A Generalization of a Two-Parameter Quantization for the Group $GL_2(k)$

We generalize a well-known two-parameter quantization for the group $GL_2(k)$ (over an arbitrary field $k$). Specifically, a certain class of Hopf algebras is constructed containing that quantization. The algebras are constructed given an arbitrary coalgebra and an arbitrary pair of its commuting anti-isomorphisms, and are defined by quadratic relations. They are densely linked to the compact quantum groups introduced by Woronowicz. We give examples of Hopf algebras that can be rowed up to the two-parameter quantization for $GL_2(k)$.