Central Hopf monads and braided commutative algebras

Let [Formula: see text] be a braided tensor category and [Formula: see text] a tensor category equipped with a braided tensor functor [Formula: see text]. For any exact indecomposable [Formula: see text]-module category [Formula: see text], we explicitly construct a right adjoint of the action functor [Formula: see text] afforded by [Formula: see text]. Here, [Formula: see text] is Müger’s centralizer of the subcategory [Formula: see text] inside the center [Formula: see text], also known as the relative center [20, 22]. The construction is parallel to the one presented by Shimizu [31], but using the relative (co)end [4] rather than the usual (co)end. This adjunction is monadic, and thus for the Hopf monad [Formula: see text], associated to it, there is a monoidal equivalence [Formula: see text] If [Formula: see text] is the right adjoint of [Formula: see text] then [Formula: see text] is the braided commutative algebra constructed in [21]. As a consequence of our construction of these algebras, in terms of the right adjoint to [Formula: see text], we can provide a recipe to compute them when [Formula: see text] is the category of finite-dimensional representations of a finite-dimensional Hopf algebra [Formula: see text] obtained by bosonization, and choosing an arbitrary [Formula: see text]-module category [Formula: see text]. We show an explicit example in the case of Taft algebras.