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.

pdf (645 kb)   ps (296 kb)   tex (215 kb)

References

  1. Batchelor M.T., Guan X.-W., Oelkers N., Tsuboi Z., Integrable models and quantum spin ladders: comparison between theory and experiment for the strong coupling ladder compounds, Adv. Phys. 56 (2007), 465-543, cond-mat/0512489.
  2. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B., Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Comm. Math. Phys. 177 (1996), 381-398, hep-th/9412229.
  3. Bazhanov V.V., Reshetikhin N., Restricted solid-on-solid models connected with simply laced algebras and conformal field theory, J. Phys. A: Math. Gen. 23 (1990), 1477-1492.
  4. Beck J., Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (1994), 555-568, hep-th/9404165.
  5. Berenstein A., Fomin S., Zelevinsky A., Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 1-52, math.RT/0305434.
  6. Caracciolo R., Gliozzi R., Tateo R., A topological invariant of RG flows in 2D integrable quantum field theories, Internat. J. Modern Phys. B 13 (1999), 2927-2932, hep-th/9902094.
  7. Chari V., Pressley A., Quantum affine algebras, Comm. Math. Phys. 142 (1991), 261-283.
  8. Chari V., Pressley A., Quantum affine algebras and their representations, in Representations of Groups (Banff, AB, 1994), CMS Conf. Proc., Vol. 16, Amer. Math. Soc., Providence, RI, 1995, 59-78, hep-th/9411145.
  9. Chari V., Pressley A., Minimal affinizations of representations of quantum groups: the simply laced case, J. Algebra 184 (1996), 1-30, hep-th/9410036.
  10. Di Francesco P., Kedem R., Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property, Lett. Math. Phys. 89 (2009), 183-216, arXiv:0803.0362.
  11. Di Francesco P., Kedem R., Positivity of the T-system cluster algebra, arXiv:0908.3122.
  12. Dorey P., Dunning C., Tateo R., The ODE/IM correspondence, J. Phys. A: Math. Theor. 40 (2007), R205-R283, hep-th/0703066.
  13. Dorey P., Pocklington A., Tateo R., Integrable aspects of the scaling q-state Potts models. II. Finite-size effects, Nuclear Phys. B 661 (2003), 464-513, hep-th/0208202.
  14. Drinfel'd V., Hopf algebras and the quantum Yang-Baxter equation, Soviet. Math. Dokl. 32 (1985), 254-258.
  15. Drinfel'd V., A new realization of Yangians and quantized affine algebras, Soviet. Math. Dokl. 36 (1988), 212-216.
  16. Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, math.RT/0104151.
  17. Fomin S., Zelevinsky A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), 63-121, math.RA/0208229.
  18. Fomin S., Zelevinsky A., Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), 977-1018, hep-th/0111053.
  19. Fomin S., Zelevinsky A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112-164, math.RA/0602259.
  20. Frenkel E., Reshetikhin N., The q-characters of representations of quantum affine algebras and deformations of W-algebras, in Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemp. Math., Vol. 248, Amer. Math. Soc., Providence, RI, 1999, 163-205, math.QA/9810055.
  21. Frenkel E., Szenes A., Thermodynamic Bethe ansatz and dilogarithm identities. I, Math. Res. Lett. 2 (1995), 677-693, hep-th/9506215.
  22. Gliozzi F., Tateo R., Thermodynamic Bethe ansatz and three-fold triangulations, Internat. J. Modern Phys. A 11 (1996), 4051-4064, hep-th/9505102.
  23. Geiss C., Leclerc B., Schröer J., Cluster algebra structures and semicanonical bases for unipotent groups, math.RT/0703039.
  24. Gromov N., Kazakov V., Vieira P., Finite volume spectrum of 2D field theories from Hirota dynamics, arXiv:0812.5091.
  25. Hernandez D., The Kirillov-Reshetikhin conjecture and solutions of T-systems, J. Reine Angew. Math. 596 (2006), 63-87, math.QA/0501202.
  26. Hernandez D., Representations of quantum affinizations and fusion product, Transform. Groups 10 (2005), 163-200, math.QA/0312336.
  27. Hernandez D., Drinfeld coproduct, quantum fusion tensor category and applications, Proc. Lond. Math. Soc. (3) 95 (2007), 567-608, math.QA/0504269.
  28. Hernandez D., The Kirillov-Reshetikhin conjecture: the general case, arXiv:0704.2838.
  29. Hernandez D., Leclerc B., Cluster algebras and quantum affine algebras, talk presented by B. Leclerc at Workshop "Lie Theory" (MSRI, Berkeley, March 2008).
  30. Hernandez D., Leclerc B., Cluster algebras and quantum affine algebras, arXiv:0903.1452.
  31. Hutchins H.C., Weinert H.J., Homomorphisms and kernels of semifields, Period. Math. Hungar. 21 (1990), 113-152.
  32. Inoue R., Iyama O., Kuniba A., Nakanishi T., Suzuki J., Periodicities of T-systems and Y-systems, Nagoya Math. J., to appear, arXiv:0812.0667.
  33. Jimbo M., A q-difference analogue of U(^g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69.
  34. Jing N., Quantum Kac-Moody algebras and vertex representations, Lett. Math. Phys. 44 (1998), 261-271.
  35. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  36. Keller B., Cluster algebras, quiver representations and triangulated categories, arXiv:0807.1960.
  37. Kirillov A.N., Identities for the Rogers dilogarithm function connected with simple Lie algebras, J. Soviet Math. 47 (1989), 2450-2459.
  38. Kirillov A.N., Reshetikhin N., Representations of Yangians and multiplicities of the inclusion of the irreducible components of the tensor product of representations of simple Lie algebras, J. Soviet Math. 52 (1990), 3156-3164.
  39. Klassen T.R., Melzer E., Purely elastic scattering theories and their ultraviolet limits, Nuclear Phys. B 338 (1990), 485-528.
  40. Klümper A., Pearce P.A., Conformal weights of RSOS lattice models and their fusion hierarchies, Phys. A 183 (1992), 304-350.
  41. Krichever I., Lipan O., Wiegmann P., Zabrodin A., Quantum integrable models and discrete classical Hirota equations. Comm. Math. Phys. 188 (1997), 267-304.
  42. Kuniba A., Nakanishi T., Spectra in conformal field theories from the Rogers dilogarithm, Internat. J. Modern Phys. A 7 (1992), 3487-3494, hep-th/9206034.
  43. Kuniba A., Nakanishi T., Suzuki J., Functional relations in solvable lattice models. I. Functional relations and representation theory, Internat. J. Modern Phys. A 9 (1994), 5215-5266, hep-th/9309137.
  44. Kuniba A., Nakanishi T., Suzuki J., Functional relations in solvable lattice models. II. Applications, Internat. J. Modern Phys. A 9 (1994), 5267-5312, hep-th/9310060.
  45. Kuniba A., Suzuki J., Functional relations and analytic Bethe ansatz for twisted quantum affine algebras, J. Phys. A: Math. Gen. 28 (1995), 711-722, hep-th/9408135.
  46. Miki K., Representations of quantum toroidal algebra Uq(sln+1,tor) (n ≥ 2), J. Math. Phys. 41 (2000), 7079-7098.
  47. Nakajima H., Quiver varieties and finite dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), 145-238, math.QA/9912158.
  48. Nakajima H., t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory 7 (2003), 259-274, math.QA/0009231.
  49. Nakajima H., Quiver varieties and cluster algebras, arXiv:0905.0002.
  50. Noumi M., Yamada Y., Affine Weyl groups, discrete dynamical systems and Painlevé equations, Comm. Math. Phys. 199 (1998), 281-295, math.QA/9804132.
  51. Noumi M., Yamada Y., Birational Weyl group action arising from a nilpotent Poisson algebra, in Proc. of the Nagoya 1999 International Workshop "Physics and Combinatorics 1999" (Nagoya), World Sci. Publ., River Edge, NJ, 2001, 287-319.
  52. Ravanini R., Valleriani A., Tateo R., Dynkin TBA's, Internat. J. Modern Phys. A 8 (1993), 1707-1727, hep-th/9207040.
  53. Runkel I., Perturbed defects and T-systems in conformal field theory, J. Phys. A: Math. Theor. 41 (2008), 105401, 21 pages, arXiv:0711.0102.
  54. Tsuboi Z., Solutions of discretized affine Toda field equations for A(1)n, B(1)n, C(1)n, D(1)n, A(2)n and D(2)n+1, J. Phys. Soc. Japan 66 (1997), 3391-3398, solv-int/9610011.
  55. Tsuboi Z., Solutions of the T-system and Baxter equations for supersymmetric spin chains, Nuclear Phys. B 826 (2010), 399-455, arXiv:0906.2039.
  56. Varagnolo M., Vasserot E., Schur duality in the toroidal setting, Comm. Math. Phys. 182 (1996), 469-483, q-alg/9506026.
  57. Volkov A.Yu., On the periodicity conjecture for Y-systems, Comm. Math. Phys. 276 (2007), 509-517, hep-th/0606094.
  58. Zamolodchikov Al.B., On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991), 391-394.


Previous article   Next article   Contents of Volume 5 (2009)