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 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 examine their corresponding quantum $K$-matrices; this yields solutions to the qRE. We also establish that the reflective algebra $R_H(\Bbbk)$ is an initial object in the category of quasitriangular $H$-comodule algebras, where $\Bbbk$ is the ground field. The case when $H$ is the Drinfeld double of a finite group is illustrated.