Braided Lie Bialgebras

This thesis is related to the theory of braided Lie bialgebras, and it is divided into four parts.First, we give some basic definitions and lemmas from the theory of braided tensor category, and study Lie algebras, Lie coalgebras and Lie bialgebras in it.Second, We make use of braided diagrams to construct double bicross sum of braided Lie bialgebras. We define matched pair of braided Lie algebras, braided Lie coalgebras and YD braided Lie bialgebras, the result from which we get is, we can turn a matched pair of YD braided Lie bialgebras into an ordinary braided Lie bialgebra. This generalize the conception of Majid's matched pair of Lie algebras and the theorems relating to it. We also generalize Majid's bisum theory of Lie bialgebras.In order to find relations between double cross sum of Lie bialgebras and double cross product of Hopf algebras, we have also studied universal enveloping algebra of the double cross sum of a matched pair of Lie bialgebras. We prove that the universal enveloping algebra of the double cross sum of a matched pair of Lie bialgebras is isomorphic to the double cross product of their universal enveloping algebras. We study conditions when this isomorphism can be happen. This isomorphism seems also to be a Hopf algebra isomorphism.As an example of the above theory, we construct the Drinfeld double of a braided Lie bialgebra. It turns out to be a quasitriangular braided Lie bialgebra.In the end, we prove the bosonization theorem for braided Lie bialgebras living in the module category of a quasitriangular braided Lie bialgebra and in the comodule category of a coquasitriangular braided Lie bialgebra.