Lusztig has constructed a Frobenius morphism for quantum groups at an
β-th root of unity, which gives an integral lift of the Frobenius map on
universal enveloping algebras in positive characteristic. Using the Hall
algebra we give a simple construction of this map for the positive part of the
quantum group attached to an arbitrary Cartan datum in the nondivisible case.Comment: Final version: typos corrected and exposition improve