Reflective centers of module categories and quantum K-matrices

TL;DR

This paper introduces the reflective center of module categories as a categorical framework to construct solutions to the quantum reflection equation, linking algebraic structures with quantum integrable systems.

Contribution

It defines the reflective center for module categories, relates it to reflective algebras, and explores their properties and applications in quantum K-matrices.

Findings

Reflective center is a braided module category associated with a module category.

Reflective algebra $R_H(A)$ is explicitly constructed and shown to be quasitriangular.

Reflective algebra $R_H(K)$ is an initial object in the category of quasitriangular $H$-comodule algebras.

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.