On the duality of generalized Lie and Hopf algebras

TL;DR

This paper explores the duality between generalized Lie and Hopf algebras, establishing an abstract version of Michaelis' theorem and applying it to Hopf group-(co)algebras.

Contribution

It introduces a framework for lifting adjoint pairs of braided monoidal functors to categories of Hopf algebras, leading to a generalized duality theorem.

Findings

Established an abstract Michaelis' theorem for Hopf algebras.

Derived natural isomorphisms between dual Lie algebras and primitive elements.

Applied the theory to Turaev's Hopf group-(co)algebras.

Abstract

We show how, under certain conditions, an adjoint pair of braided monoidal functors can be lifted to an adjoint pair between categories of Hopf algebras. This leads us to an abstract version of Michaelis' theorem, stating that given a Hopf algebra $H$, there is a natural isomorphism of Lie algebras $Q(H)^*\cong P(H^\circ)$, where $Q(H)^*$ is the dual Lie algebra of the Lie coalgebra of indecomposables of $H$, and $P(H^\circ)$ is the Lie algebra of primitive elements of the Sweedler dual of $H$. We apply our theory to Turaev's Hopf group-(co)algebras.