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Lusztig has constructed a Frobenius morphism for quantum groups at an -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. To George Lusztig on his 60th birthday.doi:10.1215/00127094-2010-036 fatcat:cxm3nibg3jgurihy6pzebi3u7y