Tensor categories (and their module categories) take center stage in many interactions between algebra and low-dimensional topology inspired by physics. If we are lucky enough, a useful tensor category can be described neatly as the representation category of a suitable Hopf algebra, and any further embellishments of the situation can be described in terms of that Hopf algebra; in fact one can argue that this is exactly what Hopf algebras are for. We visit a recent installment of such a connection. Laugwitz-Walton-Yakimov introduce the "reflective center" of a module category over a braided tensor category (setting aside for this abstract any technical requirements of course). The notion of a braided tensor category is closely related to (representations of) the Artin braid group (of type A), and in the same fashion the braided module category constructed by Laugwitz-Walton-Yakimov is related to the Artin braid group of type B which is to a Weyl group of type B what the Artin braid group of type A is to the symmetric group. If the braided tensor category is the module category of a Hopf algebra H, and the module category is described by an algebra with an H-action, Laugwitz-Walton-Yakimov construct a new algebra with H-action describing the "reflective center". This "reflective algebra" is parallel in a sense to the famous Drinfeld double construction producing, from any Hopf algebra, a new Hopf algebra whose module category is braided. After explaining what all this is about, we humbly redo the reflective algebra construction of Laugwitz-Walton-Yakimov in a more conceptual (and slightly selfish) way, making full use of Majid's "transmuted" Hopf algebra and a peculiar and often overlooked structure the latter enjoys.