On the duality of generalized Lie and Hopf algebras. (Q2445831)

In the classical Hopf algebra setting, Michaelis' theorem states that if \(H\) is a Hopf algebra, there is a natural isomorphism of Lie algebras between the dual Lie algebra of the Lie coalgebra of indecomposable elements of \(H\), and the Lie algebra of primitive elements of Sweedler's dual of \(H\). This theorem is extended to the setting of Hopf algebras in a braided monoidal category, the Sweedler dual being replaced by a contravariant monoidal functor. Applied in the category of vector spaces \(\mathtt{Vect}\), we obtain Michaelis' theorem; applied in the category \(\mathtt{Fam}(\mathtt{Vect})\) of families of \(\mathtt{Vect}\), we obtain an application to Turaev's Hopf group-coalgebras.