Generalized Onsager Algebras

  • Jasper V. StokmanEmail author
Open Access


Let \(\mathfrak {g}(A)\) be the Kac-Moody algebra with respect to a symmetrizable generalized Cartan matrix A. We give an explicit presentation of the fix-point Lie subalgebra \(\mathfrak {k}(A)\) of \(\mathfrak {g}(A)\) with respect to the Chevalley involution. It is a presentation of \(\mathfrak {k}(A)\) involving inhomogeneous versions of the Serre relations, or, from a different perspective, a presentation generalizing the Dolan-Grady presentation of the Onsager algebra. In the finite and untwisted affine case we explicitly compute the structure constants of \(\mathfrak {k}(A)\) in terms of a Chevalley type basis of \(\mathfrak {k}(A)\). For the symplectic Lie algebra and its untwisted affine extension we explicitly describe the one-dimensional representations of \(\mathfrak {k}(A)\).


Kac-Moody algebra Chevalley involution Fix-point Lie subalgebra Onsager algebra Dolan-Grady relations Inhomogeneous Serre relations Split symmetric pairs 

Mathematics Subject Classification (2010)

17B67 81R10 



I thank Stefan Kolb, Pascal Baseilhac, Samuel Belliard and Nicolai Reshetikhin for useful discussions on the topic of the paper. Shortly after the appearance on the arXiv of this paper Xinhong Chen, Ming Lu and Weiqiang Wang informed me that they have obtained an explicit presentation of Kolb’s coideal subalgebra associated to a quasi-split Kac-Moody quantum symmetric pair using different techniques. Their results have appeared now in the preprint [3].


  1. 1.
    Balagovic, M., Kolb, S.: The bar involution for quantum symmetric pairs. Represent. Theory 19, 186–210 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baseilhac, P., Belliard, S.: Generalized q-Onsager algebras and boundary affine Toda field theories. Lett. Math. Phys. 93, 213–228 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, X., Lu, M., Wang, W.: A Serre presentation for the ι quantum groups. arXiv:1810.12475
  4. 4.
    Gabber, O., Kac, V.G.: On defining relations of certain infinite-dimensional Lie algebras. Bull. Amer. Math. Soc. (N.S.) 5, 185–189 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Date, E., Usami, K.: On an analog of the Onsager algebra of type \(D_{n}^{(1)}\). In: Kac-Moody Lie Algebras and Related Topics, 43–51, Contemp. Math., p. 343. Amer. Math. Soc., Providence (2004)Google Scholar
  6. 6.
    Davies, B.: Onsager’s algebra and superintegrability. J. Phys. A 23, 2245–2261 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Davies, B.: Onsager’s algebra and the Dolan-Grady condition in the non-self-dual case. J. Math Phys. 32, 2945–2950 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dolan, L., Grady, M.: Conserved charges from self-duality. Phys. Rev. D (3) 25, 1587–1604 (1982)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Garland, H.: Arithmetic theory of loop algebras. J. Algebra 53, 480–551 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hartwig, B., Terwilliger, P.: The tetrahedron algebra, the Onsager algebra, and the \(\mathfrak {sl}_{2}\) loop algebra. J. Algebra 308, 840–863 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory Graduate Texts in Math, vol. 9. Springer, New York (1972)CrossRefGoogle Scholar
  12. 12.
    Kac, V.G.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge Univ. Press (1990)Google Scholar
  13. 13.
    Knapp, A.W.: Lie Groups Beyond an Introduction, Progress in Math., vol. 140. Birkhäuser (1996)Google Scholar
  14. 14.
    Kolb, S.: Radial part calculations for \(\widehat {\mathfrak {sl}}_{2}\) and the Heun KZB heat equation. Int. Math. Res. Not. IMRN 23, 12941–12990 (2015)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kolb, S.: Quantum symmetric Kac-Moody pairs. Adv. Math. 267, 395–469 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Letzter, G.: Coideal subalgebras and quantum symmetric pairs. In: New Directions in Hopf Algebras (Cambridge), MSRI Publications, vol. 43, pp. 117–166. Cambridge Univ. Press (2002)Google Scholar
  17. 17.
    Letzter, G.: Cartan subalgebras of quantum symmetric pair coideals. Represent. Theory 23, 88–153 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. (2) 65, 117–149 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Perk, J.H.H.: Star-triangle relations, quantum Lax pairs, and higher genus curves,. In: Proc. Sympos. Pure Math., vol. 49, pp. 341–354. Amer. Math. Soc., Providence (1989)Google Scholar
  20. 20.
    Reshetikhin, N., Stokman, J.V.: Vector-valued Harish-Chandra series and their applications, in preparationGoogle Scholar
  21. 21.
    Roan, S.S.: Onsager’s algebra, loop algebra and chiral Potts model, preprint MPI 91-70, Max Planck Institute for Math. Bonn (1991)Google Scholar
  22. 22.
    Uglov, D.B., Ivanov, I.T.: \(\mathfrak {sl}(N)\) Onsager’s algebra and integrability. J. Stat. Phys. 82, 87–113 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.KdV Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations