The Lie algebra of homeomorphisms of the circle
The authors define and study an infinite-dimensional Lie algebra \(\text{homeo}_+\) directly related to the topological Lie group \(\text{Homeo}_+\) of all orientation preserving homeomorphisms of the circle. They use the second-named author's notion of universal decorated Teichmüller theory [Adv. Math. 98, No. 2, 143-215 (1993; Zbl 0772.30040)] to give Fréchet coordinates on \(\text{Homeo}_+\) modulo the group or real fractional linear transformations. The vector fields on the circle arising from this quotient group are then extended by the Lie algebra \(\text{sl}_2\), giving their \(\text{homeo}_+\). The approach is fundamentally arithmetic, the basic deformations being indexed by the modular group of integer fractional linear transformations.
scientific article; zbMATH DE number 1268552
scientific article