Reflection equation- and FRT-type algebras

We introduce two types of algebras which include respectively the well known reflection equation (RE) and Faddeev-Reshetikhin-Takhtayan algebras associated with a quasitriangular Hopf algebra H. We show that these two types of algebras are twist-equivalent. It follows that a RE algebra is a module algebra over a twisted tensor square of H. We present some applications to the equivariant quantization.