On Lie Algebras in the Category of Yetter - Drinfeld Modules

The category of Yetter-Drinfeld modules YD K over a Hopf algebra K (with bijektive antipode over a field k) is a braided monoidal category. If H is a Hopf algebra in this category then the primitive elements of H do not form an ordinary Lie algebra anymore. We introduce the notion of a (generalized) Lie algebra in YD K such that the set of primitive elements P (H) is a Lie algebra in this sense. Also the Yetter-Drinfeld module of derivations of an algebra A in YD K is a Lie algebra. Furthermore for each Lie algebra in YD K there is a universal enveloping algebra which turns out to be a Hopf algebra in YD K .