Braided commutative algebras over quantized enveloping algebras

We produce braided commutative algebras in braided monoidal categories by generalizing Davydov's full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal centers $\mathcal{Z}_{\mathcal{B}}(\mathcal{C})$ from algebras in $\mathcal{B}$-central monoidal categories $\mathcal{C}$, where $\mathcal{B}$ is an arbitrary braided monoidal category; Davydov's (and previous works of others) take place in the special case when $\mathcal{B}$ is the category of vector spaces $\mathbf{Vect}$ over a field $\Bbbk$. Since key examples of relative monoidal centers are suitable representation categories of quantized enveloping algebras, we supply braided commutative module algebras over such quantum groups. One application of our work is that we produce Morita invariants for algebras in $\mathcal{B}$-central monoidal categories. Moreover, for a large class of $\mathcal{B}$-central monoidal categories, our braided commutative algebras arise as a braided version of centralizer algebras. This generalizes the fact that centers of algebras in $\mathbf{Vect}$ serve as Morita invariants. Many examples are provided throughout.