Reflection equation, twist, and equivariant quantization

We prove that the reflection equation (RE) algebra ℒ_R associated with a finite dimensional representation of a quasitriangular Hopf algebra H is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show that ℒ_R is a module algebra over the twisted tensor square H^R⊗H and the double D(H). We define FRT- and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces.