Infinite Dimensional Lie Algebras April 4, 2005

Gabriel C. Drummond-Cole

April 4, 2005

Last time we discussed the notion of highest weight module, which is a representation V generated by a single vector of some wheight Λ such that e_{i}v = 0 for all i.

We discussed that every such representation, assuming it has a weight decomposition, is the direct sum ⨁_{μ∈Λ−Q_{+}} V [μ] where Q_{+} = {∑ n_{i}α_{i}, n_{i} ≥ 0}. We discussed that among other examples we have the so-called Verma modules, freely generated by the highest weight. This may be reducible so we define L_{Λ}, the irreducible quotient. This works the same way for Kac-Moody algebras, finite dimensional algebras, and affine Lie algebras. Remember for 𝔰𝔩(2) the Verma module is an infinite chain and when the highest weight is an integer you can find a submodule and break off a finite dimensional irreducible quotient which is a highest weight module.

Just considering these is too restrictive; it’s not closed under product. So we consider the category 𝒪 which is representations V with weight decomposition V = ⨁_{Λ∈P (V)} V [Λ], where P (V) = {Λ ∈ 𝔥̂^{∗}|V [Λ] ≠ 0}. subject to

1. dim V [Λ] < ∞

2. P (V) is a subset of a finite union of sets of the form Λ_{i} − Q_{+}.

Sometimes this is called the category of highest weight modules; this is not literally true since this includes sums.

Let me state some properties.

1. Every highest weight module is in 𝒪.

2. 𝒪 is closed under ⊕, submodules, and quotients.

3. Define 𝒪_{k} = {V ∈ 𝒪|K|_{V} = k Id} for k ∈ ℂ. If Kv = kv you want to show that xv satisfies the same condition. So Kxv = xKv = kxv for all x ∈ 𝔤. So it makes sense to talk about representations of a fixed level; these are called representations of level k.

Then 𝒪 = ⨁ 𝒪_{k}. This means that a representation of 𝒪 can be written as a direct sum of representations from the different levels, and Hom between different levels is trivial.

So to study 𝒪 we need only study what happens at 𝒪_{k} for arbitrary k.

4. If V ∈ 𝒪 then P (V) is “bounded from above.” What does this mean?

(a) Re (Λ, ρ̂) is bounded from above. Let me remind you that ρ̂ is defined so that ⟨ρ̂, α_{i}^{√}⟩ = 1, ρ̂ = ρ + h^{√}K. This is one way to make this notion precise.

(b) If you take some weight Λ ∈ P (V), and then you take (Λ + Q_{+}) ∩ P (V), this is finite.

The proof is rather obvious. The second part (of the last one) follows from the first. To go up you are adding things in Q_{+}; this makes the inner product with ρ̂ go up, so if it’s bounded from above, we know we can only do this finitely many times.

For the first part, the argument is essentially that ρ̂ has to be one of the highest weights, and there are only finitely many.

Lemma 1 Irreducible representations in 𝒪 are exactly L_{Λ}, Λ ∈ 𝔥̂^{∗}.

This is easy to believe; here is a formal proof. Assume V is irreducible. Since the set of weights is bounded from above, there exists a weight Λ ∈ P (V) such that Λ + α_{i} ∉ P (V). Therefore, if you take v ∈ V [Λ], this will be a highest weight vector in the sense that e_{i}v = 0.

Consider the submodule generated by v. If the representation is irreducible, the vector must generate all of V. So V is a highest weight representation. This must be unique because these are nonisomorphic for different Λ.

The structure of this category is very complicated. 𝔰𝔩(2) gives you a description of the category. A lot is known about it, but I don’t want to. I’d have to go into perverse sheaves and things.

This does, though, have a fairly simple subcategory, of finite dimensional modules. This category gives you a fairly nice piece. So what about for affine Lie algebras. You cannot expect finite dimensional modules. We want something essentially as nice; there is an appropriate notion.

Definition 1 A representation V ∈ 𝒪 is integrable if for any v ∈ V the 𝔰𝔩(2)_{i}-submodule generated by v is finite dimensional. Here 𝔰𝔩(2)_{i} = ⟨e_{i}, f_{i}, h_{i}⟩.

Later we’ll see a more invariant definition. Let me make a picture for 𝔰̂𝔩(2).

[...]

The notation is that this category is 𝒪^{int} and similarly I can define 𝒪_{k}^{int}. Integrable modules almost always means integrable and from the category 𝒪. I didn’t even define the others.

Let me write as a lemma that

Lemma 2 𝒪^{int} is closed under ⊕, submodules, and quotients

First, I would like to explain, I don’t have a single example of such a module. Verma modules are not integrable, since f_{1} acts freely on v and is infinite dimensional.

We do have examples. For 𝔤 then every finite dimensional module is integrable. As a matter of fact, if 𝔤 is finite dimensional, integrables are always finite dimensional.

But examples for affien Lie algebras will have to wait. Let me start with some properties. This is probably the most important result I’m going to prove today.

Theorem 1 If V ∈ 𝒪^{int} then the set of weights P (V) is Ŵ -invariant.

Proof. It suffices to show that P (V) is invariant under simple reflections s_{i}, i = 0, …, r. Now, if v ∈ V [Λ], let’s say n_{i} = ⟨Lambda, h_{i}⟩, the weight of v considered as a module over 𝔰𝔩(2)_{i}. For one thing it should be integer, since it generates a finite dimensional module over 𝔰𝔩(2).

If n_{i} ≥ 0 then consider

f_{i}^{n_{i}}v ∈ V [Λ − n_{i}α_{i}] = V [s_{i}Λ].

By well-known results for 𝔰𝔩(2) this gives an isomorphism V [Λ] ≅ V [S_{i}Λ].

Let me now write a simple corollary, a couple, actually.

Corollary 1 1. If V ∈ 𝒪^{int}, v ∈ V then 𝔰𝔩(2)_{α}-module generated by v is finite dimensonal, 𝔰𝔩(2)_{α} = ⟨e_{α}, f_{α}, h_{α}⟩ for α ∈ R̂_{re}.

This shows in the definition of integrability I could have stated in terms of all the roots, not just the simple ones.

2. For V ∈ 𝒪^{int},

P (V) ⊂ P = {Λ ∈ 𝔥̂^{∗}|⟨Λ, h_{i}⟩ ∈ ℤ}
= {λ + kΛ_{0} + aδ|⟨λ, h_{i}⟩ ∈ ℤ, i = 1, …, r, ⟨λ + kΛ_{0}, −θ^{√} + K⟩ ∈ ℤ =
{λ + kΛ_{0} + aδ|λ ∈ P, k ∈ ℤ}

This reminds us of the finite dimensional case.

3. For 𝔤, 𝒪^{int} are precisely the finite dimensional modules. This is one reason this name is not used in finite dimensional cases.

Proof. Let v ∈ V [Λ]. Then 𝔰𝔩(2)_{α}v ∈ ⨁_{k∈ℤ} V [Λ + kα].

Assuming that α is positive, sooner or later the right hand side is zero for some k greater than zero by assumption. The same is true for negative roots by invariance under S_{α}.

For the third one, one direction I already proved. Well, there is nothing to prove. A finite dimensional module is integrable. The difficult part is proving the opposite direction. It suffices to show that P (V) is finite. Let me draw a picture.

[...]

Since the set of weights is bounded from above, P (V) ∩ C_{+} is finite, where C_{+} is the positive Weyl chamber. By Weyl group invariance, there are only finitely many weights in any Weyl chamber. Since there are only finitely many, we’re done.