Connected braided Hopf algebras

We prove a braided version of Kostant–Cartier–Milnor–Moore theorem: The category of connected τ-cocommutative (τ2=id) braided Hopf algebras over a field of zero characteristic is equivalent via the enveloping construction to the category of generalized Lie algebras. This statement includes an embedding of any generalized Lie algebra into its universal enveloping algebra, the isomorphism H U(PrimH), and primitive generation results. The embedding theorem may be derived from the Poincaré–Birkhoff–Witt theorem for quadratic algebras of Koszul type (Remark C). We also provide a direct proof that uses neither Koszul cohomologies nor the algebraic deformation theory. We consider the primitive generation problem for subalgebras, biideals and homomorphic images of connected braided Hopf algebras in much more general context, when the braiding is not necessary involutive, and even it is not necessary invertible.