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Yangian Characters and Classical \(\mathcal{W}\)-Algebras

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Conformal Field Theory, Automorphic Forms and Related Topics

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 8))

Abstract

The Yangian characters (or q-characters) are known to be closely related to the classical \(\mathcal{W}\)-algebras and to the centers of the affine vertex algebras at the critical level. We make this relationship more explicit by producing families of generators of the \(\mathcal{W}\)-algebras from the characters of the Kirillov–Reshetikhin modules associated with multiples of the first fundamental weight in types B and D and of the fundamental modules in type C. We also give an independent derivation of the character formulas for these representations in the context of the RTT presentation of the Yangians. In all cases the generators of the \(\mathcal{W}\)-algebras correspond to the recently constructed elements of the Feigin–Frenkel centers via an affine version of the Harish-Chandra isomorphism.

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Notes

  1. 1.

    Our V i essentially coincides with the operator \(\overline{V }_{i}[1]\) in the notation of [11, Sect. 7.3.4], which is associated with the Langlands dual Lie algebra \(^{L}\mathfrak{g}\).

References

  1. Arnaudon, D., Avan, J., Crampé, N., Frappat, L., Ragoucy, E.: R-matrix presentation for super-Yangians Y (osp(m | 2n)). J. Math. Phys. 44, 302–308 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arnaudon, D., Molev, A., Ragoucy, E.: On the R-matrix realization of Yangians and their representations. Annales Henri Poincaré 7, 1269–1325 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brundan, J., Kleshchev, A.: Representations of shifted Yangians and finite W-algebras. Mem. Am. Math. Soc. 196(918) (2008)

    Google Scholar 

  4. Chervov, A.V., Molev, A.I.: On higher order Sugawara operators. Int. Math. Res. Not. 2009, no. 9, 1612–1635

    Google Scholar 

  5. Chervov, A., Talalaev, D.: Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence. arXiv:hep-th/0604128

    Google Scholar 

  6. Chervov, A., Falqui, G., Rubtsov, V.: Algebraic properties of Manin matrices 1. Adv. Appl. Math. 43, 239–315 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dixmier, J.: Algèbres Enveloppantes. Gauthier-Villars, Paris (1974)

    MATH  Google Scholar 

  8. Drinfeld, V.G.: Quantum groups. In: International Congress of Mathematicians (Berkeley, 1986), pp. 798–820. American Mathematical Society, Providence (1987)

    Google Scholar 

  9. Drinfeld, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg–de Vries type. J. Sov. Math. 30, 1975–2036 (1985)

    Article  Google Scholar 

  10. Feigin, B., Frenkel, E.: Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras. Int. J. Mod. Phys. A 7(suppl. 1A), 197–215 (1992)

    Article  MATH  Google Scholar 

  11. Frenkel, E.: Langlands Correspondence for Loop Groups. Cambridge Studies in Advanced Mathematics, vol. 103. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  12. Frenkel, E., Mukhin, E.: Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras. Commun. Math. Phys. 216, 23–57 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Frenkel, E., Mukhin, E.: The Hopf algebra \(\mathrm{Rep}\,U_{q}\widehat{\mathfrak{g}\mathfrak{l}}_{\infty }\). Selecta Math. 8, 537–635 (2002)

    Google Scholar 

  14. 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, 1998). Contemporary Mathematics, vol. 248, pp. 163–205. American Mathematical Society, Providence (1999)

    Google Scholar 

  15. Hernandez, D.: The Kirillov–Reshetikhin conjecture and solutions of T-systems. J. Reine Angew. Math. 596, 63–87 (2006)

    MATH  MathSciNet  Google Scholar 

  16. Iorgov, N., Molev, A.I., Ragoucy, E.: Casimir elements from the Brauer–Schur–Weyl duality. J. Algebra 387, 144–159 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Isaev, A.P., Molev, A.I., Ogievetsky, O.V.: A new fusion procedure for the Brauer algebra and evaluation homomorphisms. Int. Math. Res. Not. 2012, no. 11, 2571–2606

    Google Scholar 

  18. Knight, H.: Spectra of tensor products of finite-dimensional representations of Yangians. J. Algebra 174, 187–196 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kuniba, A., Suzuki, J.: Analytic Bethe ansatz for fundamental representations of Yangians. Commun. Math. Phys. 173, 225–264 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kuniba, A., Ohta, Y., Suzuki, J.: Quantum Jacobi–Trudi and Giambelli formulae for U q (B r (1)) from the analytic Bethe ansatz. J. Phys. A 28, 6211–6226 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kuniba, A., Okado, M., Suzuki, J., Yamada, Y.: Difference L operators related to q-characters. J. Phys. A 35, 1415–1435 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kuniba, A., Nakanishi, T., Suzuki, J.: T-systems and Y -systems in integrable systems. J. Phys. A 44(10), 103001 (2011)

    Article  MathSciNet  Google Scholar 

  23. Molev, A.: Yangians and Classical Lie Algebras. Mathematical Surveys and Monographs, vol. 143. American Mathematical Society, Providence (2007)

    Google Scholar 

  24. Molev, A.I.: Feigin–Frenkel center in types B, C and D. Invent. Math. 191, 1–34 (2013)

    Google Scholar 

  25. Molev, A.I., Ragoucy, E.: The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for \(\widehat{\mathfrak{g}\mathfrak{l}}_{m\vert n}\). Moscow Math. J. 14, 83–119 (2014)

    MATH  Google Scholar 

  26. Mukhin, E., Young, C.A.S.: Path description of type B q-characters. Adv. Math. 231, 1119–1150 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mukhin, E., Young, C.A.S.: Extended T-systems. Selecta Math. (N.S.) 18, 591–631 (2012)

    Google Scholar 

  28. Mukhin, E., Tarasov, V., Varchenko, A.: Bethe eigenvectors of higher transfer matrices. J. Stat. Mech. Theory Exp. 2006, no. 8, P08002, 44 pp

    Google Scholar 

  29. Mukhin, E., Tarasov, V., Varchenko, A.: Bethe algebra of Gaudin model, Calogero-Moser space and Cherednik algebra. Int. Math. Res. Not. 2014, no. 5, 1174–1204

    Google Scholar 

  30. Nakai, W., Nakanishi, T.: Paths, tableaux and q-characters of quantum affine algebras: the C n case. J. Phys. A 39, 2083–2115 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Nakai, W., Nakanishi, T.: Paths and tableaux descriptions of Jacobi–Trudi determinant associated with quantum affine algebra of type D n . J. Algebraic Combin. 26, 253–290 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Nakajima, H.: t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras. Represent. Theory 7, 259–274 (2003)

    MATH  MathSciNet  Google Scholar 

  33. Nazarov, M.L.: Quantum Berezinian and the classical Capelli identity. Lett. Math. Phys. 21, 123–131 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  34. Okounkov, A.: Quantum immanants and higher Capelli identities. Transform. Groups 1, 99–126 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  35. Reshetikhin, N.Yu., Takhtajan, L.A., Faddeev, L.D.: Quantization of Lie groups and Lie algebras. Leningrad Math. J. 1, 193–225 (1990)

    MATH  MathSciNet  Google Scholar 

  36. Talalaev, D.V.: The quantum Gaudin system. Funct. Anal. Appl. 40, 73–77 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  37. Zamolodchikov, A.B., Zamolodchikov, Al.B.: Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models. Ann. Phys. 120, 253–291 (1979)

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia

    A. I. Molev

  2. Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN, 46202-3216, USA

    E. E. Mukhin

Authors
  1. A. I. Molev
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  2. E. E. Mukhin
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Correspondence to A. I. Molev .

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Editors and Affiliations

  1. University of Heidelberg, Department of Mathematics, Heidelberg, Germany

    Winfried Kohnen

  2. University of Heidelberg, Department of Mathematics, Heidelberg, Germany

    Rainer Weissauer

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Molev, A.I., Mukhin, E.E. (2014). Yangian Characters and Classical \(\mathcal{W}\)-Algebras. In: Kohnen, W., Weissauer, R. (eds) Conformal Field Theory, Automorphic Forms and Related Topics. Contributions in Mathematical and Computational Sciences, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43831-2_10

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