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.
- Arnold-Liouville integrability
- Cluster algebras
- Twisted dynkin diagrams
- Zamolodchikov periodicity