### Description

This talk will be on recent joint work with Robert Laugwitz and Milen Yakimov (arXiv:2307.14764) that is motivated by obtaining solutions to the quantum reflection equation (qRE). To start, given a braided monoidal category C and C-module category M, we introduce a version of the Drinfeld center Z(C) of C adapted for M. We refer to this category as the "reflective center" E_C(M) of M. Just like Z(C) is a canonical braided monoidal category attached to C, we show that E_C(M) is a canonical braided module category attached to M. When C is the category of modules over a quasitriangular Hopf algebra H, and M is the category of modules over an H-comodule algebra A, we show that E_C(M) is equivalent to a category of modules over an explicit algebra, which we call the "reflective algebra" R_H(A) of A. Here, R_H(A) is akin to Drinfeld double of H. We show that reflective algebras are quasitriangular H-comodule algebras, and examine their corresponding quantum K-matrices (which are solutions to the qRE).