Reflective centers of module categories and quantum K-matrices

Abstract Our work is motivated by obtaining solutions to the quantum reflection equation (qRE) by categorical methods. To start, given a braided monoidal category ${\mathcal {C}}$ and ${\mathcal {C}}$ -module category ${\mathcal {M}}$ , we introduce a version of the Drinfeld center ${\mathcal {Z}}({\mathcal {C}})$ of ${\mathcal {C}}$ adapted for ${\mathcal {M}}$ ; we refer to this category as the reflective center ${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$ of ${\mathcal {M}}$ . Just like ${\mathcal {Z}}({\mathcal {C}})$ is a canonical braided monoidal category attached to ${\mathcal {C}}$ , we show that ${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$ is a canonical braided module category attached to ${\mathcal {M}}$ ; its properties are investigated in detail. Our second goal pertains to when ${\mathcal {C}}$ is the category of modules over a quasitriangular Hopf algebra H , and ${\mathcal {M}}$ is the category of modules over an H -comodule algebra A . We show that the reflective center ${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$ here is equivalent to a category of modules over an explicit algebra, denoted by $R_H(A)$ , which we call the reflective algebra of A . This result is akin to ${\mathcal {Z}}({\mathcal {C}})$ being represented by the Drinfeld double ${\operatorname {Drin}}(H)$ of H . We also study the properties of reflective algebras. Our third set of results is also in the Hopf setting above. We show that reflective algebras are quasitriangular H -comodule algebras, and we examine their corresponding quantum K -matrices; this yields solutions to the qRE. We also establish that the reflective algebra $R_H(\mathbb {k})$ is an initial object in the category of quasitriangular H -comodule algebras, where $\mathbb {k}$ is the ground field. The case when H is the Drinfeld double of a finite group is illustrated.