Algebras and Representation Theory

, Volume 22, Issue 1, pp 177–209 | Cite as

Several realizations of Fock modules for toroidal \(\ddot {U}_{q,d}(\mathfrak {sl}_{n})\)

  • Alexander TsymbaliukEmail author


In this paper, we relate the well-known Fock representations of \(\ddot {U}_{q,d}(\mathfrak {sl}_{n})\) to the vertex, shuffle, and ‘L-operator’ representations of \(\ddot {U}_{q,d}(\mathfrak {sl}_{n})\). These identifications generalize those for the quantum toroidal algebra of \(\mathfrak {gl}_{1}\), which were recently established in Feigin et al. (J. Phys. A 48(24), 244001, 2015).


Quantum toroidal algebras Shuffle algebras L operators Fock module Vertex module 

Mathematics Subject Classification (2010)

20G42 17B37 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



I would like to thank P. Etingof, B. Feigin, M. Finkelberg, and A. Negut for many stimulating discussions over the years. I am indebted to the anonymous referee for insightful comments on the first version of the paper, which led to a better exposition of the material.

I would like to thank the Max Planck Institute for Mathematics in Bonn for the hospitality and support in June 2015, where part of this project was carried out. The author also gratefully acknowledges support from the Simons Center for Geometry and Physics, Stony Brook University, at which most of the research for this paper was performed, as well as Yale University, where the final version of this paper was completed.

This work was partially supported by the NSF Grants DMS–1502497, DMS–1821185.


  1. 1.
    Beck, J.: Braid group action and quantum affine algebras. Comm. Math. Phys. 165(3), 555–568 (1994). arXiv:hep-th/9404165 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chari, V., Jing, N.: Realization of level one representations of \(U_q(\widehat {\mathfrak {g}})\) at a root of unity. Duke Math. J. 108 (1), 183–197 (2001). arXiv:math/9909118 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Damiani, I.: From the Drinfeld realization to the Drinfeld-Jimbo presentation of affine quantum algebras: injectivity. Publ. Res. Inst. Math. Sci. 51(1), 131–171 (2015). arXiv:1407.0341 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Drinfeld, V.: Quantum Groups, vol. 1, 2, pp. 798–820. Amer. Math. Soc., Providence (1987)Google Scholar
  5. 5.
    Drinfeld, V.: A new realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36(2), 212–216 (1988)Google Scholar
  6. 6.
    Ding, J., Iohara, K.: Generalization of drinfeld quantum affine algebras. Lett. Math. Phys. 41(2), 181–193 (1997). arXiv:q-alg/9608002 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Frenkel, I., Jing, N.: Vertex representations of quantum affine algebras. Proc. Nat. Acad. Sci. U.S.A. 85(24), 9373–9377 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Feigin, B., Jimbo, M., Miwa, T., Mukhin, E.: Representations of quantum toroidal \(\mathfrak {gl}_{n}\). J. Algebra 380, 78–108; (2013). arXiv:1204.5378 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Feigin, B., Jimbo, M., Miwa, T., Mukhin, E.: Quantum toroidal \(\mathfrak {gl}_{1}\) and Bethe ansatz. J. Phys. A 48 (24), 244001, 27 (2015). arXiv:1502.07194 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Feigin, B., Tsymbaliuk, A.: Bethe subalgebras of \(U_{q}(\widehat {\mathfrak {gl}}_{n})\) via shuffle algebras. Selecta Math. 122 (2), 979–1011 (2016). arXiv:1504.01696 CrossRefzbMATHGoogle Scholar
  11. 11.
    Feigin, B., Tsymbaliuk, A.: Bethe ansatz for quantum toroidal \(\ddot {U}_{q,d}(\mathfrak {sl}_n)\) via shuffle algebras, in preparationGoogle Scholar
  12. 12.
    Grossé, P.: On quantum shuffle and quantum affine algebras. J. Algebra 318 (2), 495–519 (2007). arXiv:math/0107176 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hernandez, D.: Representations of quantum affinizations and fusion product. Transform. Groups 10(2), 163–200 (2005). arXiv:math/0312336 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jantzen, J.: Lectures on Quantum Groups, Graduate Studies in Mathematics, vol. 6, p. viii+ 266. American Mathematical Society, Providence, RI (1996)Google Scholar
  15. 15.
    Jimbo, M.: A q-analogue of \(U({\mathfrak {gl}}(n + 1))\), Hecke algebra and the Yang-Baxter equation. Lett. Math. Phys. 11(3), 247–252 (1986)Google Scholar
  16. 16.
    Jing, N.: On drinfeld realization of quantum affine algebras. In: Monster and Lie Algebras, Columbus, OH, 1996, vol. 7, pp. 195–206. Ohio State University Mathematical Research Institute Publications, de Gruyter, Berlin (1998). arXiv:q-alg/9610035
  17. 17.
    Kassel, C., Rosso, M., Turaev, V.: Quantum groups and knot invariants, panoramas et synthèses [panoramas and syntheses], 5. Société Mathématique de France, Paris, vi+ 115pp (1997)Google Scholar
  18. 18.
    Miki, K.: Toroidal braid group action and an automorphism of toroidal algebra \(U_{q}(\mathfrak {sl}_{n + 1,tor}) (n\geq 2)\). Lett. Math. Phys. 47(4), 365–378 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Miki, K.: Representations of quantum toroidal algebra \(U_{q}(\mathfrak {sl}_{n + 1,\text {tor}}) (n\geq 2)\). J. Math. Phys. 41(10), 7079–7098 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Negut, A.: Quantum toroidal and shuffle algebras, R-matrices and a conjecture of Kuznetsov. preprint, arXiv:1302.6202 (2013)
  21. 21.
    Saito, Y.: Quantum toroidal algebras and their vertex representations. Publ. Res. Inst. Math. Sci. 34(2), 155–177 (1998). arXiv:q-alg/9611030 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tsymbaliuk, A.: The affine Yangian of \(\mathfrak {gl}_{1}\) revisited. Adv. Math. 304(2), 583–645 (2017). arXiv:1404.5240 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Varagnolo, M., Vasserot, E.: Schur duality in the toroidal setting. Commun. Math. Phys. 182(2), 469–483 (1996). arXiv:q-alg/9506026 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Personalised recommendations