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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?

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There is some confusion about exactly what you mean here by "Weyl module for $\mathfrak{g}$". The term "Weyl module" was invented by Carter-Lusztig to deal with situations in prime characteristic where you have a finite dimensional highest weight module (for an algebraic group) which fails to be completely reducible. There are also analogues in the work of Chari and others in different settings. In characteristic 0, are your "Weyl modules" identical with the finite dimensional simple modules? And what do you refer to in positive characteristic? – Jim Humphreys Jun 18 '12 at 13:36
Exactly, I am talking about this highest weight modules. In characteristic zero they are identical with the finite dimensional simple modules. In positive characteristic I refer to the Weyl module for the hyperalgebra (the tensor product over $\mathbb Z$ of an integral form of $\frak g$ with an algebraically closed field of characteristic $p$). However, I am also interested on the analogues of Chari for loop algebras! – Angelo Jun 18 '12 at 17:12
As Jim notes below, this question is, unfortunately, difficult to parse because you're using nonstandard terminology. For example, "Weyl module for $\mathfrak g$" does not mean the same thing as "Weyl module for the hyperalgebra in positive characteristic." I would interpret "Weyl module for $\mathfrak g$" as "Weyl module for the first Frobenius kernel," in which case there are things you can say about projectivity; I would direct you to Jantzen's book "Representations of Algebraic Groups" for more on projective modules for Frobenius kernels. – Chuck Hague Jun 19 '12 at 15:59
@Chuck: I agree that the question is difficult to parse, but I don't agree that the terminology "Weyl module for the first Frobenius kernel" is used. Those highest weight modules (of fixed $p$-power dimension) usually get called something like "baby Verma modules" and are compared in subtle ways to the actual Weyl modules for the algebraic group. Their dimensions are usually quite different and they are constructed in different settings. – Jim Humphreys Jun 19 '12 at 21:16
up vote 7 down vote accepted

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.

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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"...)

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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.

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Thanks, Wilberd! I had a vague memory of some category like that; probably I read about Polo's theorem in your Lecture Notes a while back, but I didn't remember much about it. – George McNinch Jun 20 '12 at 12:44
@Wilberd: I was interpreting the question as asking about a category of modules in which all Weyl modules are projective (at least those living in the category). Your category won't usually have that property. Also, it's probably more accurate to say that the category depends on the single weight $\lambda$ rather than on the corresponding Weyl module. But this type of bounded category is certainly natural enough. – Jim Humphreys Jun 20 '12 at 16:09

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.

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Weyl modules in characteristic zero are of course projective. – Bruce Westbury Jun 16 '12 at 11:04
A finite dimension $\mathfrak{g}$-module $M$ is a projective object in the category of all finite dimensional $\mathfrak{g}$-modules whenever $\mathfrak{g}$ is a (semi)simple Lie alegbra over a field of characteristic 0. This is just a consequence of the fact that every finite dimensional $\mathfrak{g}$-module is completely reducible (under the indicated hypothesis on $\mathfrak{g}$). – George McNinch Jun 20 '12 at 0:19

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