A poisson?lie structure on the diffeomorphism group of a circle

Abstract

Starting from the Gelfand-Fuks-Virasoro cocycle on the Lie algebraX(S1) of the vector fields on the circleS1 and applying the standard procedure described by Drinfel'd in a finite dimension, we obtain a classicalr-matrix (i.e. an elementr ∈X(S1) ∧X(S1) satisfying the classical Yang-Baxter equation), a Lie bialgebra structure onX(S1), and a sort of Poisson-Lie structure on the group\(\widetilde{Diff}(S^1 )\) of diffeomorphisms. Quantizations of such Lie bialgebra structures may lead to ‘quantum diffeomorphism groups’.