Abstract. Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in literature recently. We study Homstructures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that (certain classes of) Hom-algebras coincide with algebras in this monoidal category. Similar properties for Hom-coalgebras, Hopf and Lie algebras hold.

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) * ∼ = P (H • ), where Q(H) * is the dual Lie algebra of the Lie coalgebra of indecomposables of H, and P (H • ) is the Lie algebra of primitive elements of the Sweedler dual of H. We apply our theory to Turaev's Hopf group-(co)algebras.

Abstract. In this short note we study Lie algebras in the framework of symmetric monoidal categories. After a brief review of the existing work in this field and a presentation of earlier studied and new examples, we examine which functors preserve the structure of a Lie algebra.

Abstract. We give a description of the category of restricted Lie algebras over a field k of prime characteristic by means of monadic decomposition of the functor that computes the kvector space of primitive elements of a k-bialgebra.

We study dualities between Lie algebras and Lie coalgebras, and their respective (co)representations. To allow a study of dualities in an infinite-dimensional setting, we introduce the notions of Lie monads and Lie comonads, as special cases of YB-Lie algebras and YB-Lie coalgebras in additive monoidal categories. We show that (strong) dualities between Lie algebras and Lie coalgebras are closely related to (iso)morphisms between associated Lie monads and Lie comonads. In the case of a duality between two Hopf algebras -in the sense of Takeuchi-we recover a duality between a Lie algebra and a Lie coalgebra -in the sense defined in this note-by computing the primitive and the indecomposables elements, respectively.