## $$T$$-systems and $$Y$$-systems for quantum affinizations of quantum Kac-Moody algebras.(English)Zbl 1201.17009

The $$T$$-systems and $$Y$$-systems appear in various aspects for integrable systems. They are related to each other by certain changes of variables [A. Klümper and P. A. Pearce, Physica A 183, No. 3, 304–350 (1992; Zbl 1201.82009); A. Kuniba, T. Nakanishi and J. Suzuki, Int. J. Mod. Phys. A 9, No. 30, 5215–5266 (1994; Zbl 0985.82501)]. Originally, the $$T$$-systems are systems of relations among the Kirillov–Reshetikhin modules [A. N. Kirillov, J. Sov. Math. 47, No. 2, 2450–2459 (1989); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 164, 121–133 (1987; Zbl 0685.33014); A. N. Kirillov, N. Yu. Reshetikhin, J. Sov. Math. 52, No. 3, 3156-3164 (1990); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 160, 211–221 (1987; Zbl 0900.16047)] in the Grothendieck rings of modules over quantum affine algebras and Yangians. The $$T$$ and $$Y$$-systems are also regarded as relations among variables for cluster algebras (see P. Di Francesco and R. Kedem, Positivity of the $$T$$-system cluster algebra, arXiv:0908.3122). Recently the $$T$$-systems are generalized by D. Hernandez [Proc. Lond. Math. Soc. (3) 95, No. 3, 567–608 (2007; Zbl 1133.17010)] to the quantum affinizations of a wide class of quantum Kac–Moody algebras.
In the paper under the review the authors introduce the corresponding $$Y$$-systems and establish a relation between $$T$$ and $$Y$$-systems. They 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.

### MSC:

 17B37 Quantum groups (quantized enveloping algebras) and related deformations 13F60 Cluster algebras

### Keywords:

T-systems; Y-systems; quantum groups; cluster algebras
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