Witt and Virasoro algebras as lie bialgebras

Abstract

We give a countably infinite number of Lie coalgebra structures on the Witt algebra W= Der k[ x] over a field k, and on the Virasoro algebras W 1=Der k[ x, x −1] and V= W 1⊕ kc with central charge c. These come from certain solutions of the classical Yang-Baxter equation, and yield Lie bialgebra structures in each case. For k of characteristic 0, we show that these Lie coalgebra structures on W are mutually non-isomorphic, using an analysis of the locally finite part of W. We also discuss the Lie bialgebra duals of each of these constructions, which can be identified with linearly recursive sequences (one-sided or two-sided).