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

Abstract

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∧ 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.