A POISSON-LIE STRUCTURE ON THE DIFFEOMORPHISM GROUP OF A CIRCLE

Starting from the Gelfand-Fuks-Virasoro cocycle on the Lie algebra X(S 1) of the vector fields on the circle S1 and applying the standard pro cedure described by Drinfel'd in a finite dimension, we obtain a class ical r-matrix (i.e. an element r is-an-element-of X(S1) AND X(S1) sati sfying the classical Yang-Baxter equation), a Lie bialgebra structure on X(S1), and a sort of Poisson-Lie structure on the group Diff(S1) of diffeomorphisms. Quantizations of such Lie bialgebra structures may l ead to 'quantum diffeomorphism groups'.