F<inf>q</inf> [M<inf>n</inf>], F<inf>q</inf> [GL<inf>n</inf>] and F<inf>q</inf> [SL<inf>n</inf>] as quantized hyperalgebras

Abstract

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 Fq [GLn] |q = 1 ≅ UZ (gln*). We describe explicitly its specializations at roots of 1, say ε, and the associated quantum Frobenius (epi)morphism from Fε [GLn] to F1 [GLn] ≅ UZ (gln*), 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]. © 2007 Elsevier Inc. All rights reserved.