Yangian Double Of The Classical Lie Algebras

This paper mainly studies the theory of the Yangian double for the classical Lie algebras.The first chapter is the introduction.We mainly introduce the research status of the Yangian double and the main work of this paper.This is the foundation of the whole paper.In Chapter 2,we review some related definitions and properties about the Lie algebra and Hopf algebra,which will be useful for later discussion.We introduce the definition of Lie algebra,Lie subalgebra,ideal,g-module,universal enveloping algebra,Hopf algebra,as well as the PBW theorem for the universal enveloping algebra.In Chapter 3,we study the algebra DYh(gln),which is called the Yangian double of type A.We construct the Drinfeld realization of DYh(gln)with the use of the Gauss decomposition of the generator matrix T±(u).At the same time we also give the Drinfeld realization of DYh(sln),where DYh(sln)is a subalgebra of DYh(gln).Then we construct a family of central elements of the completed algebra DYh(gln)at the critical level by normalizing the R-matrix.As a corollary,we obtain an explicit formula of invariants of the vacuum module over DYh(gln).At last,we also prove an analogue of Harish-Chandra homomorphism.In Chapter 4,we develop the theory of DYh(gN),where gN is the simple Lie algebra of types B,C,D.We give the definition of DYh(gN)by using the RTT relation together with a unitarity condition.Then we construct its Drinfeld generators using the Gauss decomposition and prove that these two constructions are isomorphic.Moreover,we also give a PBW theorem using the ordered monomials in the generators ?ij(r)and c.Each of these results has its counterpart in type A,so it tells us that it is very important to develop the theory in types B,C,D.Finally,we construct level one vertex representation of the Yangian double DYh(gN)in terms of bosons as well.