Braided-Lie bialgebras

We introduce braided Lie bialgebras as the infinitesimal version of braided groups. They are Lie algebras and Lie coalgebras with the coboundary of th e Lie cobracket an infinitesimal braiding. We provide theorems of transmuta tion, Lie biproduct, bosonisation and double-bosonisation relating braided Lie bialgebras to usual Lie bialgebras. Among the results, the kernel of an y split projection of Lie bialgebras is a braided-Lie bialgebra. The Kirill ov-Kostant Lie cobracket provides a natural braided-Lie bialgebra on any co mplex simple Lie algebra, as the transmutation of the Drinfeld-Sklyanin Lie cobracket. Other nontrivial braided-Lie bialgebras are associated to the i nductive construction of simple Lie bialgebras along the C and exceptional series.