Chapter

Infinite Dimensional Groups with Applications

Volume 4 of the series Mathematical Sciences Research Institute Publications pp 125-135

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

  • Roe GoodmanAffiliated withRutgers University
  • , Nolan R. WallachAffiliated withRutgers University

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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.