F<inf>q</inf>[M<inf>2</inf>], F<inf>q</inf>[GL<inf>2</inf>], and F <inf>q</inf>[SL<inf>2</inf>] as quantized hyperalgebras

Abstract

Within the quantum function algebra Fq[SL2], we study the subset script F signq[SL2]introduced in Gavarini (1998a)of all elements of Fq[SL2] which are ℤ[q, q-1]-valued when paired with script U signq(ζI 2), the unrestricted ℤ[q, q-1]-integral form of Uq(ζI2) introduced by De Concini, Kac, and Procesi. In particular, we yield a presentation of it by generators and relations, and a nice ℤ[q, q-1]-spanning set (of PBW type). Moreover, we give a direct proof that script F signq[SL2] is a Hopf subalgebra of Fq[SL2], and that script F signq[SL 2]|q=1 ≅ UZ(ζI2*). We describe explicitly its specializations at roots of 1, say ε, and the associated quantum Frobenius (epi)morphism (also introduced in Gavarini, 1998a) from script F signε[SL2] to script F sign 1[SL2] ≅Uℤ(ζI 2*). The same analysis is done for script F sign q[GL2], with similar results, and also (as a key, intermediate step) for script F signq[M2].