Y-system and T-system are two related classes of algebraic relations associated with affine Lie algebras and can be considered as encoding certain integrable systems. Y-system can be considered as a system of difference equation?s for a finite set of commuting variables , ,
The elements of the T-system satisfy discrete Hirota equations.
For reviews see
Y-system has a remarkable connection to cluster algebras:
Y-systems are relevant for integrability phenomena in superstring theory and in relation to study of spectrum of N=4 SUSY Zang-Mills theory. See survey
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