  The set of primitive elements of a Hopf algebra in the braided category of
group graded vector spaces (with a commutative group) carry the structure of a
generalized Lie algebra. In particular the graded derivations of an associative
algebra carry this Lie algebra structure. The Lie multiplications consist of
certain n-ary partially defined multiplications satisfying generalized
antisymmetry and Jacobi identities. This generalizes the concept of Lie super
algebras and Lie color algebras.
  We show that universal enveloping algebras in the braided category exist.
They are (braided) Hopf algebras. This explains many constructions of
noncommutative noncocommutative Hopf algebras in the literature.
