Lie monads and dualities

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 indecomposable elements, respectively.