Quantum Frobenius II

Question

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' q-sphere, and how the quantum Frobenius is defined upon it.

Answer 1

In general, the idea of the Kumar-Littelmann 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 scheme-theoretic 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 1-dimensional $U_q(b)$-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' q-sphere, 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 characteristic-0 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).

Answer 2

There are two nice papers by Kevin McGerty on quantum Frobenius that I would recommend, arXiv:math/0601150 and 0511697