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 W-1, 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 boolean AND b, In particular, for the one-sided Witt algebra W-1 = 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 W-1 up to isomorphism. We prove the analogous result for a class of Lie subalgebras of W which includes W-1. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 17B37; 17B68.