Presentation by Borel subalgebras and Chevalley generators for quantum enveloping algebras

We provide an alternative approach to the Faddeev–Reshetikhin–Takhtajan presentation of the quantum group U_q(g), with L-operators as generators and relations ruled by an R-matrix. We look at U_q(g) as being generated by the quantum Borel subalgebras U_q(b_+) and U_q(b_-), and use the standard presentation of the latter as quantum function algebras. When g = gl_n , these Borel quantum function algebras are generated by the entries of a triangular q-matrix. Thus, eventually, U_q(gl_n) is generated by the entries of an upper triangular and a lower triangular q-matrix, which share the same diagonal. The same elements generate over k[q,q^{-1}] the unrestricted k[q,q^{-1}]-integral form of U_q(gl_n) of De Concini and Procesi, which we present explicitly, together with a neat description of the associated quantum Frobenius morphisms at roots of 1. All this holds, mutatis mutandis, for g = sl_n too.