On Lie Algebras in the Category of Yetter-Drinfeld Modules

Abstract

The category of Yetter—Drinfeld modules \(YD_K^K \) over a Hopf algebra K (with bijective 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 \(YD_K^K \) 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 \(YD_K^K \) is a Lie algebra. Furthermore for each Lie algebra in \(YD_K^K \) there is a universal enveloping algebra which turns out to be a Hopf algebra in \(YD_K^K \).