The Lie Algebra of Homeomorphisms of the Circle

AbstractWe define and study an infinite-dimensional Lie algebrahomeo+which is shown to be naturally associated to thetopologicalLie grouphomeo+of all orientation-preserving homeomorphisms of the circle. Roughly, we rely on the universal decorated Teichmüller theory developed before as motivation to provide Fréchet coordinates on the homogeneous space given byhomeo+modulo the group of real fractional linear transformations, whose corresponding vector fields on the circle we then extend by the usual Lie algebrasl2of real traceless two-by-two matrices in order to definehomeo+. Surprisingly,homeo+turns out to be equal to the algebra of all vector fields on the circle which are “piecewisesl2” in the obvious sense. It is evidently important to consider the relationship between our new Fréchet coordinates and the usual trigonometric functions on the circle, and we undertake here both natural infinitesimal calculations. We finally apply some further previous work in order to give sufficient conditions on the Fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory