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 a quantized analog of the multicone over a flag varietywas 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' qsphere, and how the quantum Frobenius is defined upon it.
2 Answers
In general, the idea of the KumarLittelmann paper is the following: For a semisimple group G, set $V := \displaystyle \bigoplus_{n \geq 0} H^0(\lambda)$, where $\lambda$ is a fixed regular dominant weight for G. Then $V$ is the projective coordinate ring of $G/B$ under the embedding $G/B \hookrightarrow \mathbb P( V( \lambda ) )$, where $V$ is the Weyl module for $G$ of highest weight $\lambda$. In particular, one can obtain the sheaf of regular functions on $G/B$ in the natural way from $V$.
Now, the (absolute) Frobenius morphism on the flag variety $G/B$ induces an automorphism of $V$ as an $\mathbb F_p$vector space, and in fact the converse holds: the appropriate $p^{th}$power morphism $V \to V$ (which is just the morphism of taking $p^{th}$ powers of sections) induces the Frobenius morphism on $G/B$ (this is the process called "sheafification" in their paper, cf section 6). The point of the paper is now that one can define a module (let's call it $V'$) for the quantum group associated to $G$ such that upon base change, $V'$ becomes $V$. Furthermore, Lusztig's Frobenius morphism induces a morphism $V \to V'$ (which they call $Fr^*$) which, upon base change, becomes exactly the desired $p^{th}$power morphism $V \to V$.
Let me give an explicit example for $\mathbb P^1$. In this case, $\mathbb P^1$ is the flag variety of $G = SL_2$. Since the weights of $SL_2$ are parametrized by integers, I'll write $H^0(n)$ for the global sections of the corresponding line bundle on $G/B$ (which is just a complicated way of saying that $H^0(n) = H^0( \mathbb P^1, O(n) )$, where that $O$ should be a \mathscr O but that doesn't seem to work). Then in this case, we can take $V = \displaystyle \bigoplus_{n \geq 0} H^0(n)$, and $V$ is just $k[x, y]$. The schemetheoretic Frobenius morphism on $G/B$ is induced by the natural $p^{th}$power morphism $V \to V$, $\; s \mapsto s^{ \otimes p }$ (which is just the natural $p^{th}$power morphism on the ring $k[x, y]$). We now quantize this picture: Set $$V' := \bigoplus_{n \geq 0} H^0( X, \chi_{n}^\xi ) ,$$ where here I'm using their notation from the paper (note that the "X" should be a mathfrak X as in the paper, but somehow I can't do mathfrak here). That is, $H^0( X, \chi_{n}^\xi )$ is the induction functor from $U_q(b)$ modules to $U_q(g)$ modules, applied to the 1dimensional $U_q(b)$module $\chi_{n}^\xi$ (cf section 2 of the paper). The point is that $V'$ is a quantized version of $V$, and Lusztig's Frobenius morphism induces a morphism $Fr^* : V \to V'$ that, upon base change, becomes the $p^{th}$power morphism $V \to V$.
(As for the Podles' qsphere, I don't know what that is, so I can't speak to that part of your question).
(Edit: I realized that there is a slight white lie in what I wrote above, namely that the morphism Fr* initially isn't quite a morphism from $V$ to $V'$, but from a characteristic0 version of $V$ to $V'$; one only gets $V$ after base change to positive characteristic. Kumar and Littelmann first construct Fr* in characteristic 0. Morally, though, one can ignore this issue on a first pass; it's a bit confusing because Fr* appears in various incarnations, both before and after base change).

$\begingroup$ I intentionally left the "upon base change" business vague in my answer above, but I'll be a bit more precise here: The induction functor from $U_q(b)$ to $U_q(g)$modules can be applied after specialization to a root of unity and tensoring with a field of positive characteristic, and in that case this quantized induction functor actually agrees with the classical induction functor, i.e. it is just the induction functor from modules for the group $B$ to the group $G$, cf Proposition 5.1 in KumarLittelmann. $\endgroup$ Nov 17, 2009 at 23:30
There are two nice papers by Kevin McGerty on quantum Frobenius that I would recommend, arXiv:math/0601150 and 0511697