Abstract
Within the quantum function algebra F q [SL 2], we study the subset ℱ q [SL 2]—introduced in Gavarini (1998a)—of all elements of F q [SL 2] which are Z opf [q, q −1]-valued when paired with 𝒰 q (𝔰 𝔩 2), the unrestricted ℤ [q, q −1] -integral form of U q (𝔰 𝔩 2) 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 ℱ q [SL 2] is a Hopf subalgebra of F q [SL 2], and that . We describe explicitly its specializations at roots of 1, say ϵ, and the associated quantum Frobenius (epi)morphism (also introduced in Gavarini, 1998a) from ℱ ϵ[SL 2] to . The same analysis is done for ℱ q [GL 2], with similar results, and also (as a key, intermediate step) for ℱ q [M 2].
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