On the duality of generalized Lie and Hopf algebras

Abstract

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.

Lie monads and dualities

“…this Michaelis pair is always strong. In [8], we generalize this result in a setting of additive symmetric monoidal categories, so that it applies in particular to Turaev's Hopf group coalgebras.…”

Pre-rigid Monoidal Categories

“…is bijective for every object T in C. The notion of pre-rigidity, in its original form, stems from [GV1] although, as we will see, it turns out to be equivalent to the definition of weak dual given in [DP]. A basic fact is that a (right) rigid monoidal category is right closed.…”

“…An adjoint pair of functors (L, R) between monoidal categories A and B such that R is a lax monoidal functor (or, equivalently, L is colax monoidal) is called liftable if the induced functor R = Alg(R) : Alg(A) → Alg(B) between the respective categories of algebra objects has a left adjoint and if the functor L = Coalg(L) : Coalg(B) → Coalg(A) between the respective categories of coalgebra objects has a right adjoint. If A and B come both endowed with a braiding, it is shown in [GV1,Theorem 2.7] that such a liftable pair of functors (L, R) gives rise to an adjunction between the respective categories of bialgebra objects…”

“…[Mic]) is proven for a particular class of liftable pairs of adjoint functors between symmetric monoidal categories, the main application being a version of this theorem for Turaev's Hopf group-(co)algebras (cf. [GV1,Theorem 4.16]).…”

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Liftable pairs of adjoint functors between braided monoidal categories in the sense of [GV1] provide auto-adjunctions between the associated categories of bialgebras. Motivated by finding interesting examples of such pairs, we study general pre-rigid monoidal categories.

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