# Quantum Frobenius

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'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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