h-Adic quantum vertex algebras associated with rational R-matrix in types B, C and D

  • Marijana Butorac
  • Naihuan Jing
  • Slaven KožićEmail author


We introduce the h-adic quantum vertex algebras associated with the rational R-matrix in types B, C and D, thus generalizing Etingof–Kazhdan’s construction in type A. Next, we construct the algebraically independent generators of the center of the h-adic quantum vertex algebra in type B at the critical level, as well as the families of central elements in types C and D. Finally, as an application, we obtain commutative subalgebras of the dual Yangian and the families of central elements of the appropriately completed double Yangian at the critical level, in types B, C and D.


Quantum vertex algebra Double Yangian Center at the critical level 

Mathematics Subject Classification

17B37 17B69 



The first author is partially supported by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund—the Competitiveness and Cohesion Operational Programme (Grant KK. The second author is supported by Simons Foundation Grant No. 523868 and National Natural Science Foundation of China Grant No. 11531004. The part of the research was carried out during the second author’s visit to the Department of Mathematics at the Faculty of Science, University of Zagreb. He would like to thank the Department for the hospitality during his visit.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. 1.
    Arnaudon, D., Molev, A., Ragoucy, E.: On the \(R\)-matrix realization of Yangians and their representations. Ann. Henri Poincaré 7, 1269–1325 (2006). arXiv:math/0511481 [math.QA]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brauer, R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. 38, 854–872 (1937)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Etingof, P., Kazhdan, D.: Quantization of Lie bialgebras. V, Selecta Math. (N.S.) 6, 105–130 (2000). arXiv:math/9808121 [math.QA]MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Frappat, L., Jing, N., Molev, A., Ragoucy, E.: Higher Sugawara operators for the quantum affine algebras of type \(A\). Commun. Math. Phys. 345, 631–657 (2016). arXiv:1505.03667 [math.QA]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Frenkel, E.: Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathematics, vol. 103. Cambridge University Press, Cambridge (2007)Google Scholar
  7. 7.
    Gardini, M.: Quantum vertex algebras. Ph.D. thesis, Sapienza – University of Rome (2018)Google Scholar
  8. 8.
    Isaev, A.P., Molev, A.I.: Fusion procedure for the Brauer algebra. St. Petersburg Math. J. 22, 437–446 (2011). arXiv:0812.4113 [math.RT]MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Isaev, A.P., Molev, A.I., Ogievetsky, O.V.: A new fusion procedure for the Brauer algebra and evaluation homomorphisms. Int. Math. Res. Not. 2571–2606 (2012). arXiv:1101.1336 [math.RT]
  10. 10.
    Jing, N., Kožić, S., Molev, A., Yang, F.: Center of the quantum affine vertex algebra in type \(A\). J. Algebra 496, 138–186 (2018). arXiv:1603.00237 [math.QA]MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jing, N., Liu, M., Molev, A.: Isomorphism between the \(R\)-matrix and Drinfeld presentations of Yangian in types \(B\), \(C\) and \(D\). Commun. Math. Phys. 361, 827–872 (2018). arXiv:1705.08155 [math.QA]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jing, N., Liu, M., Yang, F.: Double Yangians of classical types and their vertex representations. arXiv:1810.06484 [math.QA]
  13. 13.
    Jucys, A.: On the Young operators of the symmetric group. Lietuvos Fizikos Rinkinys 6, 163–180 (1966)MathSciNetGoogle Scholar
  14. 14.
    Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kožić, S.: Quasi modules for the quantum affine vertex algebra in type \(A\). Commun. Math. Phys. 365, 1049–1078 (2019). arXiv:1707.09542 [math.QA]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Li, H.-S.: Nonlocal vertex algebras generated by formal vertex operators. Selecta Math. (N. S.) 11, 349–397 (2005). arXiv:math/0502244 [math.QA]MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li, H.-S.: \(\hbar \)-adic quantum vertex algebras and their modules. Commun. Math. Phys. 296, 475–523 (2010). arXiv:0812.3156 [math.QA]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, H.-S.: \(\phi \)-Coordinated quasi-modules for quantum vertex algebras. Commun. Math. Phys. 308, 703–741 (2011). arXiv:0906.2710 [math.QA]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Molev, A.I.: Feigin–Frenkel center in types \(B\), \(C\) and \(D\). Invent. Math. 191, 1–34 (2013). arXiv:1105.2341 [math.RT]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Molev, A.: Sugawara Operators for Classical Lie Algebras, Mathematical Surveys and Monographs, vol. 229. American Mathematical Society, Providence (2018)CrossRefzbMATHGoogle Scholar
  21. 21.
    Reshetikhin, NYu., Semenov-Tian-Shansky, M.A.: Central extensions of quantum current groups. Lett. Math. Phys. 19, 133–142 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zamolodchikov, A.B., Zamolodchikov, AlB: Factorized \(S\)-matrices in two dimensions as the exact solutions of certain relativistic quantum field models. Ann. Phys. 120, 253–291 (1979)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of RijekaRijekaCroatia
  2. 2.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  3. 3.Department of Mathematics, Faculty of ScienceUniversity of ZagrebZagrebCroatia

Personalised recommendations