Positive-Energy Representations of the Group of Diffeomorphisms of the Circle

  • Roe Goodman
  • Nolan R. Wallach
Conference paper
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 4)

Abstract

Let D be the group of orientation-preserving diffeomorphisms of the circle S1. Then D is Fréchet Lie group with Lie algebra (d) the smooth real vector fields on S1. Let d be the subalgebra of real vector fields with finite Fourier series. This lecture outlines a proof that every infinitesimally unitary projective positive-energy representation of d integrates to a continuous projective unitary representation of D. This result was conjectured by V. Kac.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [G-W1]
    Goodman, R., and Wallach, N. R., Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle, J. fur reine und angewandte Math. (Crelles J.), Vol. 347(1984), 69–133.MathSciNetMATHGoogle Scholar
  2. [G-W2]
    Goodman, R., and Wallach, N. R., Projective Unitary Positive-Energy Representations of Diff(S1), J. Functional Analysis (to appear).Google Scholar
  3. [Ka1]
    Kac, V. G., Highest weight representations of infinite dimensional Lie algebras. Proceedings of ICM, Helsinki(1978), 299-304.Google Scholar
  4. [Ka2]
    Kac, V. G., Some problems on infinite-dimensional Lie algebras and their representations, in “Lie Algebras and Related Topics.” Lecture Notes in Mathematics, Vol. 933, Berlin, Heidelberg, New York: Springer 1982.Google Scholar
  5. [Ne]
    Nelson, E., Time-ordered operator products of sharp-time quadratic forms, J. Functional Analysis 11(1972), 211–219.MathSciNetMATHCrossRefGoogle Scholar
  6. [Se]
    Segal, G., Unitary representations of some infinite dimensional groups, Commun. Math. Phys. 80(1981), 301–342.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Roe Goodman
    • 1
  • Nolan R. Wallach
    • 1
  1. 1.Rutgers UniversityNew BrunswickUSA

Personalised recommendations