Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Yangians and quantum loop algebras

  • 280 Accesses

  • 20 Citations


Let \(\mathfrak{g }\) be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra \(U_\hbar (L\mathfrak g )\) of \(\mathfrak{g }\) degenerates to the Yangian \({Y_\hbar (\mathfrak g )}\). We strengthen this result by constructing an explicit algebra homomorphism \(\Phi \) from \(U_\hbar (L\mathfrak g )\) to the completion of \({Y_\hbar (\mathfrak g )}\) with respect to its grading. We show moreover that \(\Phi \) becomes an isomorphism when \({U_\hbar (L\mathfrak g )}\) is completed with respect to its evaluation ideal. We construct a similar homomorphism for \(\mathfrak{g }=\mathfrak{gl }_n\) and show that it intertwines the actions of \(U_\hbar (L\mathfrak gl _{n})\) and \(Y_\hbar (\mathfrak gl _{n})\) on the equivariant \(K\)-theory and cohomology of the variety of \(n\)-step flags in \({\mathbb{C }}^d\) constructed by Ginzburg–Vasserot.

This is a preview of subscription content, log in to check access.


  1. 1.

    \( \widetilde{U}^0\) is isomorphic to the subalgebra \(U^0\) of \(U_\hbar (L\mathfrak g )\) generated by \(\{H _{i,r}\}_{i\in \mathbf{I },r\in \mathbb Z }\) by the PBW Theorem for \(U_\hbar (L\mathfrak g )\) [1], but we shall not need this fact.

  2. 2.

    note that the expansions in \(z^ {\pm 1}\) are related by the symmetry \(\vartheta _m(\zeta ^{-1})=\vartheta _{-m}(\zeta )\).

  3. 3.

    our conventions are adapted to [15, 34] They differ from those of [25] by the permutation \(e_{i,r}\leftrightarrow f_{i,r}\) and the relabelling \(\theta _{i,r}\leftrightarrow h_{i,r}\).

  4. 4.

    Note the difference between \(J(v)\) and the function \(G(v)\) used in Sect. 4.1 for constructing the solutions for simple Lie algebras: \(J(v) = G(v) + \frac{v}{2}\).


  1. 1.

    Beck, J.: Convex bases of PBW type for quantum affine algebras. Commun. Math. Phys. 165, 193–199 (1994)

  2. 2.

    Chari, V., Hernandez, D.: Beyond Kirillov-Reshetikhin modules. In: Quantum Affine Algebras, Extended Affine Lie Algebras, and Their Applications, pp. 49–81, Contemp. Math., vol. 506, Am. Math. Soc. (2010)

  3. 3.

    Chari, V., Pressley, A.: Yangians and \(R\)-matrices. Enseign. Math. 36, 267–302 (1990)

  4. 4.

    Chari, V., Pressley, A.: Quantum affine algebras. Commun. Math. Phys. 142, 261–283 (1991)

  5. 5.

    Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)

  6. 6.

    Cherednik, I.: Affine extensions of Knizhnik-Zamolodchikov equations and Lusztig’s isomorphisms. Special functions (Okayama, 1990), 63–77, ICM-90 Satell. Springer, Conf. Proc. 1991

  7. 7.

    Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Birkhäuser, Boston (1997)

  8. 8.

    Ding, J., Frenkel, I.: Isomorphism of two realizations of the quantum affine algebra \({U}_q(\widehat{\mathfrak{gl}(n)})\). Commun. Math. Phys. 156, 277–300 (1993)

  9. 9.

    Drinfeld, V.G.: Hopf algebras and the quantum Yang-Baxter equation. Soviet Math. Dokl. 32, 254–258 (1985)

  10. 10.

    Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, (Berkeley, 1986), pp. 798–820, Am. Math. Soc. (1987)

  11. 11.

    Drinfeld, V.G.: A new realization of Yangians and quantum affine algebras. Soviet Math. Dokl. 36, 212–216 (1988)

  12. 12.

    Drinfeld, V.G.: Personal Communication (2009)

  13. 13.

    Gautam, S., Toledano Laredo, V.: Yangians and quantum loop algebras—II (in preparation)

  14. 14.

    Ginzburg, V.: Geometric methods in the representation theory of Hecke algebras and quantum groups. Notes by Vladimir Baranovsky. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 514, Representation theories and algebraic geometry (Montreal, PQ, 1997), pp. 127–183, Kluwer, Dordrecht (1998)

  15. 15.

    Ginzburg, V., Vasserot, E.: Langlands reciprocity for affine quantum groups of type \({A}_n\). Int. Math. Res. Not. IMRN, 67–85 (1993)

  16. 16.

    Guay, N.: Affine Yangians and deformed double current algebras in type A. Adv. Math. 211, 436–484 (2007)

  17. 17.

    Guay, N., Ma, X.: From quantum loop algebras to Yangians. J. Lond. Math. Soc. (to appear). doi:10.1112/jlms/jds021, published, online July 3

  18. 18.

    Guay, N., Nakajima, H.: Personal Communication (2012)

  19. 19.

    Jantzen, J.C.: Lectures on Quantum Groups. Graduate Studies in Mathematics, 6. American Mathematical Society (1996)

  20. 20.

    Jing, N.: Quantum Kac-Moody algebras and vertex representations. Lett. Math. Phys. 44, 261–271 (1998)

  21. 21.

    Kassel, C.: Quantum Groups, Graduate Texts in Mathematics. Springer, Berlin (1995)

  22. 22.

    Levendorskii, S.Z.: On generators and defining relations of Yangians. J. Geom. Phys. 12, 1–11 (1992)

  23. 23.

    Levendorskii, S.Z.: On PBW bases for Yangians. Lett. Math. Phys. 27, 37–42 (1993)

  24. 24.

    Lusztig, G.: Affine Hecke algebras and their graded versions. J. Am. Math. Soc. 2, 599–635 (1989)

  25. 25.

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

  26. 26.

    Nakajima, H.: Quiver varieties and finite dimensional representations of quantum affine algebras. J. Am. Math. Soc. 14, 145–238 (2001)

  27. 27.

    Olshanskii, G.I.: Representations of infinite-dimensional classical groups, limits of enveloping algebras, and Yangians, Topics in representation theory, Adv. Soviet Math., 2, Am. Math. Soc., 1–66

  28. 28.

    Tarasov, V.O.: The structure of quantum L-operators for the \(R\)-matrix of the XXZ-model. Theor. Math. Phys. 61, 1065–1072 (1984)

  29. 29.

    Tarasov, V.O.: Irreducible monodromy matrices for an \(R\)-matrix of the XXZ-model, and lattice local quantum Hamiltonians. Theor. Math. Phys. 63, 440–454 (1985)

  30. 30.

    Toledano Laredo, V.: A Kohno-Drinfeld theorem for quantum Weyl groups. Duke Math. J. 112(3), 421–451 (2002)

  31. 31.

    Toledano Laredo, V.: Quasi-Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups. Int. Math. Res. Pap. IMRP (2008), art. ID rpn009, 167 p

  32. 32.

    Toledano Laredo, V.: The trigonometric Casimir connection of a simple Lie algebra. J. Algebra 329, 286–327 (2011)

  33. 33.

    Varagnolo, M.: Quiver varieties and Yangians. Lett. Math. Phys. 53, 273–283 (2000)

  34. 34.

    Vasserot, E.: Affine quantum groups and equivariant K-theory. Transform. Groups 3, 269–299 (1998)

Download references


We are very grateful to Ian Grojnowski from whom we learned that the quantum loop algebra and Yangian should be isomorphic after appropriate completions. His explanations and friendly insistence helped us overcome our initial doubts. We are also grateful to V. Drinfeld for showing us a proof that finite-dimensional representations separate elements of the Yangian and allowing us to reproduce it in Sect. 8. We would also like to thank N. Guay for sharing a preliminary version of the preprint [17] and E. Vasserot for useful discussions.

Author information

Correspondence to Valerio Toledano Laredo.

Additional information

Both authors were supported by NSF grants DMS-0707212 and DMS-0854792.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gautam, S., Toledano Laredo, V. Yangians and quantum loop algebras. Sel. Math. New Ser. 19, 271–336 (2013). https://doi.org/10.1007/s00029-012-0114-2

Download citation


  • Affine quantum groups
  • Yangian
  • Quantum loop algebra

Mathematics Subject Classification (1991)

  • 17B37 (17B67 · 82B23 · 14F43)