Advertisement

Coadjoint Representation of Virasoro-type Lie Algebras and Differential Operators on ensor-densities

  • Valentin Yu. Ovsienko
Part of the DMV Seminar Band book series (OWS, volume 31)

Abstract

We discuss the geometrical nature of the coadjoint representation of the Virasoro algebra and some of its generalizations. The isomorphism of the coadjoint representation of the Virasoro group to the Diff(S1)-action on the space of Sturm-Liouville operators was discovered by A.A. Kirillov and G. Segal. This deep and fruitful result relates this topic to the classical problems of projective differential geometry (linear differential operators, projective structures on S1 etc.) The purpose of this talk is to give a detailed explanation of the A.A. Kirillov method [14] for the geometric realization of the coadjoint representation in terms of linear differential operators. Kirillov’s method is based on Lie superalgebras generalizing the Virasoro algebra. One obtains the Sturm-Liouville operators directly from the coadjoint representation of these Lie superalgebras. We will show that this method is universal. We will consider a few examples of infinite-dimensional Lie algebras and show that the Kirillov method can be applied to them. This talk is purely expository: all the results are known.

Keywords

Poisson Structure Projective Structure Linear Differential Operator Schwarzian Derivative Virasoro Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries type equation, Invent. Math. 50:3 (1987) 219–248.CrossRefGoogle Scholar
  2. [2]
    R. Bott, On the characteristic classes of groups of diffeomorphisms, Enseign. Math. 23:3–4 (1977), 209–220.MathSciNetzbMATHGoogle Scholar
  3. [3]
    E. Caftan, Lecons sur la theorie des espaces a connexion projective, Gauthier-Villars, Paris, 1937.Google Scholar
  4. [4]
    V.G. Drinfel’d & V.V. Sokolov, Lie algebras and equations of Korteweg-De Vries type, J. Soviet Math. 30 (1985), 1975–2036.CrossRefzbMATHGoogle Scholar
  5. [5]
    C. Duval & V. Ovsienko, Space of second order linear differential operators as a module over the Lie algebra of vector fields, Advances in Math. 132 (1997), no.2, 316–333.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    H. Gargoubi & V. Ovsienko, Space of linear differential operators on the real line as a module over the Lie algebra of vector fields, IMRN (1996), N.5, 235–251.MathSciNetCrossRefGoogle Scholar
  7. [7]
    I.M. Gel’fand & L.A. Dikii, A family of hamiltonian structures connected with integrable nonlinear differential equations in: I.M. Gel’fand collected papers (S.G. Gindikin et al, eds), Vol. 1, Springer, 1987, 625–646.Google Scholar
  8. [8]
    I.M. Gel’fand & D.B. Fuchs, Cohomology of the Lie algebra of vector fields on the circle, Funct. Anal. Appl. 2:4 (1968), 342–343.CrossRefGoogle Scholar
  9. [9]
    P. Gordan, Invariantentheorie, Teubner, Leipzig, 1887.Google Scholar
  10. [10]
    L. Guieu, Nombre de rotation, structures geometriques sur un cercle et groupe de Bott-Virasoro, Ann. Inst. Fourier 46 (1996), no.4, 971–1009.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    L. Guieu & V. Ovsienko, Structures symplectiques sur les espaces de courbes projectives et affines, J. Geom. Phys. 16 (1995) 120–148.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    P. Ya. Grozman, Classification of bilinear invariant operators over tensor fields, Funct. Anal. Appl., 14 (1980), 58–59.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    A.A. Kirillov, Infinite dimensional Lie groups: their orbits, invariants and representations. The geometry of moments, Lect. Notes in Math., 970, Springer-Verlag (1982) 101–123.Google Scholar
  14. [14]
    A.A. Kirillov, Orbits of the group of diffeomorphisms of a circle and local superalgebras, Funct. Anal Appl., 15:2 (1980) 135–137.CrossRefGoogle Scholar
  15. [15]
    A.A. Kirillov, ähler structure on K -orbits of the group of diffeomorphisms of a circle, Funct. Anal. Appl. 21:2 (1987) 122–125.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    A.A. Kirillov & D.V. Yuriev, Kiihler geometry of the infinite-dimensional homogeneous space M =Diff+(S’)/Rot(S’), Funct. Anal. Appl. 21:4 (1987) 248–294.CrossRefGoogle Scholar
  17. [17]
    B. Kostant & S. Sternberg, The Schwarzian derivative and the conformal geometry of the Lorentz hyperboloid, in: Quantum Theories and Geometry (M. Cahen and M. Flato eds.) Kluwer, 1988, 113–125.CrossRefGoogle Scholar
  18. [18]
    N.H. Kuiper, Locally projective spaces of dimension one, Michigan Math. J., 2 (1954) 95–97.MathSciNetzbMATHGoogle Scholar
  19. [19]
    V.F. Lazutkin & T F Pankratova, Normal forms and versal deformations for Hill’s equations, Funct. Anal. Appl 9:4 (1975), 306–311.MathSciNetCrossRefGoogle Scholar
  20. [20]
    P.B.A. Lecomte, P. Mathonet & E. Tousset, Comparison of some modules of the Lie algebra of vector fields, Indag. Math. (N.S.) 7 (1996), no.4, 461–471.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    D.A. Leites, B.A. Feigin, New Lie superalgebras of string theories, Group-Theoretic Methods in Physics, v.1, Moscow (1983) 269–273.Google Scholar
  22. [22]
    P. Marcel, V. Ovsienko & C. Roger, Extension of the Virasoro and Neveu-Schwarz algebras and generalized Sturm-Liouville operators, Lett. Math. Phys. 40 (1997), no.1, 31–39.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    V. Ovsienko, Classification of third-order linear differential equations and symplectic sheets of the Gel fand-Dikii bracket, Math. Notes, 47:5 (1990) 465–470.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    O. Ovsienko & V. Ovsienko, Lie derivative of order n on a line. Tensor meaning of the Gelfand-Dickey bracket, Adv. in Soviet Math., 2, 1991.Google Scholar
  25. [25]
    V. Ovsienko, C. Roger, Extension of Virasoro group and Virasoro algebra by modules of tensor densities on S l, Funct. Anal Appl. 30 (1996), no.4, 290–291.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    C. Roger, Extensions centrales d’algebres et de groupes de Lie de dimension infinie, algebre de Virasoro et generalisations, Rep. Math. Phys. 35 (1995), no.2–3, 225–266.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    G.B. Segal, Unitary representations of some infinite dimensional groups, Comm Math. Phys., 80:3 (1981) 301–342.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    G.B. Segal, The geometry of the KdV equation in: Trieste Conference on topological methods in quantum field theories ¡X W Nahm & al, eds ¡X World Scientific (1990), 96–106.Google Scholar
  29. [29]
    E.J. Wilczynski, Projective differential geometry of curves and ruled surfaces, Leipzig ¡X Teubner, 1906.Google Scholar
  30. [30]
    E. Witten, Coadjoint orbits of the Virasoro group, Comm. Math. Phys., 114:1, (1988) 1–53.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    P. Marcel, Extensions of the Neveu-Schwarz Lie superalgebra, Comm. Math. Phys. 207 (1999), no. 2, 291–306.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    P. Marcel, Generalizations of the Virasoro algebra and matrix Sturm-Lionville operators, J. Geom. Phys. 36 (2000), 211–222.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Valentin Yu. Ovsienko

There are no affiliations available

Personalised recommendations