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

Summary

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

Keywords/Phrases

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

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Affiliation

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

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