Central Hopf Monads and Braided Commutative Algebras

Let $V$ be a braided tensor category and $C$ a tensor category equipped with a braided tensor functor $G:V\to Z(C)$. For any exact indecomposable $C$-module category $M$, we explicitly construct a right adjoint of the action functor $\rho:Z^V(C)\to C^*_{M}$ afforded by $M$. Here $Z^V(C)$ is the M\"uger's centralizer of the subcategory $G(V)$ inside the center $Z^V(C)$, also known as the relative center. The construction is parallel to the one presented by K. Shimizu, but using instead the relative coend end. This adjunction turns out to be monadic, thus inducing Hopf monads $T_{V}: C\to C$, such that there is a monoidal equivalence of categories $C_{T_{V}}\simeq Z^V(C).$ If $\bar{\rho}: C^*_{ M}\to Z^V(C)$ is the right adjoint of $\rho,$ then $\bar{\rho}(Id_{M})$ is the braided commutative algebra constructed in [R. Laugwitz and C. Walton. Braided commutative algebras over quantized enveloping algebras, Transform. Groups 26(3) (2021), 957--993]. As a consequence of our construction of these algebras, in terms of the right adjoint to $\rho$, we can provide a recipe to compute them when $C=Rep(H\# T)$ is the category of finite-dimensional representations of a finite-dimensional Hopf algebra $H\# T$ obtained by bosonization, and choosing an arbitrary $Rep(H\# T)$-module category $M$. We show an explicit example in the case of Taft algebras.