Pbw Deformations of Braided Symmetric Algebras and a Milnor-moore Type Theorem for Braided Bialgebras

Braided bialgebras were defined by mimicking the definition of bialgebras in a braided category; see [Ta]. In this paper we are interested in those braided bialgebras that are connected as a coalgebra, and such that, up to multiplication by a certain scalar, their braiding restricted to the primitive part is a Hecke operator. To every braided bialgebra as above we associate a braided Lie algebra. Conversely, for each braided Lie algebra we construct a braided bialgebra, namely its enveloping algebra. Braided symmetric and exterior algebras are examples of enveloping algebras (they correspond to a trivial Lie bracket). We show many of the properties of ordinary enveloping algebras still hold in the braided case: the graded associated (with respect to the standard filtration) is a braided symmetric algebra; the coalgebra counterpart is isomorphic to the coalgebra structure of a braided symmetric algebra; the coradical filtration and the standard filtration are identical. As in the classical case, braided symmetric and exterior algebras are Koszul algebras. The proof of this fact is based on a new characterization of Koszul algebras. These properties are used to prove a Milnor-Moore type theorem for infinitesimally cocommutative connected braided bialgebras (see Theorem 5.5). We apply our result to three different classes of braided bialgebras: connected bialgebras in the category of comodules over a coquasitriangular cosemisimple Hopf algebra (e.g. superbialgebras and enveloping algebras of color Lie algebras), bialgebras with dual Chevalley property (in the same braided category) and finite dimensional braided bialgebras in the category of Yetter-Drinfeld modules.