nLab Y-system

33## Overview

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 $Y_i$, $i\in I$,

$$Y_i(t+1)Y_i(t-1)=\prod_{j\neq i}\bigl(Y_j(t)+ 1\bigr)^{-a_{ij}}$$

The elements of the T-system satisfy discrete Hirota equations.

Literature

For reviews see

• Atsuo Kuniba, Tomoki Nakanishi, Junji Suzuki, T-systems and Y-systems in integrable systems, J. Phys. A44 103001 (2011) doi; T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras, SIGMA 5 (2009), 108, 23 pages

Y-system has a remarkable connection to cluster algebras:

• Bernhard Keller, Cluster algebras, quiver representations and triangulated categories, in: Triangulated categories, 76–160, London Math. Soc.

  Lecture Note Ser 375 arXiv:0807.1960

• Sergey Fomin, Andrei Zelevinsky, Y-systems and generalized associahedra, Ann. Math. 158 977-1018 (2003) doi

Y-systems are relevant for integrability phenomena in superstring theory and in relation to study of spectrum of N=4 SUSY Yang-Mills theory. See survey

• Stijn J. van Tongeren, Integrability of the $AdS_5 \times S^5$ superstring and its deformations, J. Phys. A: Math. Theor. 47 (2014) 433001 arXiv:1310.4854

On Q- and T-system

• Philippe Di Francesco, Discrete integrable systems, positivity, and continued fraction rearrangements, Lett Math Phys 96 (2011) 299–324 doi

category: physics