  We introduce braided Lie bialgebras as the infinitesimal version of braided
groups. They are Lie algebras and Lie coalgebras with the coboundary of the Lie
cobracket an infinitesimal braiding. We provide theorems of transmutation, Lie
biproduct, bosonisation and double-bosonisation relating braided Lie bialgebras
to usual Lie bialgebras. Among the results, the kernel of any split projection
of Lie bialgebras is a braided-Lie bialgebra. The Kirillov-Kostant Lie
cobracket provides a natural braided-Lie bialgebra on any complex simple Lie
algebra $g$, as the transmutation of the Drinfeld-Sklyanin Lie cobracket. Other
nontrivial braided-Lie bialgebras are associated to the inductive construction
of simple Lie bialgebras along the $C$ and exceptional series.
