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

SIGMA 9 (2013), 054, 28 pages      arXiv:1208.4821      https://doi.org/10.3842/SIGMA.2013.054
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Extended T-System of Type G2

Jian-Rong Li a and Evgeny Mukhin b
a) Department of Mathematics, Lanzhou University, Lanzhou 730000, P.R. China
b) Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA

Received April 03, 2013, in final form August 16, 2013; Published online August 22, 2013

Abstract
We prove a family of 3-term relations in the Grothendieck ring of the category of finite-dimensional modules over the affine quantum algebra of type G2 extending the celebrated T-system relations of type G2. We show that these relations can be used to compute classes of certain irreducible modules, including classes of all minimal affinizations of type G2. We use this result to obtain explicit formulas for dimensions of all participating modules.

Key words: quantum affine algebra of type G2; minimal affinizations; extended T-systems; q-characters; Frenkel-Mukhin algorithm.

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References

  1. Chari V., Pressley A., Quantum affine algebras, Comm. Math. Phys. 142 (1991), 261-283.
  2. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  3. Chari V., Pressley A., Minimal affinizations of representations of quantum groups: the nonsimply-laced case, Lett. Math. Phys. 35 (1995), 99-114, hep-th/9410036.
  4. 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.
  5. Chari V., Pressley A., Factorization of representations of quantum affine algebras, in Modular Interfaces (Riverside, CA, 1995), AMS/IP Stud. Adv. Math., Vol. 4, Amer. Math. Soc., Providence, RI, 1997, 33-40.
  6. Cherednik I.V., A new interpretation of Gel'fand-Tzetlin bases, Duke Math. J. 54 (1987), 563-577.
  7. Drinfel'd V.G., A new realization of Yangians and of quantum affine algebras, Soviet Math. Dokl. 36 (1988), 212-216.
  8. Frenkel E., Mukhin E., Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), 23-57, math.QA/9911112.
  9. 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.
  10. Hernandez D., The Kirillov-Reshetikhin conjecture and solutions of T-systems, J. Reine Angew. Math. 596 (2006), 63-87, math.QA/0501202.
  11. Hernandez D., On minimal affinizations of representations of quantum groups, Comm. Math. Phys. 276 (2007), 221-259, math.QA/0607527.
  12. Hernandez D., Leclerc B., Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2010), 265-341, arXiv:0903.1452.
  13. Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T-systems and Y-systems, dilogarithm identities, and cluster algebras I: type Br, Publ. Res. Inst. Math. Sci. 49 (2013), 1-42, arXiv:1001.1880.
  14. Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T-systems and Y-systems, dilogarithm identities, and cluster algebras II: types Cr, F4, and G2, Publ. Res. Inst. Math. Sci. 49 (2013), 43-85, arXiv:1001.1881.
  15. Kirillov A.N., Reshetikhin N.Yu., 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.
  16. 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.
  17. Kuniba A., Nakanishi T., Suzuki J., T-systems and Y-systems in integrable systems, J. Phys. A: Math. Theor. 44 (2011), 103001, 146 pages, arXiv:1010.1344.
  18. Mukhin E., Young C.A.S., Extended T-systems, Selecta Math. (N.S.) 18 (2012), 591-631, arXiv:1104.3094.
  19. Mukhin E., Young C.A.S., Path description of type B q-characters, Adv. Math. 231 (2012), 1119-1150, arXiv:1103.5873.
  20. Nakajima H., t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory 7 (2003), 259-274, math.QA/0204185.
  21. Nakajima H., Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math. (2) 160 (2004), 1057-1097, math.QA/0105173.
  22. Nazarov M., Tarasov V., Representations of Yangians with Gelfand-Zetlin bases, J. Reine Angew. Math. 496 (1998), 181-212, q-alg/9502008.

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