**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.

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*, 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 Zang-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

category: physics

Last revised on September 17, 2023 at 14:03:26. See the history of this page for a list of all contributions to it.