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For a finite dimensional semisimple Lie algebra and a root of unity, Lusztigdefined an infinite dimensional quantum group of divided powers. Under certainrestrictions on the order of the root of unity, he constructed a Frobeniushomomorphism with finite dimensional Hopf kernel and with image the universalenveloping algebra.  In this article we define and describe the Frobenius homomorphism forarbitrary roots of unity by systematically using the theory of Nicholsalgebras. In several new exceptional cases the Frobenius-Lusztig kernel isassociated to a different Lie algebra than the initial Lie algebra. Moreover,the Frobenius homomorphism often switches short and long roots, and may produceLie algebras in a symmetrically braided category.

A Frobenius homomorphism for Lusztig's quantum groups over arbitrary roots of unity