Some remarks on Lie bialgebra structures on simple complex Lie algebras

Abstract

Let g be a complex simple Lie algebra with root system ▵. We prove that any classical double D (g) is graded by ▵.As a consequence of this fact we obtain that D (g)≅g⊗A, where A is a unital com-muative associative algebra of dimension 2. Therefore we have two possibilities for A nilpotent and semisimple. The first case leads to solutions of CYBE and the second case leads to solutions of mCYBE. We obtain an explicit description of Lie bialgebra structures on g in both cases.