Classical Yang- Baxter Equation and Low Dimensional Triangular Lie Bialgebras Over Arbitrary Field

Abstract

Let $ L $ be a Lie algebra over arbitrary field $ k $ with dim $ L $ =3 and dim $ L' $ =2. All solutions of constant classical Yang- Baxter equation (CYBE) in Lie algebra $ L $ are obtained and the necessary conditions which $ (L,[\ ],\Delta_{r}, r) $ is a coboundary (or triangular) Lie bialgebra are given.