The Lie algebra of homeomorphisms of the circle

We define and study an infinite-dimensional Lie algebra homeo(+). which is shown to be naturally associated to the topological Lie group Homeo(+) of a ll orientation-preserving homeomorphisms of the circle. Roughly, we rely on the universal decorated Teichmuller theory developed before as motivation to provide Frechet coordinates on the homogeneous space given by Homeo(+). modulo the group of real fractional linear transformations, whose correspon ding vector fields on the circle we then extend by the usual Lie algebra sl (2) of real traceless two-by-two matrices in order to define homeo(+). Surp risingly, homeo(+) turns out to be equal to the algebra of all vector field s on the circle which are "piecewise sl(2)" in the obvious sense. It is evi dently important to consider the relationship between our new Frechet coord inates and the usual trigonometric functions on the circle, and we undertak e here both natural infinitesimal calculations. We finally apply some furth er previous work in order to give sufficient conditions on the Fourier coef ficients of a certain class of homeomorphisms of the circle which arises na turally in topology and number theory. (C) 1998 Academic Press.