Characters of representations (of semisimple Lie algebras)

Posted February 8, 2010 by Akhil Mathew
Categories: algebra, representation theory
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Let {X} be a representation of a semisimple Lie algebra {\mathfrak{g}}, a Cartan subalgebra {\mathfrak{h}}, and some choice of splitting {\Phi = \Phi^+ \cup \Phi^-} on the roots.

Recall from the representation theory of finite groups that to each representation of a finite group {G} one can associate a character function {\chi}, and that the ring generated by the characters is the Grothendieck ring of the semisimple tensor category {Rep(G)}. There is something similar to be said for semisimple Lie algebras. So, assume {\mathfrak{h}} acts semisimply on {X} and that the weight spaces are finite-dimensional, and set formally

\displaystyle \mathrm{ch}(X) := \sum_{\lambda} \dim X_{\lambda} e(\lambda).

In other words, we define the character so as to include all the information on the size of the weight spaces at once.

It is necessary, however, to define what {e(\lambda)} for {\lambda \in \mathfrak{h}^{\vee}}. Basically, it is just a formal symbol; {\mathrm{ch}(X)} can more rigorously be thought of as a function {\mathfrak{h}^{\vee} \rightarrow \mathbb{Z}_{\geq 0}}. Nevertheless, we want to think of {e(\lambda)} as a formal exponential in a sense; we want to have {e(\lambda) e(\lambda') = e(\lambda + \lambda')}. The reason is that we can tensor two representations, and we want to talk about multiplying to characters.

I now claim that the above condition on {X} makes sense for {X \in \mathcal{O}}, the BGG category. This will follow because it is true for highest weight modules and we have:

Proposition 1 If {M \in \mathcal{O}}, then there is a finite filtration on {M} whose quotients are highest weight modules. Read the rest of this post »

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When is the simple quotient of a Verma module finite-dimensional?

Posted February 7, 2010 by Akhil Mathew
Categories: algebra, representation theory
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Today’s is going to be a long post, but an important one.  It tells us precisely what weights are allowed to occur as highest weights in finite-dimensional representations of a semisimple Lie algebra. 

Dominant integral weights

Let {V} be a finite-dimensional simple representation of a semisimple Lie algebra {\mathfrak{g}}, with Cartan subalgebra {\mathfrak{h}}, and root space decomposition {\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}}. Suppose given a base {\Delta} and a corresponding division {\Phi = \Phi^+ \cup \Phi^-}.

For each {\alpha \in \Phi^+}, choose {X_{\alpha} \in \mathfrak{g}_{\alpha}, Y_{\alpha} \in \mathfrak{g}_{-\alpha}} such that {[X_{\alpha},Y_{\alpha}] = H_{\alpha}} and {X_{\alpha}, Y_{\alpha}, H_{\alpha}} generate a subalgebra {\mathfrak{s}_{\alpha}} isomorphic to {\mathfrak{sl}_2}.

Consider the weight space decomposition

\displaystyle V = \bigoplus_{\beta \in \Pi} V_{\beta}

where {\Pi} denotes the set of weights of {V}. Then if {\beta \in \Pi} is the weight associated to a highest weight vector, {\beta(H_{\alpha})} is necessarily a nonnegative integer by the representation theory of {\mathfrak{sl}_2}. In other words,

\displaystyle <\beta, \alpha> := 2 \frac{ (\beta, \alpha)}{(\alpha, \alpha)} \in \mathbb{Z}_{\geq 0}.

Any weight {\beta} satisfying that identity for all {\alpha \in \Phi^+} is called dominant integral. We have shown that the highest weight of a finite-dimensional simple {\mathfrak{g}}-representation is necessarily dominant integral. In fact, given a dominant integral weight, we can actually construct such a finite-dimensional simple module.

The set of merely integral weights—those {\beta} with {<\beta, \alpha> \in \mathbb{Z}} for {\alpha \in \Phi}—form a lattice, spanned by vectors {\lambda_i} such that {<\lambda_i, \delta_j> = \delta_{ij}}, where the last {\delta_{ij}} is the Kronecker delta.

Theorem 1 The unique simple quotient of the Verma module {V(\beta)} is finite-dimensional if and only if {\beta} is dominant integral. Read the rest of this post »

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Highest weight vectors and Verma modules

Posted February 6, 2010 by Akhil Mathew
Categories: algebra, representation theory
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So, let’s suppose that we have a splitting of the roots {\Phi = \Phi^+ \cup \Phi^-}, as before, associated to a semisimple Lie algebra {\mathfrak{g}} and a Cartan subalgebra {\mathfrak{h}}. Recall that a vector {v \in V} for a representation {V} of {\mathfrak{g}} (not necessarily finite-dimensional!) is called a highest weight vector if {v} is annihilated by the nilpotent algebra {\mathfrak{n} = \bigoplus_{\alpha \in \Phi^+}}.

Let {V} be a highest weight module, generated by a highest weight vector {v}. We proved before, using a PBW basis for {U\mathfrak{g}}, that {V} is the direct sum of its finite-dimensional weight spaces—in particular, {\mathfrak{h}} acts semisimply, which is not a priori obvious since {V} is finite-dimensional—and so is any subrepresentation. The highest weight space is one-dimensional.  Now I am actually going to talk about them in a bit more detail.

Proposition 1 {V} is indecomposable and has a unique maximal submodule and unique simple quotient.

 

Indeed, let {W,W' \subset V} be any proper submodules; we will prove {W + W' \neq V}. If either contains {v}, then it is all of {V}. So we may assume both don’t contain {v}; by the above fact that {W,W'} decompose into weight spaces, they have no vectors of weight the same as {v}. So neither does {W + W'}, which means that {W+W' \neq V}.

We can actually take the sum of all proper submodules of {V}; the above argument shows that this sum does not contain {v} (and has no vectors with nonzero {v}-component). The rest of the proposition is now clear.

There is an important category, the BGG category {\mathcal{O}}, defined as follows: {X \in \mathcal{O}} if {X} is a representation of {\mathfrak{g}} on which {\mathfrak{n}} acts locally nilpotently (i.e., each {x \in X} is annihilated by some power of {\mathfrak{n}} in {U\mathfrak{g}}), {\mathfrak{h}} acts semisimply, and {X} is finitely generated over the enveloping algebra {U\mathfrak{g}}. I’m hoping to say a few things about category {\mathcal{O}} in the future, but for now, what we’ve seen is that highest weight modules belong to it. It is in fact a theorem that any object in {\mathcal{O}} has a filtration whose quotients are highest weight modules.

Proposition 2 Any simple highest weight modules of the same weight are isomorphic. Read the rest of this post »

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The Weyl group

Posted February 6, 2010 by Akhil Mathew
Categories: algebra, representation theory
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Recall that the reflections {s_{\alpha} \in Aut(E)} for {\alpha \in \Phi} preserve {\Phi}. They generate a group {W} called the Weyl group. Moreover, since {\Phi} spans {E}, the map {W \rightarrow S_{\Phi}}, the symmetric group on {\Phi}, is injective. So {W} is a finite group of orthogonal isomorphisms of {E}, i.e. leaving invariant the bilinear form {(\cdot, \cdot)}.

Everything here actually makes sense for root systems in general, but we are restricting ourselves to the case of a root system associated to semisimple Lie algebra and a Cartan subalgebra.  The only difference is that one has to prove the result on maximal strings (which was proved in the case of Lie algebras here), though it can be done for root systems in general.

Now choose a base {\Delta} for {\Phi} and a corresponding partition {\Phi = \Phi^+ \cup \Phi^-}; the {s_{\delta}} for {\delta \in \Delta} are called simple reflections.

The goal we are aiming for is the following theorem, which gives a large amount of information about the Weyl group.

Theorem 1 {W} acts simply transitively on Weyl chambers and on bases. Any root can be moved by an element of the Weyl group into a given base. {W} is generated by the simple reflections. Read the rest of this post »

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Positive and simple roots

Posted February 5, 2010 by Akhil Mathew
Categories: algebra, representation theory
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So, suppose given a root system {\Phi} in a euclidean space {E}, which arises from a semisimple Lie algebra and a Cartan subalgebra as before. The first goal of this post is to discuss the “splitting”

\displaystyle \Phi = \Phi^+ \cup \Phi^-

(disjoint union) in a particular way, into positive and negative roots, and the basis decomposition into simple roots. Here {\Phi^- = - \Phi^+}.

To do this, choose {v \in E} such that {(v, \alpha) \neq 0} for {\alpha \in \Phi}. Then define {\Phi^+} to be those roots {\alpha} with {(v,\alpha)>0} and {\Phi^-} those with {(v,\alpha) < 0}. This was easy. We talked about positive and negative roots before using a real-valued linear functional, which here is given by an inner product anyway.

Bases

OK. Next, I claim it is possible to choose a linearly independent set {\Delta \subset \Phi^+} such that every root is a combination

\displaystyle \alpha = \sum k_i \delta_i, \quad \delta_i \in \Delta, \ k_i \in \mathbb{Z}

with all the {k_i \geq 0} or all the {k_i \leq 0}.

Then {\Delta} will be called a base. It is not unique, but I will show how to construct this below. Read the rest of this post »

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The roots form a root system

Posted February 5, 2010 by Akhil Mathew
Categories: algebra, representation theory
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I’m keeping the same notation as all the previous posts here on semisimple Lie algebras.

Consider the real vector space

\displaystyle E = \sum_{\alpha \in \Phi} \mathbb{R} \alpha \subset \mathfrak{h}^{\vee}.

I claim that the form {(\cdot, \cdot)} (obtained by the isomorphism {\mathfrak{h}^{\vee} \rightarrow \mathfrak{h}} induced by the Killing form and the Killing form itself) is actually an inner product making {E} into a euclidean space. To see this, we will check that {(\alpha, \alpha) > 0} for all {\alpha}. Indeed:

\displaystyle (\alpha, \alpha) = B(T_{\alpha}, T_{\alpha})

where {B} is the Killing form, by definition.

Now

\displaystyle B(T_{\alpha}, T_{\alpha}) = \mathrm{Tr}_{\mathfrak{g}} ( \mathrm{ad} T_{\alpha}^2) = \sum_{\beta \in \Phi} \mathrm{Tr}_{\mathfrak{g}_{\beta}} ( \mathrm{ad} T_{\alpha}^2) .

Now {T_{\alpha}} acts by the scalar {\beta(T_{\alpha}) = (\beta, \alpha)} on {\mathfrak{g}_{\beta}}, so after dividing by {(\alpha, \alpha)^2}, this becomes

\displaystyle (\alpha, \alpha)^{-1} = \sum_{\beta \in \Phi} \left( \frac{ (\beta, \alpha)}{(\alpha, \alpha ) } \right)^2.

But as we showed yesterday, {\frac{ (\beta, \alpha)}{(\alpha, \alpha )} \in \mathbb{Q}}, so the sum in question is actually positive. This proves one half of:

Proposition 1 {E} is a euclidean space and {\mathfrak{h}^{\vee} = E \oplus iE}. Read the rest of this post »

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The root space decomposition II

Posted February 4, 2010 by Akhil Mathew
Categories: algebra, representation theory
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OK, now we’ve gotten some of the basic facts about the root space decomposition down. So, as usual {\mathfrak{g}} is a semisimple Lie algebra and {\mathfrak{h}} a Cartan subalgebra; we have the decomposition {\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}}, where {\Phi \subset \mathfrak{h}^{\vee}} is the root system. For each {\alpha \in \Phi}, we can choose a pair of vectors {X_{\alpha} \in \mathfrak{g}_{\alpha}< Y_{\alpha} \in \mathfrak{g}_{-\alpha}, H_{\alpha} \in \mathfrak{h}}. Then {X_{\alpha}, Y_{\alpha}, H_{\alpha}} generate a subalgebra {\mathfrak{s}_{\alpha} \subset \mathfrak{g}} which is isomorphic to {\mathfrak{sl}_2}. Here {\alpha(H_{\alpha})=2} and {H_{\alpha}} is a multiple of {T_{\alpha}}, which in turn is the dual to {\alpha} under the Killing form that identifies {\mathfrak{h} \simeq \mathfrak{h}^{\vee}}.

That was a lightning review of where we are; if you’ve missed something, check back at this post.

The notation {\mathfrak{s}_{\alpha}} suggests that the algebra should only depend on {\alpha} and not on the particular choice of {X_{\alpha}, Y_{\alpha}} (but {H_{\alpha}} is uniquely determined from {\alpha(H_{\alpha})=2} and {H_{\alpha} \in \mathbb{C} T_{\alpha}}). Indeed, this is the case, and it follows from

Proposition 1 When {\alpha \in \Phi}, {\mathfrak{g}_{\alpha}} is one-dimensional.

 

Choose any {\mathfrak{s}_{\alpha}} coming from suitable {X_{\alpha}, Y_{\alpha}} and {H_{\alpha}}. We have a representation of {\mathfrak{s}_{\alpha}} on

\displaystyle V := \bigoplus_{\mathbb{Z} \alpha} \mathfrak{g}_{\alpha}

(recall {\mathfrak{g}_0 = \mathfrak{h}}) and we can apply the representation theory of {\mathfrak{sl}_2} to it. Read the rest of this post »

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Interlude on sl2

Posted February 3, 2010 by Akhil Mathew
Categories: algebra, representation theory
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I talked about the Lie algebra {\mathfrak{sl}_2} a while back.  Now I’m going to do it more properly, and using the tools developed.  This is going to feature prominently in some of the proofs in the sequel.

Now, let’s see how all this works for the familiar case of {\mathfrak{sl}_2}, with its usual generators {H,X,Y}. This is a simple Lie algebra in fact. To see this, let’s consider the ideal {I} of {\mathfrak{sl}_2} generated by some nonzero vector {aX + bH + cY}; I claim it is all of {\mathfrak{sl}_2}.

Consider the three cases {a \neq 0, b \neq 0, c \neq 0}:

First, assume {a} or {c} is nonzero. Bracketing with {H}, and again, gives

\displaystyle -2aX + 2 c Y \in I , \ (-2)^2 a X + 2^2 cY \in I, \ (-2)^3 a X + 2^3 cY \in I.

Using a vanderMonde invertibility of this system of linear equations, we find that either {X} or {Y} belongs to {I}. Say {X} does, for definiteness; then {H = [X,Y] \in I} too; from this, {Y = -\frac{1}{2} [H,Y] \in I} as well. Thus {I = \mathfrak{sl}_2}.

If {a=c=0}, then from {b \neq 0}, we find {H \in I}, which implies {X = \frac{1}{2}[H,X] \in I} and similarly for {Y}. Thus {I= \mathfrak{sl}_2}.

I claim now that the algebra {\mathbb{C} H} is in fact a Cartan subalgebra. Indeed, it is easily checked to be maximal abelian. Moreover, since {H} acts by a diagonalizable operator on the faithful representation on {\mathbb{C}^2}, it follows that {H \in \mathfrak{sl}_2} is (abstractly) semisimple. Read the rest of this post »

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The weight space decomposition

Posted February 3, 2010 by Akhil Mathew
Categories: algebra, representation theory
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Recall that in the representation theory of {\mathfrak{sl}_2}, one considered an element {H} and its action on a representation {V}. We looked for its largest eigenvalue and the corresponding highest weight vector.

There is something along the same lines to be done here for arbitrary semisimple Lie algebras, though it is much more complicated (and interesting).   I’m only going to scratch the surface today.

Let {\mathfrak{g}} be a semisimple Lie algebra and {\mathfrak{h}} a Cartan subalgebra. Then {\mathfrak{h}} is to play the role of {H} in {\mathfrak{sl}_2}; the {X,Y} matrices in {\mathfrak{sl}_2} are now replaced by the root space decomposition

\displaystyle \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}.

We know that {\mathfrak{h}} acts on a representation {V} of {\mathfrak{g}} by commuting semisimple transformations, so we can write

\displaystyle \mathfrak{h} = \bigoplus_{\beta \in \mathfrak{h}^{\vee}} V_{\beta}

where {V_{\beta} := \{ v \in V: hv = \beta(h) v \ \forall h \in \mathfrak{h} \}}. These are called the weight spaces, and the {\beta} are called weights.

Now

\displaystyle g_{\alpha} V_{\beta} \subset V_{\alpha + \beta }

by an analog of the “fundamental calculation,” proved as follows. Let {h \in \mathfrak{h}, x \in \mathfrak{g}_{\alpha}, v \in V_{\beta}}. Then

\displaystyle h (x v) =xh(v) + [h,x] v = x (\alpha(h)) v + \beta(h) x v = (\alpha + \beta)(h) xv. Read the rest of this post »

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The root space decomposition

Posted February 2, 2010 by Akhil Mathew
Categories: algebra, representation theory
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Let {\mathfrak{g}} be a semisimple Lie algebra over {\mathbb{C}} and {\mathfrak{h}} a Cartan subalgebra.

Given {\alpha \in \mathfrak{h}^{\vee}}, we can define a subspace of {\mathfrak{g}}

\displaystyle \mathfrak{g}_{\alpha} = \{ x \in \mathfrak{g}: (\mathrm{ad} H)x = \alpha(H) x , \ \forall H \in \mathfrak{h} \}.

The nonzero {\alpha} that occur with {\mathfrak{g}_{\alpha} \neq 0} are called roots, and they form a set {\Phi}. Because {\mathfrak{h}} acts on {\mathfrak{g}} by commuting diagonalizable operators (by semisimplicity of the elements of {\mathfrak{h}}), it follows by simultaneous diagonalization, that

\displaystyle \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}.

Recall that {\mathfrak{g}_0 = \mathfrak{h}}, because a Cartan subalgebra is maximal abelian.

This is called the root space decomposition. A simple but important property is that {[\mathfrak{g}_{\alpha}, \mathfrak{g}_{\beta}] \subset \mathfrak{g}_{\alpha + \beta}}; this is checked because the {\mathrm{ad} H} are derivations.

The root space decomposition is highly useful in studying simple representations of {\mathfrak{g}}.

I shall collect here a few facts about it.

Proposition 1 {\mathfrak{g}_{\alpha}, \mathfrak{g}_{\beta}} are orthogonal under the Killing form unless {\alpha + \beta = 0}.

 

This follows by a familiar argument, in view of {[\mathfrak{g}_{\alpha}, \mathfrak{g}_{\beta}] \subset \mathfrak{g}_{\alpha + \beta}}. Read the rest of this post »

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