Quivers with additive labelings: Classification and algebraic entropy

Pavel Galashin, Pavlo Pylyavskyy

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We show that Zamolodchikov dynamics of a recurrent quiver has zero algebraic entropy only if the quiver has a weakly sub- additive labeling, and conjecture the converse. By assigning a pair of generalized Cartan matrices of affine type to each quiver with an addi- tive 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.

Original languageEnglish (US)
Pages (from-to)2057-2135
Number of pages79
JournalDocumenta Mathematica
StatePublished - 2019

Bibliographical note

Publisher Copyright:
© Deutsche Mathematiker Vereinigung.


  • Arnold-Liouville integrability
  • Cluster algebras
  • T-system
  • Twisted dynkin diagrams
  • Zamolodchikov periodicity


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