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 $YD_K^K$.