Formula omitted], [formula omitted] and [formula omitted] as quantized hyperalgebras

Abstract

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 ( gl n ) , 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 q [ GL n ] | q = 1 ≅ U Z ( 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 ] ≅ U Z ( 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 ] .