We define and study an infinite-dimensional Lie algebra homeo + which is shown to be naturally associated to the topological Lie group Homeo + of all orientationpreserving homeomorphisms of the circle. Roughly, we rely on the universal decorated Teichmu ller theory developed before as motivation to provide Fre chet coordinates on the homogeneous space given by Homeo + modulo the group of real fractional linear transformations, whose corresponding 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 + . Surprisingly, homeo + turns out to be equal to the algebra of all vector fields on the circle which are ``piecewise sl 2 '' in the obvious sense. It is evidently important to consider the relationship between our new Fre chet 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.