On Primitively Generated Braided Bialgebras

The main aim of this paper is to investigate the structure of primitively generated braided bialgebras A with respect to the braided vector space P consisting of their primitive elements. When the Nichols algebra of P is obtained dividing out the tensor algebra T(P) by the two-sided ideal generated by its primitive elements of degree at least two, we show that A can be recovered as a sort of universal enveloping algebra of P. One of the main applications of our construction is the description, in terms of universal enveloping algebras, of braided bialgebras whose associated graded coalgebra is a quadratic algebra.