Fq[Mn], Fq[GLn] and Fq[SLn] as quantized hyperalgebras

Within the quantum function algebra Fq[GLn], we study the subset —introduced in [F. Gavarini, Quantization of Poisson groups, Pacific J. Math. 186 (1998) 217–266]—of all elements of Fq[GLn] which are -valued when paired with , the unrestricted -integral form of 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 is a Hopf subalgebra of Fq[GLn], and that . We describe explicitly its specializations at roots of 1, say ε, and the associated quantum Frobenius (epi)morphism from to , also introduced in [F. Gavarini, Quantization of Poisson groups, Pacific J. Math. 186 (1998) 217–266]. The same analysis is done for and (as key step) for .