In what sense does Lusztig's quantum Frobenius, defined on a quantum enveloping algebra, generalise the classical Frobenius mapping on a variety over a finite field?



Quantum Frobenius
There is one sense in which I'd say that Lusztig's Frobenius morphism is a generalization of the Frobenius morphism on a variety: In [this](http://arxiv.org/abs/math/0005246) paper, Kumar and Littelmann show that Lusztig's quantum Frobenius morphism induces a Frobenius morphism on a quantized analog of the multicone over a flag variety (which they call a "lift of the Frobenius morphism to characteristic 0"). Upon specialization and base change this morphism becomes the standard Frobenius morphism on the flag variety.


