The periodicity conjecture for pairs of Dynkin diagrams

Bernhard Keller

doi:https://doi.org/10.4007/annals.2013.177.1.3

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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@book {Happel88, MRKEY = {0935124}, AUTHOR = {Happel, Dieter}, TITLE = {Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras}, SERIES = {London Math. Soc. Lecture Note Series}, VOLUME = {119}, PUBLISHER = {Cambridge Univ. Press}, ADDRESS = {Cambridge}, YEAR = {1988}, PAGES = {x+208}, ISBN = {0-521-33922-7}, MRCLASS = {16A46 (16A48 16A62 16A64 18E30)}, MRNUMBER = {0935124}, MRREVIEWER = {Alfred G. Wiedemann}, DOI = {10.1017/CBO9780511629228}, ZBLNUMBER = {0635.16017}, }
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@article {Henriques07, MRKEY = {2335700}, AUTHOR = {Henriques, Andr{é}}, TITLE = {A periodicity theorem for the octahedron recurrence}, JOURNAL = {J. Algebraic Combin.}, FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal}, VOLUME = {26}, YEAR = {2007}, NUMBER = {1}, PAGES = {1--26}, ISSN = {0925-9899}, CODEN = {JAOME7}, MRCLASS = {05E99 (05A15 05C70 05E15 52C07)}, MRNUMBER = {2335700}, MRREVIEWER = {Joel Kamnitzer}, DOI = {10.1007/s10801-006-0045-0}, ZBLNUMBER = {1125.05106}, }
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@article {HernandezLeclerc10, MRKEY = {2682185}, AUTHOR = {Hernandez, David and Leclerc, Bernard}, TITLE = {Cluster algebras and quantum affine algebras}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {154}, YEAR = {2010}, NUMBER = {2}, PAGES = {265--341}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {17B37 (13F60)}, MRNUMBER = {2682185}, MRREVIEWER = {Jan E. Grabowski}, DOI = {10.1215/00127094-2010-040}, ZBLNUMBER = {05788166}, }
[InoueIyamaKellerKunibaNakanishi10a] R. Inoue, O. Iyama, B. Keller, A. Kuniba, and T. Nakanishi, Periodicities of $T$ and $Y$-systems, dilogarithm identities, and cluster algebras I: type $B_r$.

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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}, }
[InoueIyamaKellerKunibaNakanishi10b] R. Inoue, O. Iyama, B. Keller, A. Kuniba, and T. Nakanishi, Periodicities of $T$ and $Y$-systems, dilogarithm identities, and cluster algebras II: types $C_r$, $F_4$, and $G_2$.

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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}, }
[InoueIyamaKunibaNakanishiSuzuki10] R. Inoue, O. Iyama, A. Kuniba, T. Nakanishi, and J. Suzuki, "Periodicities of $T$-systems and $Y$-systems," Nagoya Math. J., vol. 197, pp. 59-174, 2010.

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@article {InoueIyamaKunibaNakanishiSuzuki10, MRKEY = {2649278}, 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.}, FJOURNAL = {Nagoya Mathematical Journal}, VOLUME = {197}, YEAR = {2010}, PAGES = {59--174}, ISSN = {0027-7630}, MRCLASS = {17B37 (13F60 81R12 82B23)}, MRNUMBER = {2649278}, MRREVIEWER = {Jan E. Grabowski}, ZBLNUMBER = {05704436}, }
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@article {IyamaYoshino08, MRKEY = {2385669}, AUTHOR = {Iyama, Osamu and Yoshino, Yuji}, TITLE = {Mutation in triangulated categories and rigid {C}ohen-{M}acaulay modules}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {172}, YEAR = {2008}, NUMBER = {1}, PAGES = {117--168}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {16G20 (13C14 18E30)}, MRNUMBER = {2385669}, MRREVIEWER = {Wolfgang Rump}, DOI = {10.1007/s00222-007-0096-4}, ZBLNUMBER = {1140.18007}, }
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@book {Kac90, MRKEY = {1104219}, AUTHOR = {Kac, Victor G.}, TITLE = {Infinite-dimensional {L}ie algebras}, EDITION = {Third}, PUBLISHER = {Cambridge Univ. Press}, ADDRESS = {Cambridge}, YEAR = {1990}, PAGES = {xxii+400}, ISBN = {0-521-37215-1; 0-521-46693-8}, MRCLASS = {17B65 (17B67 17B68 58F07)}, MRNUMBER = {1104219}, DOI = {10.1017/CBO9780511626234}, ZBLNUMBER = {0716.17022}, }
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@article {Kedem08, MRKEY = {2452184}, AUTHOR = {Kedem, Rinat}, TITLE = {{$Q$}-systems as cluster algebras}, JOURNAL = {J. Phys. A}, FJOURNAL = {Journal of Physics. A. Mathematical and Theoretical}, VOLUME = {41}, YEAR = {2008}, NUMBER = {19}, PAGES = {194011, 14}, ISSN = {1751-8113}, MRCLASS = {17B37 (13A99 81R50 82B23)}, MRNUMBER = {2452184}, MRREVIEWER = {Laurent Demonet}, DOI = {10.1088/1751-8113/41/19/194011}, ZBLNUMBER = {1141.81014}, }
[Keller09b] B. Keller, "Algèbres amassées et applications (d’après Fomin-Zelevinsky, $\ldots$)," in Séminaire Bourbaki. Vol. 2009/2010. Exposés 1012–1026, Soc. Math. France, 2011, vol. 339, p. exp. no. 1014, vii, 63-90.

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@incollection{Keller09b, MRKEY = {2906350}, AUTHOR = {Keller, Bernhard}, TITLE = {Algèbres amassées et applications (d'après {F}omin-{Z}elevinsky, {$\ldots$})}, 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}, ISBN = {978-2-85629-326-3}, MRCLASS = {13F60}, MRNUMBER = {2906350}, }
[Keller08c] B. Keller, Cluster algebras, quiver representations and triangulated categories.

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@misc{Keller08c, author={Keller, Bernhard}, TITLE = {Cluster algebras, quiver representations and triangulated categories}, ARXIV={0807.1960}, SORTYEAR={2013}, }
[Keller93] $Go to document$ B. Keller, "A remark on tilting theory and DG algebras," Manuscripta Math., vol. 79, iss. 3-4, pp. 247-252, 1993.

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@article {Keller93, MRKEY = {1223020}, AUTHOR = {Keller, Bernhard}, TITLE = {A remark on tilting theory and {DG} algebras}, JOURNAL = {Manuscripta Math.}, FJOURNAL = {Manuscripta Mathematica}, VOLUME = {79}, YEAR = {1993}, NUMBER = {3-4}, PAGES = {247--252}, ISSN = {0025-2611}, CODEN = {MSMHB2}, MRCLASS = {18E30 (16D90)}, MRNUMBER = {1223020}, MRREVIEWER = {Daniel Simson}, DOI = {10.1007/BF02568343}, ZBLNUMBER = {0810.16006}, }
[Keller05] $Go to document$ B. Keller, "On triangulated orbit categories," Doc. Math., vol. 10, pp. 551-581, 2005.

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@article {Keller05, MRKEY = {2184464}, AUTHOR = {Keller, Bernhard}, TITLE = {On triangulated orbit categories}, JOURNAL = {Doc. Math.}, FJOURNAL = {Documenta Mathematica}, VOLUME = {10}, YEAR = {2005}, PAGES = {551--581}, ISSN = {1431-0635}, MRCLASS = {18E30 (16G20)}, MRNUMBER = {2184464}, MRREVIEWER = {Michael S. Barot}, ZBLNUMBER = {1086.18006}, URL = {http://www.math.uiuc.edu/documenta/vol-10/17.html}, }
[Keller07a] $Go to document$ B. Keller, "Derived categories and tilting," in Handbook of Tilting Theory, Cambridge: Cambridge Univ. Press, 2007, vol. 332, pp. 49-104.

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@incollection {Keller07a, MRKEY = {2384608}, 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}, MRREVIEWER = {Jun-ichi Miyachi}, DOI = {10.1017/CBO9780511735134.005}, }
[Keller08d] $Go to document$ B. Keller, "Calabi-Yau triangulated categories," in Trends in Representation Theory of Algebras and Related Topics, A. Skowroński, Ed., Zurich: European Math. Soc., 2008, pp. 467-489.

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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}, }
[Keller10b] $Go to document$ B. Keller, "Cluster algebras, quiver representations and triangulated categories," in Triangulated Categories, Cambridge: Cambridge Univ. Press, 2010, vol. 375, pp. 76-160.

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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}, }
[Keller11b] $Go to document$ B. Keller, "Deformed Calabi–Yau completions," J. Reine Angew. Math., vol. 654, pp. 125-180, 2011.

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@article{Keller11b, author={Keller, Bernhard}, 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}, }
[KellerReiten07] $Go to document$ B. Keller and I. Reiten, "Cluster-tilted algebras are Gorenstein and stably Calabi-Yau," Adv. Math., vol. 211, iss. 1, pp. 123-151, 2007.

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@article {KellerReiten07, MRKEY = {2313531}, AUTHOR = {Keller, Bernhard and Reiten, Idun}, TITLE = {Cluster-tilted algebras are {G}orenstein and stably {C}alabi-{Y}au}, JOURNAL = {Adv. Math.}, FJOURNAL = {Advances in Mathematics}, VOLUME = {211}, YEAR = {2007}, NUMBER = {1}, PAGES = {123--151}, ISSN = {0001-8708}, CODEN = {ADMTA4}, MRCLASS = {18E30 (16D90 18G10 18G40)}, MRNUMBER = {2313531}, MRREVIEWER = {Andrey Yu. Lazarev}, DOI = {10.1016/j.aim.2006.07.013}, ZBLNUMBER = {1128.18007}, }
[KellerYang11] $Go to document$ B. Keller and D. Yang, "Derived equivalences from mutations of quivers with potential," Adv. Math., vol. 226, iss. 3, pp. 2118-2168, 2011.

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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}, }
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@misc {KontsevichSoibelman08, 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}, }
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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}, FJOURNAL = {International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology. Nuclear Physics}, VOLUME = {9}, YEAR = {1994}, NUMBER = {30}, PAGES = {5215--5266}, ISSN = {0217-751X}, MRCLASS = {82B20 (82B23)}, MRNUMBER = {1304818}, MRREVIEWER = {Henrik Johannesson}, DOI = {10.1142/S0217751X94002119}, ZBLNUMBER = {0985.82501}, }
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@article {Nakanishi11c, MRKEY = {2804544}, AUTHOR = {Nakanishi, Tomoki}, TITLE = {Dilogarithm identities for conformal field theories and cluster algebras: simply laced case}, JOURNAL = {Nagoya Math. J.}, FJOURNAL = {Nagoya Mathematical Journal}, VOLUME = {202}, YEAR = {2011}, PAGES = {23--43}, ISSN = {0027-7630}, MRCLASS = {81Txx (13F60 17Bxx 33B30)}, MRNUMBER = {2804544}, URL = {http://projecteuclid.org/euclid.nmj/1306851588}, ZBLNUMBER = {05920542}, }
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@misc{Reiten10, author={Reiten, I.}, TITLE={Tilting theory and cluster algebras}, ARXIV={1012.6014}, }
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@incollection{Ringel07, author={Ringel, Claus Michael}, TITLE = {Appendix: Some remarks concerning tilting modules and tilted algebras. {O}rigin. {R}elevance. {F}uture}, BOOKTITLE={Handbook of Tilting Theory}, SERIES={LMS Lecture Note Series}, VOLUME={332}, PUBLISHER={Cambridge Univ. Press}, ADDRESS={Cambridge}, YEAR={2007}, PAGES={413--472}, MRNUMBER = {2384619}, ZBLNUMBER = {1106.16300}, DOI = {10.1017/CBO9780511735134.016}, }
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@article {Volkov07, MRKEY = {2346398}, AUTHOR = {Volkov, Alexandre Yu.}, TITLE = {On the periodicity conjecture for {$Y$}-systems}, JOURNAL = {Comm. Math. Phys.}, FJOURNAL = {Communications in Mathematical Physics}, VOLUME = {276}, YEAR = {2007}, NUMBER = {2}, PAGES = {509--517}, ISSN = {0010-3616}, CODEN = {CMPHAY}, MRCLASS = {17B20 (20F55)}, MRNUMBER = {2346398}, MRREVIEWER = {Christof Geiß}, DOI = {10.1007/s00220-007-0343-y}, ZBLNUMBER = {1136.82011}, }
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@article {YangZelevinsky08, MRKEY = {2452619}, AUTHOR = {Yang, Shih-Wei and Zelevinsky, Andrei}, TITLE = {Cluster algebras of finite type via {C}oxeter elements and principal minors}, JOURNAL = {Transform. Groups}, FJOURNAL = {Transformation Groups}, VOLUME = {13}, YEAR = {2008}, NUMBER = {3-4}, PAGES = {855--895}, ISSN = {1083-4362}, MRCLASS = {13F99 (17B20 20F55 20G15)}, MRNUMBER = {2452619}, MRREVIEWER = {Gregoire Dupont}, DOI = {10.1007/s00031-008-9025-x}, ZBLNUMBER = {1177.16010}, }
[Zamolodchikov91] $Go to document$ A. B. Zamolodchikov, "On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories," Phys. Lett. B, vol. 253, iss. 3-4, pp. 391-394, 1991.

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@article {Zamolodchikov91, MRKEY = {1092210}, AUTHOR = {Zamolodchikov, Al. B.}, TITLE = {On the thermodynamic {B}ethe ansatz equations for reflectionless {ADE} scattering theories}, JOURNAL = {Phys. Lett. B}, FJOURNAL = {Physics Letters. B}, VOLUME = {253}, YEAR = {1991}, NUMBER = {3-4}, PAGES = {391--394}, ISSN = {0370-2693}, CODEN = {PYLBAJ}, MRCLASS = {81T40 (81U20 82B23)}, MRNUMBER = {1092210}, MRREVIEWER = {Philippe Christe}, DOI = {10.1016/0370-2693(91)91737-G}, }
[Zelevinsky07a] A. Zelevinsky, "What is $\dots$ a cluster algebra?," Notices Amer. Math. Soc., vol. 54, iss. 11, pp. 1494-1495, 2007.

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@article {Zelevinsky07a, MRKEY = {2361161}, AUTHOR = {Zelevinsky, Andrei}, TITLE = {What is {$\dots$} a cluster algebra?}, JOURNAL = {Notices Amer. Math. Soc.}, FJOURNAL = {Notices of the American Mathematical Society}, VOLUME = {54}, YEAR = {2007}, NUMBER = {11}, PAGES = {1494--1495}, ISSN = {0002-9920}, CODEN = {AMNOAN}, MRCLASS = {13A99}, MRNUMBER = {2361161}, ZBLNUMBER = {1130.13301}, }

The periodicity conjecture for pairs of Dynkin diagrams

Abstract

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