Reflective algebras over factorizable Hopf algebras

The reflective algebra (RH(A),Kref(A)), introduced by Laugwitz, Walton and Yakimov, is a quasitriangular left H-comodule algebra constructed from a quasitriangular Hopf algebra $(H,R)$ and a left H-comodule algebra A. We study the structure and factorizability of finite-dimensional reflective algebras. When $(H,R)$ is factorizable, we show that (RH(A),Kref(A)) is isomorphic, as a quasitriangular left H-comodule algebra, to A⊗ H equipped with the trivial left H-coaction on A and the universal K-matrix induced by R₂₁R. In this case, we also classify all universal K-matrices on RH(A). Finally, we prove that (RH(A),Kref(A)) is factorizable if and only if $(H,R)$ is factorizable and A is simple as an algebra.