Notes on BRST V: Highest Weight Theory

In the last posting we discussed the Lie algebra cohomology [tex]H^*(\mathfrak g, V)[/tex] for [tex]\mathfrak g[/tex] a semi-simple Lie algebra. Because the invariants functor is exact here, this tells us nothing about the structure of irreducible representations in this case. In this posting we’ll consider a different sort of example of Lie algebra cohomology, one that is intimately involved with the structure of irreducible [tex]\mathfrak g[/tex]-representations.

Structure of semi-simple Lie algebras

A semi-simple Lie algebra is a direct sum of non-abelian simple Lie algebras. Over the complex numbers, every such Lie algebra is the complexification [tex]\mathfrak g_{\mathbf C}[/tex] of some real Lie algebra [tex] \mathfrak g[/tex] of a compact, connected Lie group. The Lie algebra [tex] \mathfrak g[/tex] of a compact Lie group [tex]G[/tex] is, as a vector space, the direct sum

[tex] \mathfrak g=\mathfrak t \oplus \mathfrak g/\mathfrak t[/tex]

where [tex] \mathfrak t[/tex] is a commutative sub-algebra (the Cartan sub-algebra), the Lie algebra of [tex]T[/tex], a maximal torus subgroup of [tex]G[/tex].

Note that [tex]\mathfrak t[/tex] is not an ideal in [tex]\mathfrak g[/tex], so [tex]\mathfrak g/\mathfrak t[/tex] is not a subalgebra. [tex]\mathfrak g[/tex] is itself a representation of [tex]\mathfrak g[/tex] (the adjoint representation: [tex]\pi(X)Y= [X,Y][/tex]), and thus a representation of the subalgebra [tex]\mathfrak t[/tex]. On any complex representation [tex]V[/tex] of [tex]\mathfrak g[/tex], the action of [tex]\mathfrak t[/tex] can be diagonalized, with eigenspaces [tex]V^\lambda[/tex] labeled by the corresponding eigenvalues, given by the weights [tex]\lambda[/tex]. These weights [tex]\lambda\in\mathfrak t_{\mathbf C}^*[/tex] are defined by (for [tex]v\in V^\lambda,\ H\in \mathfrak t[/tex]):

[tex]\pi(H)v=\lambda(H)v[/tex]

Complexifying the adjoint representation, the non-zero weights of this representation are called roots, and we have

[tex]\mathfrak g_{\mathbf C}=\mathfrak t_{\mathbf C} \oplus ((\mathfrak g/\mathfrak t)\otimes\mathbf C)[/tex]

The second term on the right is the sum of the root spaces [tex]V^\alpha[/tex] for the roots [tex]\alpha[/tex]. If [tex]\alpha[/tex] is a root, so is [tex]-\alpha[/tex], and one can choose decompositions of the set of roots into “positive roots” and “negative roots” such that:

[tex]\mathfrak n^+=\bigoplus_{+\ roots\ \alpha}(\mathfrak g_{\mathbf C})^\alpha,\ \mathfrak n^-=\bigoplus_{-\ roots\ \alpha}(\mathfrak g_{\mathbf C})^\alpha[/tex]

where [tex]\mathfrak n^+[/tex] (the “nilpotent radical”) and [tex]\mathfrak n^-[/tex] are nilpotent Lie subalgebras of [tex]\mathfrak g_{\mathbf C}[/tex]. So, while [tex]\mathfrak g/\mathfrak t[/tex] is not a subalgebra of [tex]\mathfrak g[/tex], after complexifying we have decompositions

[tex](\mathfrak g/\mathfrak t)\otimes \mathbf C=\mathfrak n^+ \oplus \mathfrak n^-[/tex]

The choice of such a decomposition is not unique, with the Weyl group [tex]W[/tex] (for a compact group [tex]G[/tex], W is the finite group [tex]N(T)/T[/tex], [tex]N(T)[/tex] the normalizer of [tex]T[/tex] in [tex]G[/tex]) permuting the possible choices.

Recall that a complex structure on a real vector space [tex]V[/tex] is given by a decomposition

[tex]V\otimes \mathbf C=W\oplus\overline{W}[/tex]

so the above construction gives [tex]|W|[/tex] different invariant choices of complex structure on [tex]\mathfrak g/\mathfrak t[/tex], which in turn give [tex]|W|[/tex] invariant ways of making [tex]G/T[/tex] into a complex manifold.

The simplest example to keep in mind is [tex]G=SU(2),\ T=U(1),\ W=\mathbf Z_2,[/tex] where [tex]\mathfrak g=\mathfrak{su}(2)[/tex], and [tex]\mathfrak g_{\mathbf C}=\mathfrak{sl}(2,\mathbf C)[/tex]. One can choose [tex]T[/tex] to be the diagonal matrices, with a basis of [tex]\mathfrak t[/tex] given by

[tex]\frac{i}{2}\sigma_3=\frac{1}{2}\begin{pmatrix}i&0\\0&-i\end{pmatrix}[/tex]

and bases of [tex]\mathfrak n^+,\ \mathfrak n^-[/tex] given by

[tex]\frac{1}{2}(\sigma_1+i\sigma_2)=\begin{pmatrix}0&1\\0&0\end{pmatrix},\ \frac{1}{2}(\sigma_1-i\sigma_2)=\begin{pmatrix}0&0\\1&0\end{pmatrix}[/tex]

(here the [tex]\sigma_i[/tex] are the Pauli matrices). The Weyl group in this case just interchanges [tex]\mathfrak n^+ \leftrightarrow \mathfrak n^-[/tex].

Highest weight theory

Irreducible representations [tex]V[/tex] of a compact Lie group [tex]G[/tex] are finite dimensional and correspond to finite dimensional representations of [tex]\mathfrak g_{\mathbf C}[/tex]. For a given choice of [tex]\mathfrak n^+[/tex], such representations can be characterized by their subspace [tex]V^{\mathfrak n^+}[/tex], the subspace of vectors annihilated by [tex]\mathfrak n^+[/tex]. Since [tex]\mathfrak n^+[/tex] acts as “raising operators”, taking subspaces of a given weight to ones with weights that are more positive, this is called the “highest weight” space since it consists of vectors whose weight cannot be raised by the action of [tex]\mathfrak g_{\mathbf C}[/tex]. For an irreducible representation, this space is one dimensional, and we can label irreducible representations by the weight of [tex]V^{\mathfrak n^+}[/tex]. The irreducible representation with highest weight [tex]\lambda[/tex] is denoted [tex]V_{\lambda}[/tex]. Note that this labeling depends on the choice of [tex]\mathfrak n^+[/tex].

Getting back to Lie algebra cohomology, while [tex]H^*(\mathfrak g, V)=0[/tex] for an irreducible representation [tex]V[/tex], the Lie algebra cohomology for [tex]\mathfrak n^+[/tex] is more interesting, with [tex]H^0(\mathfrak n^+, V)=V^{\mathfrak n^+}[/tex], the highest weight space. [tex]\mathfrak t[/tex] acts not just on [tex]V[/tex], but on the entire complex [tex]C(\mathfrak n^+, V)[/tex], in such a way that the cohomology spaces [tex]H^i(\mathfrak n^+,V)[/tex] are representations of [tex]\mathfrak t[/tex], so can be characterized by their weights.

For an irreducible representation [tex]V_\lambda[/tex], one would like to know which higher cohomology spaces are non-zero and what their weights are. The answer to this question involves a surprising “[tex]\rho[/tex] – shift”, a shift in the weights by a weight [tex]\rho[/tex], where

[tex]\rho=\frac{1}{2}\sum_{+\ roots} \alpha[/tex]

half the sum of the positive roots. This is a first indication that it might be better to work with spinors rather than with the exterior algebra that is used in the Koszul resolution used to define Lie algebra cohomology. Much more about this in a later posting.

One finds that [tex]dim\ H^*(\mathfrak n^+,V_\lambda)=|W|[/tex], and the weights occuring in [tex]H^i(\mathfrak n^+,V_\lambda)[/tex] are all weights of the form [tex]w(\lambda +\rho)-\rho[/tex], where [tex]w\in W[/tex] is an element of length [tex]i[/tex]. The Weyl group can be realized as a reflection group action on [tex]\mathfrak t^*[/tex], generated by one reflection for each “simple” root. The length of a Weyl group element is the minimal number of reflections necessary to realize it. So, in dimension 0, one gets [tex]H^0(\mathfrak n^+, V_\lambda)=V^{\mathfrak n^+}[/tex] with weight [tex]\lambda[/tex], but there is also higher cohomology. Changing one’s choice of [tex]\mathfrak n^+[/tex] by acting with the Weyl group permutes the different weight spaces making up [tex]H^*(\mathfrak n^+, V)[/tex]. For an irreducible representation, to characterize it in a manner that is invariant under change in choice of [tex]\mathfrak n^+[/tex], one should take the entire Weyl group orbit of the [tex]\rho[/tex] – shifted highest weight [tex]\lambda[/tex], i.e. the set of weights

[tex]\{w(\lambda +\rho),\ w\in W\}[/tex]

In our [tex]G=SU(2)[/tex] example, highest weights can be labeled by non-negative half integral values (the “spin” [tex]s[/tex] of the representation)

[tex]s=0,\frac{1}{2},1,\frac{3}{2}\2,\cdots[/tex]

with [tex]\rho=\frac{1}{2}[/tex]. The irreducible representation [tex]V_s[/tex] is of dimension [tex]2s+1[/tex], and one finds that [tex]H^0(\mathfrak n^+,V_s)[/tex] is one-dimensional of weight [tex]s[/tex], while [tex]H^1(\mathfrak n^+,V_s)[/tex] is one-dimensional of weight [tex]-s-1[/tex].

The character of a representation is given by a positive integral combination of the weights

[tex]char(V)=\sum_{weights\ \omega} (dim\ V^\omega)\omega[/tex]

(here [tex]V^\omega[/tex] is the [tex]\omega[/tex] weight space). The Weyl character formula expresses this as a quotient of expressions involving weights taken with both positive and negative integral coefficients. The numerator and denominator have an interpretation in terms of Lie algebra cohomology:

[tex]char(V)=\frac{\chi(H^*(\mathfrak n^+, V))}{\chi(H^*(\mathfrak n^+, \mathbf C))}[/tex]

Here [tex]\chi[/tex] is the Euler characteristic: the difference between even-dimensional cohomology (a sum of weights taken with a + sign), and odd-dimensional cohomology (a sum of weights taken with a – sign). Note that these Euler characteristics are independent of the choice of [tex]\mathfrak n^+[/tex].

The material in this last section goes back to Bott’s 1957 paper Homogeneous Vector Bundles, with more of the Lie algebra story worked out by Kostant in his 1961 Lie Algebra Cohomology and the Generalized Borel-Weil Theorem. For an expository treatment with details, showing how one actually computes the Lie algebra cohomology in this case, for U(n) see chapter VI.3 of Knapp’s Lie Groups, Lie Algebras and Cohomology, or for the general case see chapter IV.9 of Knapp and Vogan’s Cohomological Induction and Unitary Representations.