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

Within the quantum function algebra Fq[GLn], we study the subset Fq[GLn]—introduced in [F. Gavarini, Quantization of Poisson groups, Pacific J. Math. 186 (1998) 217–266]—of all elements of Fq[GLn] which are Z[q,q−1]-valued when paired with Uq(gln), the unrestricted Z[q,q−1]-integral form of Uq(gln) 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 Fq[GLn] 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 Fε[GLn] to , also introduced in [F. Gavarini, Quantization of Poisson groups, Pacific J. Math. 186 (1998) 217–266]. The same analysis is done for Fq[SLn] and (as key step) for Fq[Mn].