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, 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.
