F_q[M_n], F_q[GL_n] and F_q[SL_n] as quantized hyperalgebras

Within the quantum function algebra F_q[GL_n], we study the subset F'_q[GL_n]— introduced in [F. Gavarini, Quantization of Poisson groups, Pacific J. Math. 186 (1998) 217–266] — of all elements of F_q[GL_n] which are Z[q,q^{−1}]-valued when paired with U'_q(gln), the unrestricted Z[q,q^{−1}]-integral form of U_q(gl_n) introduced by De Concini, Kac and Procesi. In particular we obtain a presentation of it by generators and relations, and a PBW-like theorem. Moreover, we give a direct proof that F'_q[GL_n] is a Hopf subalgebra of F_q[GL_n], and that F'_1[GL_n] - i.e., the specialization of F'_q[GL_n] at q=1 - is isomorphic to U(gl_n^*). We describe explicitly its specializations at roots of 1, say ε, and the associated quantum Frobenius (epi)morphism from F'_ε[GL_n] to F'_1[GL_n], also introduced in [F. Gavarini, Quantization of Poisson groups, Pacific J. Math. 186 (1998) 217–266]. The same analysis is done for F_q[SL_n] and (as key step) for F_q[M_n].