Are the Weyl modules projectives?

Question

Let $\frak g$ a simple finite-dimensional complex Lie algebra.

Which categories of modules has the Weyl modules for $\frak g$ (in characteristic zero or positive) as projective objects?

It is an ample question, since we can see them in the category of finite-dimensional modules, in the category of finite-dimensional modules with bounded weights and so on.

Is there any reference for this stuff?

Answer 1

The discussion here has gotten over-complicated. For some background on the notion of "Weyl module" I should refer to my answer just posted here of an older question.

Concerning projective objects in various module categories in prime characteristic (or perhaps for quantum groups at a root of unity), it seems to be almost never true that arbitrary "Weyl modules" in such categories will be projective. Often they play instead a sort of intermediate role between simple modules and projective modules. (In the Cline-Parshall-Scott formalism of "highest weight categories", Weyl modules tend to play the role of "standard" objects.)

Note that in the category of all rational modules for a semisimple algebraic group, there are no nonzero projectives (Donkin). The idea goes back to Hochschild that in a module category for a group with additional structure (Lie or algebraic, say), the injective modules typically exist and play a more natural role in homological constructions. Jantzen's book Representations of Algebraic Groups provides a lot of evidence for this viewpoint.

I don't know enough about the spin-off concept of "Weyl module" developed since the mid-1990s by Chari and her collaborators, but here too it seems doubtful that such modules will behave like projective modules in the natural categories occurring. These Weyl modules are defined in the setting of finite dimensional modules for affine or quantum affine Lie algebras, etc.

Answer 2

Jim Humphreys has given the right commentary on Weyl modules in positive characteristic, I think. Maybe it is useful to have an explicit example to see why you shouldn't expect "projective-ness" from Weyl modules.

Let $G= \operatorname{SL}_n$ and consider the adjoint representation of $G$ acting on $\mathfrak{g} = \mathfrak{sl}_n$. When $n \not \equiv 0 \pmod{p}$, $\mathfrak{g}$ is a simple $G$-module. But when $n \equiv 0 \pmod{p}$, the identity matrix $I$ has trace 0 and the span of $I$ defines a 1 dimensional "trivial" $G$-submodule. There is a (non-split) short exact sequence $$0 \to k.I \to \mathfrak{g} \to L \to 0$$ for a self-dual $n^2-2$ dimensional simple $G$-module $L$, and in fact $\mathfrak{g}$ is the Weyl module whose highest weight is $\tilde \alpha$ (the unique root which is a dominant weight in this case).

Writing $V$ for the dual module $V = \mathfrak{g}^\vee \simeq \mathfrak{gl}_n/kI$, there is a short exact sequence $$(\flat) \quad 0 \to L \to V \to k \to 0.$$ Now, $k$ is the Weyl module whose highest weight is $0$, and the sequence $(\flat)$ is not split. So the Weyl module $k$ is not a projective object in the category of all $G$-modules.

(And for "the same" reason, $k$ won't be projective in any "interesting" category of $G$-modules. Whatever is meant by "interesting"...)

Answer 3

There is actually a mildly interesting category in which a given Weyl module is projective. The simplest Weyl module is one dimensional with trivial action. So in that case the category has to have trivial cohomology and is indeed rather uninteresting.

But now consider more generally a Weyl module with highest weight $\lambda$. Choose a Weyl group invariant inner product on the vector space spanned by the weight lattice, so that one can speak of the length of a weight. Then the appropriate category consists of the representations all whose weights have length at most equal to the length of $\lambda$.

This is Polo's theorem as treated (dually) in my Lectures on Frobenius Splittings and B-modules.

But notice that the category depends on the Weyl module.

Answer 4

I am not sure what you are looking for but you may want to look up the theory of tilting modules. Another closely related theory is Lusztig's theory of based modules.