 1012--1026},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      PUBLISHER={Soc. Math. France},
      VOLUME = {339},
      YEAR = {2011},
      PAGES = {Exp. No. 1014, vii, 63--90},
      ISSN = {0303-1179},
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      }
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      author={Keller, Bernhard},
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      JOURNAL = {Manuscripta Math.},
      FJOURNAL = {Manuscripta Mathematica},
      VOLUME = {79},
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      JOURNAL = {Doc. Math.},
      FJOURNAL = {Documenta Mathematica},
      VOLUME = {10},
      YEAR = {2005},
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      ISSN = {1431-0635},
      MRCLASS = {18E30 (16G20)},
      MRNUMBER = {2184464},
      MRREVIEWER = {Michael S. Barot},
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      AUTHOR = {Keller, Bernhard},
      TITLE = {Derived categories and tilting},
      BOOKTITLE = {Handbook of Tilting Theory},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {332},
      PAGES = {49--104},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2007},
      MRCLASS = {16G10 (14F05 16D90 18E30)},
      MRNUMBER = {2384608},
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      DOI = {10.1017/CBO9780511735134.005},
      }
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    @incollection{Keller08d,
      author = {Keller, Bernhard},
      TITLE = {{C}alabi-{Y}au triangulated categories},
      BOOKTITLE={Trends in Representation Theory of Algebras and Related Topics},
      ADDRESS={Zurich},
      EDITOR={A.~Skowroński},
      PUBLISHER={European Math. Soc.},
      YEAR={2008},
      PAGES={467--489},
      DOI = {10.4171/062-1/11},
      }
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    @incollection {Keller10b, MRKEY = {2681708},
      AUTHOR = {Keller, Bernhard},
      TITLE = {Cluster algebras, quiver representations and triangulated categories},
      BOOKTITLE = {Triangulated Categories},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {375},
      PAGES = {76--160},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2010},
      MRCLASS = {13F60 (16G20 18E30)},
      MRNUMBER = {2681708},
      MRREVIEWER = {Gregoire Dupont},
      ZBLNUMBER = {1215.16012},
      DOI = {10.1017/CBO9781139107075.004},
     }
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      TITLE={Deformed {C}alabi--{Y}au completions},
      JOURNAL = {J. Reine Angew. Math.},
      VOLUME={654},
      YEAR={2011},
      PAGES={125--180},
      TITLENOTE={with an appendix by Michel~Van den Bergh},
      MRNUMBER = {2795754},
      ZBLNUMBER = {1220.18012},
      DOI = {10.1515/crelle.2011.031},
      }
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      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {211},
      YEAR = {2007},
      NUMBER = {1},
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      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {18E30 (16D90 18G10 18G40)},
      MRNUMBER = {2313531},
      MRREVIEWER = {Andrey Yu. Lazarev},
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      }
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    @article {KellerYang11, MRKEY = {2739775},
      AUTHOR = {Keller, Bernhard and Yang, Dong},
      TITLE = {Derived equivalences from mutations of quivers with potential},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {226},
      YEAR = {2011},
      NUMBER = {3},
      PAGES = {2118--2168},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {16G20 (18E30)},
      MRNUMBER = {2739775},
      MRREVIEWER = {Alex Martsinkovsky},
      DOI = {10.1016/j.aim.2010.09.019},
      ZBLNUMBER = {05848698},
      }
  • [KontsevichSoibelman08] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations.
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      author = {Kontsevich, Maxim and Soibelman, Yan},
      TITLE = {Stability structures, motivic {D}onaldson-{T}homas invariants and cluster transformations},
      ARXIV= {0811.2435},
      SORTYEAR={2012},
     }
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    @incollection{KontsevichSoibelman10a,
      author={Kontsevich, Maxim and Soibelman, Yan},
      TITLE = {Motivic {D}onaldson-{T}homas invariants: summary of results},
      BOOKTITLE={Mirror Symmetry and Tropical Geometry},
      SERIES={Contemp. Math.},
      VOLUME={257},
      PUBLISHER={Amer. Math. Soc.},
      ADDRESS={Providence, RI},
      YEAR={2010},
      PAGES={55--89},
      MRNUMBER = {2681792},
      ZBLNUMBER = {1214.14014},
      DOI = {10.1090/conm/527/10400},
     }
  • [KunibaNakanishi92] Go to document A. Kuniba and T. Nakanishi, "Spectra in conformal field theories from the Rogers dilogarithm," Modern Phys. Lett. A, vol. 7, iss. 37, pp. 3487-3494, 1992.
    @article {KunibaNakanishi92, MRKEY = {1192727},
      AUTHOR = {Kuniba, A. and Nakanishi, T.},
      TITLE = {Spectra in conformal field theories from the {R}ogers dilogarithm},
      JOURNAL = {Modern Phys. Lett. A},
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      VOLUME = {7},
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      PAGES = {3487--3494},
      ISSN = {0217-7323},
      MRCLASS = {81T40 (33B99)},
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      MRREVIEWER = {Peter J. Forrester},
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      }
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    @article {KunibaNakanishiSuzuki94, MRKEY = {1304818},
      AUTHOR = {Kuniba, Atsuo and Nakanishi, Tomoki and Suzuki, Junji},
      TITLE = {Functional relations in solvable lattice models. {I}. {F}unctional relations and representation theory},
      JOURNAL = {Internat. J. Modern Phys. A},
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      VOLUME = {9},
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      NUMBER = {30},
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      ISSN = {0217-751X},
      MRCLASS = {82B20 (82B23)},
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      JOURNAL = {Trans. Amer. Math. Soc.},
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      MRNUMBER = {1990581},
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      DOI = {10.1090/S0002-9947-03-03320-8},
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Authors

Bernhard Keller

Université Paris Diderot -- Paris 7
UFR de Mathématiques, Case 7012
75205 Paris cedex 13
France

and

Institut de Mathématiques de Jussieu
75252 Paris
France