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In a previous question, I asked how Lusztig's quantum Frobenius generalizes the classical Frobenius map on a variety over a finite field. I was directed to a very interesting paper by Kumar and Littlemann in which was constructed a quantized analog of the multicone over a flag varietyvarietywas constructed. The response claimed that "Upon specialization and base change this morphism becomes the standard Frobenius morphism on the flag variety." I am finding the paper pretty tough going. Could someone explain, for the simple case of $\mathbb{CP}^1$, what exactly its quantum analogue is, whether it has anything to do with the standard Podles' q-sphere, and how the quantum Frobenius is defined upon it.

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# Quantum Frobenius on the Podles' Sphere?

In a previous question, I asked how Lusztig's quantum Frobenius generalizes the classical Frobenius map on a variety over a finite field. I was directed to a very interesting paper by Kumar and Littlemann in which was constructed a quantized analog of the multicone over a flag variety. The response claimed that "Upon specialization and base change this morphism becomes the standard Frobenius morphism on the flag variety." I am finding the paper pretty tough going. Could someone explain, for the simple case of $\mathbb{CP}^1$, what exactly its quantum analogue is, whether it has anything to do with the standard Podles' q-sphere, and how the quantum Frobenius is defined upon it.