PBW theorems and Frobenius structures for quantum matrices

Let G one of Mat_n(C), GL_n(C) or SL_n(C)}, let O_q(G) be the quantum function algebra - over Z[q,q^{-1}] - associated to G, and let O_e(G) be the specialisation of the latter at a root of unity \varepsilon, whose order \ell is odd. There is a quantum Frobenius morphism that embeds O(G), the function algebra of G, in O_e(G) as a central Hopf subalgebra, so that O_e(G) is a module over O(G). When G = SL_n(C), it is known by works of Brown, Gordon, and of Brown, Gordon and Stafford, that (the complexification of) such a module is free, with rank \ell^{dim(G)}. In this note I prove a PBW-like theorem for O_q(G), and I show that - when G Mat_n or GL_n - it yields explicit bases of O_e(G) over O(G). As a direct application, I prove that O_e(GL_n) and O_e(Mat_n) are free Frobenius extensions over O(GL_n) and O(Mat_n), thus extending some results of Brown, Gordon and Stroppel.