Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 108, 23 pages      arXiv:0909.4618      https://doi.org/10.3842/SIGMA.2009.108

T-Systems and Y-Systems for Quantum Affinizations of Quantum Kac-Moody Algebras

Atsuo Kuniba a, Tomoki Nakanishi b and Junji Suzuki c
a) Institute of Physics, University of Tokyo, Tokyo, 153-8902, Japan
b) Graduate School of Mathematics, Nagoya University, Nagoya, 464-8604, Japan
c) Department of Physics, Faculty of Science, Shizuoka University, Ohya, 836, Japan

Received October 05, 2009, in final form December 16, 2009; Published online December 19, 2009

Abstract
The T-systems and Y-systems are classes of algebraic relations originally associated with quantum affine algebras and Yangians. Recently the T-systems were generalized to quantum affinizations of a wide class of quantum Kac-Moody algebras by Hernandez. In this note we introduce the corresponding Y-systems and establish a relation between T and Y-systems. We also introduce the T and Y-systems associated with a class of cluster algebras, which include the former T and Y-systems of simply laced type as special cases.

Key words: T-systems; Y-systems; quantum groups; cluster algebras.

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