The Lie - Double Algebraic Structure On A Class Of Infinite Dimensional Lie Algebras

In many branches of mathematics and physics, affine Kac-Moody algebra and its representation play an important role, which are based on the single variable Laurent polynomial ring as its coordinate algebra. As a non-commutative generalization of the Laurent polynomial algebra, the affine Schr¨odinger Lie algebra has already been studied by many scholars. In this paper, we begin the the affine Schr¨odinger Lie algebra, then we discuss the Lie bialgebra structures on the derivation algebra of it.Lie bialgebra is a vector space with both the structures of Lie algebra and Lie coalgebra. In the ?rst chapter, we mainly introduce the current situations of Lie bialgebra, the purpose and the signi?cance of this investigate.In the second chapter, we ?rst recall the concept of Lie bialgebra and some basic knowledge, then we introduce the derivation algebra of the affine Schr¨odinger Lie algebra and some related results of derivation.In the third chapter, we show the main results of this paper, which is that every Lie bialgebra structures on the derivation algebra of the affine Schr¨odinger Lie algebra is a triangular coboundary Lie bialgebra.