Lie Subalgebras of Differential Operators in One Variable

  • F. J. Plaza MartínEmail author
  • C. Tejero Prieto


Let \(\mathrm{Witt}\) be the Lie algebra generated by the set \(\{L_i\,\vert \, i \in {{\mathbb {Z}}}\}\) and \(\mathrm{Vir}\) its universal central extension. Let \(\mathrm{Diff}(V)\) be the Lie algebra of differential operators on \(V={{\mathbb {C}}}(\!(z)\!)\), Open image in new window or \(V={{\mathbb {C}}}(z)\). We explicitly describe all Lie algebra homomorphisms from \(\mathfrak {sl}(2)\), \(\mathrm{Witt}\), and \(\mathrm{Vir}\) to \(\mathrm{Diff}(V)\), such that \(L_0\) acts on V as a first-order differential operator.


Witt algebra Virasoro algebra representation theory differential operators 

Mathematics Subject Classification

Primary 17B68 Secondary 81R10 



The authors would like to thank the anonymous referee for the careful reading of the paper and pointing out many typos and some inaccuracies that have helped to greatly improve this work.


  1. 1.
    Bavula, V.V.: Classification of simple sl(2)-modules and the finite-dimensionality of the module of extensions of simple sl(2)-modules. Ukr. Math. J. 42(9), 1044–1049 (1990). (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Block, R.E.: The irreducible representations of the Lie algebra \(\mathfrak{sl}(2)\) and of the Weyl algebra. Adv. Math. 39(1), 69–110 (1981)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dixmier, J.: Sur les algèbres de Weyl. Bull. Soc. Math. Fr. 96, 209–242 (1968)CrossRefGoogle Scholar
  4. 4.
    Draisma, J.: Constructing Lie algebras of first order differential operators. J. Symb. Comput. 36(5), 685–698 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Frenkel, E., Kac, V., Radul, A., Wang, W.: \({\cal{W}}_{1+\infty }\) and \({\cal{W}}({\mathfrak{gl}}(N))\) with central charge \(N\). Commun. Math. Phys. 170(2), 337–357 (1995)CrossRefGoogle Scholar
  6. 6.
    Givental, A.: Gromov–Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J. 1(4), 551–568 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gelfand, I.M., Fuchs, D.B.: Cohomology of the Lie algebra of vector fields on a circle. Funct. Anal. Appl. 2, 92–93 (1968). English translation: Funct. Anal. Appl. 2, 342–343 (1968)Google Scholar
  8. 8.
    Iohara, K., Koga, Y.: Representation Theory of the Virasoro Algebra. Springer Monographs in Mathematics. Springer, London (2011)CrossRefGoogle Scholar
  9. 9.
    Kac, V.G., Raina, A.K., Rozhkovskaya, N.: Bombay lectures on highest weight representations of infinite dimensional Lie algebras. In: Advanced Series in Mathematical Physics, 2nd Edn., 29. World Scientific Publishing Co. Pte. Ltd., Hackensack (2013) (ISBN: 978-981-4522-19-9) Google Scholar
  10. 10.
    Kamran, N., Olver, P.J.: Lie algebras of differential operators and Lie-algebraic potentials. J. Math. Anal. Appl. 145(2), 342–356 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lie, S., Theorie der Transformationsgruppen, Vols. I, II, III. Unter Mitwirkung von. F. Engel., Leipzig, 1888, 1890 (1893)Google Scholar
  12. 12.
    Mazorchuk, V.: Lectures on \(\mathfrak{sl}_{2}({\mathbb{C}})\)-modules. Imperial College Press, London (2010)zbMATHGoogle Scholar
  13. 13.
    Miller Jr., W.: Lie Theory and Special Functions, Mathematics in Science and Engineering, vol. 43. Academic Press, New York (1968)Google Scholar
  14. 14.
    Mulase, M.: Algebraic theory of the KP equations. In: Perspectives in Mathematical Physics, Conf. Proc. Lecture Notes Math. Phys., III, pp. 151–217. International Press, Cambridge (1994)Google Scholar
  15. 15.
    Plaza Martín, F.J.: Representations of the Witt algebra and Gl(n)-opers. Lett. Math. Phys. 103(10), 1079–1101 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Plaza Martín, F.J.: Algebro-geometric solutions of the generalized Virasoro constraints. SIGMA Symmetry Integrability Geom. Methods Appl. 11, 052 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Plaza Martín, F.J., Tejero Prieto, C.: Extending representations of \(\mathfrak{sl}(2)\) to Witt and Virasoro algebras. Algebr Represent. Theor. 20, 433–468 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Plaza Martín, F.J., Tejero Prieto, C.: Construction of simple non-weight \(\mathfrak{sl}(2)\)-modules of arbitrary rank. J. Algebra 472, 172–194 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Plaza Martín, F.J., Tejero Prieto, C.: Virasoro and KdV. Lett. Math. Phys. 107(5), 963–994 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pope, C.N., Romans, L.J., Shen, X.: Ideals of Kac–Moody algebras and realisations of \(W_{\infty }\). Phys. Lett. B 245(1), 72–78Google Scholar
  21. 21.
    Popovych, R.O., Boyko, V.M., Nesterenko, M.O., Lutfullin, M.W.: Realizations of real low-dimensional Lie algebras. J. Phys. A 36(26), 7337–7360 (2003)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ribault, S.: Conformal field theory on the plane. arXiv:1406.4290v3
  23. 23.
    Smirnov, Y., Turbiner, A.: Hidden \(\mathfrak{sl}(2)\)-algebra of finite-difference equations. In: Proceedings of the IV Wigner Symposium (Guadalajara, 1995), pp. 435–440, World Sci. Publ., River Edge (1996)Google Scholar
  24. 24.
    Turbiner, A.: Lie algebras and polynomials in one variable. J. Phys. A 25(18), L1087–L1093 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de Matemáticas and IUFFyMUniversidad de SalamancaSalamancaSpain

Personalised recommendations