Lie Bialgebra Structures On The Infinite-dimensional Lie Algebra

To search for the solutions of the Yang-baxter quantum equation, Drinfeld introduced the notion of Lie bialgebras in1983.In this paper we shall determine Lie bialgebra structures on the infinite-dimensional Lie algebra. The Lie algebra over a field F of characteristic0with basis and the following nonvanishing Lie brackets:We denote this Lie algebra by L. We get its bialgebra structure through some special technical, comparison and by some lengthy calculations. At the same time, we found out that its derivations are all inner derivations.further more, we prove that its bialgebra structures are triangular coboundary.In this thesis, the main contents and results are summarized as follows:Chapter1is introduction. In chapter1and chapter2, we give the basic knowledge and related concepts of the Lie bialgebra.In chapter3,we study H1(L,V).The main results are given infollowing:Theorem1H1(L,V)0,where H1(L,V)is the first cohomologygroup of the Lie algebra L with coefficients in the L-module V.In chapter4,we study the Lie bialgebra.The main results aregiven in following:Theorem2The Lie bialgebra L,[,], is triangular coboundary.