Galashin, Pavel; Pylyavskyy, Pavlo

Quivers with Additive Labelings: Classification and Algebraic Entropy

Doc. Math. 24, 2057-2135 (2019)
DOI: 10.25537/dm.2019v24.2057-2135
Communicated by Henning Krause


We show that Zamolodchikov dynamics of a recurrent quiver has zero algebraic entropy only if the quiver has a weakly subadditive labeling, and conjecture the converse. By assigning a pair of generalized Cartan matrices of affine type to each quiver with an additive labeling, we completely classify such quivers, obtaining 40 infinite families and 13 exceptional quivers. This completes the program of classifying Zamolodchikov periodic and integrable quivers.

Mathematics Subject Classification

13F60, 37K10, 05E99


cluster algebras, Zamolodchikov periodicity, T-system, Arnold-Liouville integrability, twisted Dynkin diagrams


  • 1. V. Arnold. Les méthodes mathématiques de la mécanique classique. Éditions Mir, Moscow, 1976. Traduit du russe par Djilali Embarek. zbl 0385.70001; MR0474391.
  • 2. M. P. Bellon and C.-M. Viallet. Algebraic entropy. Comm. Math. Phys., 204(2):425-437, 1999. DOI 10.1007/s002200050652; zbl 0987.37007; MR1704282; arxiv chao-dyn/9805006.
  • 3. Nicolas Bourbaki. Lie groups and Lie algebras. Chapters 4-6. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. zbl 0983.17001; MR1890629.
  • 4. J. Diller and C. Favre. Dynamics of bimeromorphic maps of surfaces. Amer. J. Math., 123(6):1135-1169, 2001. DOI 10.1353/ajm.2001.0038; zbl 1112.37308; MR1867314.
  • 5. Vlastimil Dlab and Claus Michael Ringel. Indecomposable representations of graphs and algebras. Mem. Amer. Math. Soc., 6(173):v+57, 1976. DOI 10.1090/memo/0173; zbl 0332.16015; MR0447344.
  • 6. G. Falqui and C.-M. Viallet. Singularity, complexity, and quasi-integrability of rational mappings. Comm. Math. Phys., 154(1):111-125, 1993. DOI 10.1007/BF02096835; zbl 0791.58116; MR1220950; arxiv hep-th/9212105.
  • 7. Sergey Fomin and Andrei Zelevinsky. Cluster algebras. I. Foundations. J. Amer. Math. Soc., 15(2):497-529 (electronic), 2002. DOI 10.1090/S0894-0347-01-00385-X; zbl 1021.16017; MR1887642; arxiv math/0104151.
  • 8. Sergey Fomin and Andrei Zelevinsky. \(Y\)-systems and generalized associahedra. Ann. of Math. (2), 158(3):977-1018, 2003. DOI 10.4007/annals.2003.158.977; zbl 1057.52003; MR2031858; arxiv hep-th/0111053.
  • 9. Allan P. Fordy and Andrew Hone. Discrete integrable systems and Poisson algebras from cluster maps. Comm. Math. Phys., 325(2):527-584, 2014. DOI 10.1007/s00220-013-1867-y; zbl 1344.37076; MR3148096; arxiv 1207.6072.
  • 10. Edward Frenkel and Nicolai Reshetikhin. The \(q\)-characters of representations of quantum affine algebras and deformations of \(W\)-algebras. In Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), volume 248 of Contemp. Math., pages 163-205. Amer. Math. Soc., Providence, RI, 1999. zbl 0973.17015; MR1745260; arxiv math/9810055.
  • 11. Shmuel Friedland and John Milnor. Dynamical properties of plane polynomial automorphisms. Ergodic Theory Dynam. Systems, 9(1):67-99, 1989. DOI 10.1017/S014338570000482X; zbl 0651.58027; MR0991490.
  • 12. Pavel Galashin and Pavlo Pylyavskyy. The classification of Zamolodchikov periodic quivers. Am. J. Math. 141(2):447-484, 2019. DOI 10.1353/ajm.2019.0010; zbl 07057018; MR3928042; arxiv 1603.03942.
  • 13. Pavel Galashin and Pavlo Pylyavskyy. Quivers with subadditive labelings: classification and integrability, 2016, Math. Z. DOI 10.1007/s00209-019-02374-x; arxiv 1606.04878.
  • 14. Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, and Alek Vainshtein. Integrable cluster dynamics of directed networks and pentagram maps. Adv. Math., 300:390-450, 2016. DOI 10.1016/j.aim.2016.03.023; zbl 1360.37148; MR3534837; arxiv 1406.1883.
  • 15. Max Glick. The Devron property. J. Geom. Phys., 87:161-189, 2015. DOI 10.1016/j.geomphys.2014.07.029; zbl 1331.37078; MR3282366; arxiv 1312.6881.
  • 16. F. Gliozzi and R. Tateo. Thermodynamic Bethe ansatz and three-fold triangulations. Internat. J. Modern Phys. A, 11(22):4051-4064, 1996. DOI 10.1142/S0217751X96001905; zbl 1044.82537; MR1403679; arxiv hep-th/9505102.
  • 17. Alexander B. Goncharov and Richard Kenyon. Dimers and cluster integrable systems. Ann. Sci. Éc. Norm. Supér. (4), 46(5):747-813, 2013. DOI 10.24033/asens.2201; zbl 1288.37025; MR3185352; arxiv 1107.5588.
  • 18. Alain Goriely. Integrability and nonintegrability of dynamical systems, volume 19 of Advanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River Edge, NJ, 2001. zbl 1002.34001; MR1857742.
  • 19. David Hernandez. Drinfeld coproduct, quantum fusion tensor category and applications. Proc. Lond. Math. Soc., 95(3):567-608, 2007. DOI 10.1112/plms/pdm017; zbl 1133.17010; MR2368277; arxiv math/0504269.
  • 20. Jarmo Hietarinta and Claude Viallet. Singularity confinement and chaos in discrete systems. Phys. Rev. Lett., 81:325-328, Jul 1998. DOI 10.1088/1751-8113/49/23/23LT01; zbl 1345.39005; arxiv 1512.09168.
  • 21. Victor G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, third edition, 1990. zbl 0716.17022; MR1104219.
  • 22. Bernhard Keller. The periodicity conjecture for pairs of Dynkin diagrams. Ann. of Math. (2), 177(1):111-170, 2013. DOI 10.4007/annals.2013.177.1.3; zbl 1320.17007; MR2999039; arxiv 1001.1531.
  • 23. A. N. Kirillov and N. Yu. Reshetikhin. Exact solution of the \(XXZ\) Heisenberg model of spin \(S\). Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 145(Voprosy Kvant. Teor. Polya i Statist. Fiz. 5):109-133, 191, 195, 1985. zbl 0586.35032; MR0857965.
  • 24. Harold Knight. Spectra of tensor products of finite-dimensional representations of Yangians. J. Algebra, 174(1):187-196, 1995. DOI 10.1006/jabr.1995.1123; zbl 0868.17009; MR1332866.
  • 25. A. Kuniba and T. Nakanishi. Spectra in conformal field theories from the Rogers dilogarithm. Modern Phys. Lett. A, 7(37):3487-3494, 1992. DOI 10.1142/S0217732392002895; zbl 1021.81842; MR1192727; arxiv hep-th/9206034.
  • 26. Atsuo Kuniba, Tomoki Nakanishi, and Junji Suzuki. Functional relations in solvable lattice models. I. Functional relations and representation theory. Internat. J. Modern Phys. A, 9(30):5215-5266, 1994. DOI 10.1142/S0217751X94002119; zbl 0985.82501; MR1304818; arxiv hep-th/9309137.
  • 27. Atsuo Kuniba, Tomoki Nakanishi, and Junji Suzuki. \(T\)-systems and \(Y\)-systems in integrable systems. J. Phys. A, 44(10):103001, 146, 2011. DOI 10.1088/1751-8113/44/10/103001; zbl 1222.82041; MR2773889; arxiv 1010.1344.
  • 28. Megan Leoni, Gregg Musiker, Seth Neel, and Paxton Turner. Aztec castles and the dp3 quiver. Journal of Physics A: Mathematical and Theoretical, 47(47):474011, 2014. DOI 10.1088/1751-8113/47/47/474011; zbl 1408.13057; MR3280002; arxiv 1308.3926.
  • 29. Takafumi Mase. Investigation into the role of the Laurent property in integrability. J. Math. Phys., 57(2):022703, 21, 2016. DOI 10.1063/1.4941370; zbl 1333.39009; MR3457928; arxiv 1505.01722.
  • 30. John McKay. Graphs, singularities, and finite groups. In The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), volume 37 of Proc. Sympos. Pure Math., pages 183-186. Amer. Math. Soc., Providence, R.I., 1980. zbl 0451.05026; MR0604577.
  • 31. Hiraku Nakajima. \(t\)-analogs of \(q\)-characters of Kirillov-Reshetikhin modules of quantum affine algebras. Represent. Theory, 7:259-274 (electronic), 2003. DOI 10.1090/S1088-4165-03-00164-X; zbl 1078.17008; MR1993360; arxiv math/0204185.
  • 32. E. Ogievetsky and P. Wiegmann. Factorized \(S\)-matrix and the Bethe ansatz for simple Lie groups. Phys. Lett. B, 168(4):360-366, 1986. MR0831897.
  • 33. Valentin Ovsienko, Richard Evan Schwartz, and Serge Tabachnikov. Liouville-Arnold integrability of the pentagram map on closed polygons. Duke Math. J., 162(12):2149-2196, 2013. DOI 10.1215/00127094-2348219; zbl 1315.37035; MR3102478; arxiv 1107.3633.
  • 34. Pavlo Pylyavskyy. Zamolodchikov integrability via rings of invariants. Journal of Integrable Systems, 1(1):xyw010, 2015. DOI 10.1093/integr/xyw010; zbl 1400.37092; arxiv 1506.05378.
  • 35. F. Ravanini, A. Valleriani, and R. Tateo. Dynkin TBAs. Internat. J. Modern Phys. A, 8(10):1707-1727, 1993. MR1216231.
  • 36. N. Yu. Reshetikhin. The spectrum of the transfer matrices connected with Kac-Moody algebras. Lett. Math. Phys., 14(3):235-246, 1987. DOI 10.1007/BF00416853; zbl 0636.17006; MR0919327.
  • 37. David E. Speyer. Perfect matchings and the octahedron recurrence. J. Algebraic Combin., 25(3):309-348, 2007. DOI 10.1007/s10801-006-0039-y; zbl 1119.05092; MR2317336; arxiv math/0402452.
  • 38. R. Stekolshchik. Notes on Coxeter transformations and the McKay correspondence. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2008. DOI 10.1007/978-3-540-77399-3; zbl 1202.20045; MR2388772: arxiv math/0510216.
  • 39. John R. Stembridge. Admissible \(W\)-graphs and commuting Cartan matrices. Adv. in Appl. Math., 44(3):203-224, 2010. DOI 10.1016/j.aam.2009.08.001; zbl 1230.05290; MR2593307.
  • 40. András Szenes. Periodicity of Y-systems and flat connections. Lett. Math. Phys., 89(3):217-230, 2009. DOI 10.1007/s11005-009-0332-5; zbl 1187.15016; MR2551180; arxiv math/0606377.
  • 41. È. B. Vinberg. Discrete linear groups that are generated by reflections. Izv. Akad. Nauk SSSR Ser. Mat., 35:1072-1112, 1971. MR0302779.
  • 42. Alexandre Yu. Volkov. On the periodicity conjecture for \(Y\)-systems. Comm. Math. Phys., 276(2):509-517, 2007. DOI 10.1007/s00220-007-0343-y; zbl 1136.82011; MR2346398.
  • 43. Al.B. Zamolodchikov. On the thermodynamic Bethe ansatz equations for reflectionless \(ADE\) scattering theories. Phys. Lett. B, 253(3-4):391-394, 1991. MR1092210.


Galashin, Pavel
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Pylyavskyy, Pavlo
Department of Mathematics, University of Minnesota, Minneapolis, MN 55414, USA