F_q[M_2], F_q[GL_2] and F_q[SL_2] as quantized hyperalgebras (original) (raw)

Within the quantum function algebra F q [SL 2 ], we study the subset F q [SL 2 ]introduced in [Ga1]-of all elements of F q [SL 2 ] which are Zˆq, q −1˜-valued when paired with U q (sl 2) , the unrestricted Zˆq, q −1˜-integral form of U q (sl 2) introduced by De Concini, Kac and Procesi. In particular we yield a presentation of it by generators and relations, and a nice Zˆq, 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 F q [SL 2 ]˛q =1 ∼ = U Z (sl 2 *). We describe explicitly its specializations at roots of 1, say ε, and the associated quantum Frobenius (epi)morphism (also introduced in [Ga1]) from F ε [SL 2 ] to F 1 [SL 2 ] ∼ = U Z (sl 2 *). 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 ] .