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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?

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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 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.

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It is a generalization in the following sense (everything I'm saying is in Lusztig's papers): the quantum group is a deformation of the classical enveloping algebra, and it's in fact possible to work over the ring A=Z[v,v^{-1}], not just a field: you can define the "q-divided powers" integral form of the quantum group, which is the analogue of the hyperalgebra of an algebraic group over Z. Now specializing to a ring of l-th cycltomic integers, Lusztig shows there is a surjective algebra map from this specialization to the quantum group specialized to Z, with the scalars then extended up to the ring of cyclotomic integers (at the moment I'm ignoring what happens when l isn't coprime to the "lacing number"). Note unlike in the classical, this map is not an endomorphism!

However, if l is a prime the F_l the finite field with l elements is an algebra over the ring of cyclotomic polynomials, since it's the residue field, so you can tensor up to get a map of two algebras over F_l now. It turns out these two algebras are both the hyperalgebra of the associated algebraic group over F_p, and the map you have is the (transpose of, since we're using the hyperalgebra, rather than the group) classical Frobenius map. Thus for l a prime number, the quantum Frobenius is really an integral lift of the classical map, but since the construction works for any l, not just l prime, it does more than just give a lift. Also, when l isn't coprime to the lacing number, you get a lift of one of Chevalley's exceptional isogenies, which are something like "square roots" of the Frobenius.

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It's not a generalization; it's an analogue. The map it's analogous to is that on the restricted universal enveloping algebra (thought of as differential operators on the algebraic group) given by pullback by the Frobenius.

Similarly, there's a Frobenius mapping functions on the group to the quantized function algebra which is dual (in the sense of Hopf algebras) to the Frobenius above, just as the Frobenius on the restricted Lie algebra is dual to pullback of functions by the Frobenius under the usual duality of functions on the group and invariant differential operators (the pairing is differentiate and evaluate at the identity).

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