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

Abstract

The quantized universal enveloping algebra U_q(gl(n)) has two integral forms - over Z[q,q^{-1}] - the restricted (by Lusztig) and the unrestricted (by De Concini and Procesi) one. Dually, the quantum function algebra F_q[GL(n)] has two integral forms, namely those of all elements - of F_q[GL(n)] - which take values in Z[q,q^{-1}] when paired respectively with the restricted or the unrestricted form of U_q(gl(n)). The first one is the well-known form generated over Z[q,q^{-1}] by the entries of a q-matrix and the inverse of its quantum determinant. In this paper instead we study the second integral form, say F'_q[GL(n)], i.e. that of all elements which are Z[q,q^{-1}]-valued over the unrestricted form of U_q(gl(n)). In particular we yield a presentation of it by generators and relations, and a PBW-like theorem: in short, it is an algebra of "quantum divided powers" and "quantum binomial coefficients". Moreover, we give a direct proof that F'_q[GL(n)] is a Hopf subalgebra of F_q[GL(n)], and that its specialization at q=1 is the Z-hyperalgebra over gl(n)^*, the Lie bialgebra dual to gl(n). In addition, we describe explicitly the specializations of F'_q[GL(n)] at roots of 1, and the associated quantum Frobenius (epi)morphism. The same analysis is done for F'_q[SL(n)] and (as a key step) F'_q[Mat(n)]: in fact, for the latter the strongest results are obtained. This work extends to general n>2 the results for n=2, already treated in math.QA/0411440.