F_q[M_2], F_q[GL_2] and F_q[SL_2] as quantized hyperalgebras

Within the quantum function algebra F_q[SL_2], we study the subset \cal{F}_q[SL_2] - introduced in [Ga1] - of all elements of F_q[SL_2] which are Z[q,q^{-1}]-valued when paired with \cal{U}_q(sl_2), the unrestricted Z[q,q^{-1}]-integral form of U_q(sl_2) introduced by De Concini, Kac and Procesi. In particular we yield a presentation of it by generators and relations, and a nice Z[q,q^{-1}]-spanning set (of PBW type). Moreover, we give a direct proof that \cal{F}q[SL_2] is a Hopf subalgebra of F_q[SL_2], and that \cal{F}_q[SL_2]|_{q=1} = U_Z(sl_2^*). We describe explicitly its specializations at roots of 1, say \varepsilon, and the associated quantum Frobenius (epi)morphism (also introduced in [Ga1]) from \cal{F}_\varepsilon[SL_2] to \cal{F}_1[SL_2] \cong U_Z(sl_2^*). The same analysis is done for \cal{F}_q[GL_2], with similar results, and also (as a key, intermediate step) for \cal{F}_q[Mat_2].