Classification of the Lie bialgebra structures on the Witt and Virasoro algebras

We prove that all the Lie bialgebra structures on the one sided Witt algebra W1, on the Witt algebra W and on the Virasoro algebra V are triangular coboundary Lie bialgebra structures associated to skew-symmetric solutions r of the classical Yang-Baxter equation of the form r=a b. In particular, for the one-sided Witt algebra W1=Der k[t] over an algebraically closed field k of characteristic zero, the Lie bialgebra structures discovered in Michaelis (Adv. Math. 107 (1994) 365–392) and Taft (J. Pure Appl. Algebra 87 (1993) 301–312) are all the Lie bialgebra structures on W1 up to isomorphism. We prove the analogous result for a class of Lie subalgebras of W which includes W1.