Not Even Wrong

Note: I’ve started putting together the material from these postings into a proper document, available here, which will be getting updated as time goes on. I’ll be making changes and additions to the text there, not on the blog postings. For most purposes, that will be what people interested in this subject will want to take a look at.

When a Lie group with Lie algebra [tex]\mathfrak g[/tex] acts on a manifold [tex]M[/tex], one gets two sorts of actions of [tex]\mathfrak g[/tex] on the differential forms [tex]\Omega^*(M[/tex]). For each [tex]X\in \mathfrak g [/tex] one has operators:

and

These operators satisfy the relation

[tex]di_X+i_Xd={\mathcal L}_X[/tex]

where [tex]d[/tex] is the de Rham differential [tex]d:\Omega^k(M)\rightarrow \Omega^{k+1}(M)[/tex], and the operators [tex]d, i_X, \mathcal L_X[/tex] are (super)-derivations. In general, an algebra carrying an action by operators satisfying the same relations satisfied by [tex]d, i_X, \mathcal L_X[/tex] will be called a [tex]\mathfrak g[/tex]-differential algebra. It will turn out that the Clifford algebra [tex]Cliff(\mathfrak g)[/tex] of a semi-simple Lie algebra [tex]\mathfrak g[/tex] carries not just the Clifford algebra structure, but the additional structure of a [tex]\mathfrak g[/tex]-differential algebra, in this case with [tex]\mathbf Z_2[/tex], not [tex]\mathbf Z[/tex] grading.

Note that in this section the commutator symbol will be the supercommutator in the Clifford algebra (commutator or anti-commutator, depending on the [tex]\mathbf Z_2[/tex] grading). When the Lie bracket is needed, it will be denoted [tex][\cdot,\cdot]_{\mathfrak g}[/tex].

To get a [tex]\mathfrak g[/tex]-differential algebra on [tex]Cliff(\mathfrak g)[/tex] we need to construct super-derivations [tex]i_X^{Cl}[/tex], [tex]{\mathcal L}_X^{Cl}[/tex], and [tex]d^{Cl}[/tex] satisfying the appropriate relations. For the first of these we don’t need the fact that this is the Clifford algebra of a Lie algebra, and can just define

[tex]i_X^{Cl}(\cdot)=[-\frac{1}{2}X,\cdot][/tex]

For [tex]{\mathcal L}_X^{Cl}[/tex], we need to use the fact that since the adjoint representation preserves the inner product, it gives a homomorphism

[tex]\widetilde{ad}:\mathfrak g \rightarrow \mathfrak{spin}(\mathfrak g)[/tex]

where [tex]\mathfrak{spin}(\mathfrak g)[/tex] is the Lie algebra of the group [tex]Spin(\mathfrak g)[/tex] (the spin group for the inner product space [tex]\mathfrak g[/tex]), which can be identified with quadratic elements of [tex]Cliff(\mathfrak g)[/tex], taking the commutator as Lie bracket. Explicitly, if [tex]X_a[/tex] is a basis of [tex]\mathfrak g[/tex], [tex]X_a^* [/tex] the dual basis, then

[tex]\widetilde{ad}(X)=\frac{1}{4}\sum_a X_a^*[X,X_a]_{\mathfrak g}[/tex]

and we get operators acting on [tex]Cliff(\mathfrak g)[/tex]

[tex]{\mathcal L}_X^{Cl}(\cdot)=[\widetilde{ad}(X),\cdot][/tex]

Remarkably, an appropriate [tex]d^{Cl}[/tex] can be constructed using a cubic element of [tex]Cliff(\mathfrak g)[/tex]. Let

[tex]\gamma= \frac{1}{24}\sum_{a,b}X^*_aX^*_b[X_a,X_b]_{\mathfrak g}[/tex]

then

[tex]d^{Cl}(\cdot)=[\gamma, \cdot][/tex]

[tex]d^{Cl}\circ d^{Cl}=0[/tex] since [tex]\gamma^2[/tex] is a scalar which can be computed to be [tex]-\frac{1}{48}tr\Omega_{\mathfrak g}[/tex], where [tex]\Omega_{\mathfrak g}[/tex] is the Casimir operator in the adjoint representation.

The above constructions give [tex]Cliff(\mathfrak g)[/tex] the structure of a filtered [tex]\mathfrak g[/tex]-differential algebra, with associated graded algebra [tex]\Lambda^*(\mathfrak g)[/tex]. This gives [tex]\Lambda^*(\mathfrak g)[/tex] the structure of a [tex]\mathfrak g[/tex]-differential algebra, with operators [tex]i_X, \mathcal L_X, d[/tex]. The cohomology of this differential algebra is just the Lie algebra cohomology [tex]H^*(\mathfrak g, \mathbf C)[/tex].

[tex]Cliff(\mathfrak g)[/tex] can be thought of as an algebra of operators corresponding to the quantization of an anti-commuting phase space [tex]\mathfrak g[/tex]. Classical observables are anti-commuting functions, elements of [tex]\Lambda^*(\mathfrak g^*)[/tex]. Corresponding to [tex]i_X, \mathcal L_X, d[/tex] one has both elements of [tex]\Lambda^*(\mathfrak g^*)[/tex] and their quantizations, the operators in [tex]Cliff(\mathfrak g)[/tex] constructed above.

For more details about the above, see the following references

Clifford Algebras

Clifford algebras are well-known to physicists, in the guise of matrix algebras generated by the [tex]\gamma[/tex] -matrices first used in the Dirac equation. They also have a more abstract formulation, which will be the topic of this posting. One way to think about Clifford algebras is as a “quantization” of the exterior algebra, associated with a symmetric bilinear form.

Given a vector space [tex]V[/tex] with a symmetric bilinear form [tex](\cdot,\cdot)[/tex], the associated Clifford algebra [tex]Cliff (V,(\cdot,\cdot))[/tex] can be defined by starting with the tensor algebra [tex]T^*(V)[/tex] ([tex]T^k(V)[/tex] is the k-th tensor power of [tex]V[/tex]), and imposing the relations

[tex]v\otimes w + w\otimes v = -2(v,w)1[/tex]

where [tex]v,w\in V=T^1(V),\ 1\in T^0(V)[/tex]. Note that many authors use a plus instead of a minus sign in this relation. The case of most interest in physics is [tex]V=\mathbf R^4, (\cdot,\cdot)[/tex] the Minkowski inner product of signature (3,1). The theory of Clifford algebras for real vector spaces [tex]V[/tex] is rather complicated. Here we’ll stick to complex vector spaces [tex]V[/tex], where the theory is much simpler, partially because over [tex]\mathbf C[/tex] there is, up to equivalence, only one non-degenerate symmetric bilinear form. We will suppress mention of the bilinear form in the notation, writing [tex]Cliff(V)[/tex] for [tex]Cliff(V,(\cdot,\cdot)).[/tex]

For a more concrete definition, one can choose an orthonormal basis [tex]e_i[/tex] of [tex]V[/tex]. Then [tex]Cliff(V)[/tex] is the algebra generated by the [tex]e_i[/tex], with multiplication satisfying the relations

[tex]e_i^2=-1,\ \ e_ie_j=-e_je_i\ \ (i\neq j)[/tex]

One can show that these complex Clifford algebras are isomorphic to matrix algebras, more precisely

[tex]Cliff(\mathbf C^{2n})\simeq M(\mathbf C, 2^n),\ \ \ Cliff(\mathbf C^{2n+1})\simeq M(\mathbf C, 2^n)\oplus M(\mathbf C, 2^n)[/tex]

Clifford Algebras and Exterior Algebras

The exterior algebra [tex]\Lambda^*(V)[/tex] is the algebra of anti-symmetric tensors, with product the wedge product [tex]\wedge[/tex]. This is also exactly what one gets if one takes the Clifford algebra [tex]Cliff(V)[/tex], with zero bilinear form. Multiplying a non-degenerate symmetric bilinear form [tex](\cdot,\cdot)[/tex] by a parameter [tex]t[/tex] gives for non-zero [tex]t[/tex] a Clifford algebra [tex]Cliff(V, t(\cdot,\cdot))[/tex] that can be thought of as a deformation of the exterior algebra [tex]\Lambda^*(V)[/tex]. Thinking of the exterior algebra on [tex]V[/tex] of dimension n as the algebra of functions on n anticommuting coordinates, the Clifford algebra can be thought of as a “quantization” of this, taking functions (elements of [tex]\Lambda^*(V)[/tex]) to operators (elements of [tex]Cliff(V)[/tex], matrices in this case).

While [tex]\Lambda^*(V)[/tex] is a [tex]\mathbf Z[/tex] graded algebra, [tex]Cliff(V)=Cliff^{even}(V)\oplus Cliff^{odd}(V)[/tex] is only [tex]\mathbf Z_2[/tex]-graded, since the Clifford product does not preserve degree but can change it by two when multiplying generators. The Clifford algebra is filtered by a [tex]\mathbf Z[/tex] degree, taking [tex]F_p(Cliff(V))\subset Cliff(V)[/tex] to be the subspace of elements that can be written as sums of [tex]\leq p[/tex] generators. The exterior algebra is naturally isomorphic to the associated graded algebra for this filtration

[tex]\Lambda^p(V)\simeq F_p(Cliff(V))/F_{p-1}(Cliff(V))[/tex]

[tex]\Lambda^*(V)[/tex] and [tex]Cliff(V)[/tex] are isomorphic as vector spaces. One choice of such an isomorphism is given by composing the skew-symmetrization map

[tex]v_1\wedge v_2\wedge\cdots\wedge v_p=\frac{1}{p!}\sum_{s\in S_p}sgn(s)v_{s(1)}\otimes v_{s(2)}\otimes\cdots\otimes v_{s(p)}[/tex]

with the projection [tex]T^*(V)\rightarrow Cliff(V)[/tex]. Denoting this map by q, it is sometimes called the “quantization map”. Using an orthonormal basis [tex]e_i[/tex], [tex]q[/tex] acts as

[tex]q(e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_p})=e_{i_1}e_{i_2}\cdots e_{i_p}[/tex]

The inverse [tex]\sigma=q^{-1}:Cliff(V)\rightarrow \Lambda^*(V)[/tex] is sometime called the “symbol map”.

This identification as vector spaces is known as the “Chevalley identification”. Using it, one can think of the Clifford algebra as just an exterior algebra with a different product.

Clifford Modules and Spinors

Given a Clifford algebra, one would like to classify the modules over such an algebra, the Clifford modules. Such a module is given by a vector space [tex]M[/tex] and an algebra homomorphism

[tex]\pi: Cliff(V)\rightarrow End(M)[/tex]

To specify [tex]\pi[/tex], we just need to know it on generators, and see that it satisfies

[tex]\pi(v)\pi(w) +\pi(w)\pi(v)= -2(v,w)Id[/tex]

One such Clifford module is [tex]M=\Lambda^*V[/tex], with

[tex]\pi(v)\omega=v\wedge\omega – i_v\omega[/tex]

where [tex]i_v[/tex] is contraction by [tex]v[/tex]. This gives the inverse to the quantization map (the symbol map [tex]\sigma[/tex]) as

[tex]\sigma: a\in Cliff(V)\rightarrow \pi(a)1\in \Lambda^*(V)[/tex]

[tex]\Lambda^*(V)[/tex] is not an irreducible Clifford module, and we would like to decompose it into irreducibles. For [tex]dim_{\mathbf C}V =2n[/tex] even, there will be a single such irreducible [tex]S[/tex], of dimension [tex]2^n[/tex], and the module map [tex]\pi:Cliff(V)\rightarrow End(S)[/tex] is an isomorphism. In the rest of this posting we’ll stick to the this case, for the odd dimensional case see the references mentioned at the end.

To pick out an irreducible module [tex]S\subset \Lambda^*(V)[/tex], one can begin by choosing a linear map [tex]J:V\rightarrow V[/tex] such that [tex]J^2=-1[/tex] and [tex]J[/tex] is orthogonal [tex]((Jv,Jw)=(v,w))[/tex]. Then let [tex]W_J\subset V[/tex] be the subspace on which [tex]J[/tex] acts by [tex]+i[/tex], [tex]\overline W_J[/tex] be the subspace on which [tex]J[/tex] acts by [tex]-i[/tex]. Note that [tex]V[/tex] is a complex vector space, and now has two linear maps on it that square to [tex]-1[/tex], multiplication by [tex]i[/tex], and multiplication by [tex]J[/tex]. [tex]W_J[/tex] is an isotropic subspace of [tex]V[/tex], since

[tex](v_1,v_2)=(Jv_1,Jv_2)=(iv_1,iv_2)=-(v_1,v_2)[/tex]

for any [tex]v_1,v_2\in W_J[/tex]. We now have a decomposition [tex]V=W_j\oplus \overline W_J[/tex] into two isotropic subspaces. Since the bilinear form is zero on these subspaces, we get two subalgebras of the Clifford algebra, [tex]\Lambda^*(W_J)[/tex] and [tex]\Lambda^*(\overline{W_J})[/tex]. It turns out that one can choose [tex]S\simeq \Lambda^*(W_J)[/tex].

One can make this construction very explicit by picking a particular [tex]J[/tex], for instance the one that acts on the element of an orthonormal basis by [tex]Je_{2j-1}=e_{2j},\ Je_{2j}=-e_{2j-1}[/tex] for [tex]j=1,\cdots n[/tex]. Letting [tex]w_j=e_{2j-1}+ie_{2j}[/tex] we get a basis of [tex]W_J[/tex]. To get an explicit representation of [tex]S[/tex] as a [tex]Cliff(V)[/tex] module isomorphic to [tex]\Lambda^*(\mathbf C^n)[/tex], we will use the formalism of fermionic annihilation and creation operators. These are the operators on an exterior algebra one gets from wedging by or contracting by an orthonormal vector, operators [tex]a_i^+[/tex] and [tex]a_i[/tex] for [tex]i=1,\cdots,n[/tex] satisfying

[tex]\{a_i,a_j\}=\{a^+_i,a^+_j\}=0[/tex]

[tex]\{a_i,a^+_j\}=\delta_{ij}[/tex]

In terms of these operators on [tex]\Lambda^*(\mathbf C^n)[/tex], [tex]Cliff(n)[/tex] acts by

[tex]e_{2j-1}=a_j^+-a_j[/tex]

[tex]e_{2j}=-i(a^+_j+a_j)[/tex]

The Spin Representation

The group that preserves [tex](\cdot,\cdot)[/tex] is [tex]O(n,\mathbf C)[/tex], and its connected component of the identity [tex]SO(n,\mathbf C)[/tex] has compact real form [tex]SO(n)[/tex]. [tex]SO(n)[/tex] has a non-trivial double cover, the group [tex]Spin(n)[/tex]. One can construct [tex]Spin(n)[/tex] explicitly as invertible elements in [tex]Cliff(V)[/tex] for [tex]V=\mathbf R^n[/tex], and its Lie algebra using quadratic elements of [tex]Cliff(V)[/tex], with the Lie bracket given by the commutator in the Clifford algebra.

For the even case, a basis for the Cartan subalgebra of [tex]Lie\ Spin(2n)[/tex] is given by the elements

[tex]\frac{1}{2}e_{2j-1}e_{2j}[/tex]

These act on the spinor module [tex]S\simeq\Lambda^*(\mathbf C^n)[/tex] as

[tex]\frac{1}{2}e_{2j-1}e_{2j}=-i\frac{1}{2}(a_j^+-a_j)(a_j^++a_j)=i\frac{1}{2}[a_j,a_j^+][/tex]

with eigenvalues [tex](\pm\frac{1}{2},\cdots,\pm\frac{1}{2})[/tex]. [tex]S[/tex] is not irreducible as a representation of [tex]Spin(2n)[/tex], but decomposes as [tex]S=S^+\oplus S^-[/tex] into two irreducible half-spin representations, corresponding to the even and odd degree elements of [tex]\Lambda^*(\mathbf C^n)[/tex].

With a standard choice of positive roots, the highest weight of [tex]S^+[/tex] is

[tex](+\frac{1}{2},+\frac{1}{2}\cdots,+\frac{1}{2},+\frac{1}{2})[/tex]

and that of [tex]S^-[/tex] is

[tex](+\frac{1}{2},+\frac{1}{2}\cdots,+\frac{1}{2},-\frac{1}{2})[/tex]

Note that the spinor representation is not a representation of [tex]SO(2n)[/tex], just of [tex]Spin(2n)[/tex]. However, if one restricts to the [tex]U(n)\subset SO(2n)[/tex] preserving [tex]J[/tex], then the [tex]\Lambda^*(W_J)[/tex] are the fundamental representations of this [tex]U(n)[/tex]. These representations have weights that are 0 or 1, shifted by [tex]+\frac{1}{2}[/tex] from those of the spin representation. One can’t restrict from [tex]Spin(2n)[/tex] to [tex]U(n)[/tex], but one can restrict to [tex]\tilde U(n)[/tex], a double cover of [tex]U(n)[/tex]. On this double cover the notion of [tex]\Lambda^n(\mathbf C^n)^{\frac{1}{2}[/tex] makes sense and one has, as [tex]\tilde U(n)[/tex] representations

[tex]S\otimes \Lambda^n(\mathbf C^n)^{\frac{1}{2}}\simeq\Lambda^*(\mathbf C^n)[/tex]

So, projectively, the spin representation is just [tex]\Lambda^*(\mathbf C^n)[/tex], but the projective factor is a crucial part of the story.

The above has been a rather quick sketch of a long story. For more details, a good reference is the book Spin Geometry by Lawson and Michelsohn. Chapter 12 of Segal and Pressley’s Loop Groups contains a very geometric version of the above material, in a form suitable for generalization to infinite dimensions. My notes for my graduate class also have a bit more detail, see here.

In the next posting we’ll see what happens when one chooses [tex]V=\mathfrak g[/tex], and studies the Clifford algebra [tex]Cliff(\mathfrak g)[/tex]

The Casimir element discussed in the last posting of this series is a distinguished quadratic element of the center [tex]Z(\mathfrak g)=U(\mathfrak g)^\mathfrak g[/tex] (note, here [tex]\mathfrak g[/tex] is a complex semi-simple Lie algebra), but there are others, all of which will act as scalars on irreducible representations. The information about an irreducible representation V contained in these scalars can be packaged as the so-called infinitesimal character of [tex]V[/tex], a homomorphism

[tex]\chi_V: Z(\mathfrak g)\rightarrow \mathbf C[/tex]

defined by [tex]zv=\chi_V(z)v[/tex] for any [tex]z\in Z(\mathfrak g)[/tex], [tex]v\in V[/tex]. Just as was done for the Casimir, this can be computed by studying the action of [tex]Z(\mathfrak g)[/tex] on a highest-weight vector.

Note: this is not the same thing as the usual (or global) character of a representation, which is a conjugation-invariant function on the group [tex]G[/tex] with Lie algebra [tex]\mathfrak g[/tex], given by taking the trace of a matrix representation. For infinite dimensional representations [tex]V[/tex], the character is not a function on [tex]G[/tex], but a distribution [tex]\Theta_V[/tex]. The link between the global and infinitesimal characters is given by

[tex]\Theta_V(zf)=\chi_V(z)\Theta_V(f)[/tex]

i.e. [tex]\Theta_V[/tex] is a conjugation-invariant eigendistribution on [tex]G[/tex], with eigenvalues for the action of [tex]Z(\mathfrak g)[/tex] given by the infinitesimal character. Knowing the infinitesimal character gives differential equations for the global character.

The Harish-Chandra Homomorphism

The Poincare-Birkhoff-Witt theorem implies that for a simple complex Lie algebra [tex]\mathfrak g[/tex] one can use the decomposition (here the Cartan subalgebra is [tex]\mathfrak h=\mathfrak t_{\mathbf C}[/tex])

[tex]\mathfrak g=\mathfrak h \oplus \mathfrak n^+ \oplus \mathfrak n^-[/tex]

to decompose [tex]U(\mathfrak g)[/tex] as

[tex]U(\mathfrak g) =U(\mathfrak h) \oplus (U(\mathfrak g)\mathfrak n^+ + \mathfrak n^-U(\mathfrak g))[/tex]

and show that If [tex]z\in Z(\mathfrak g)[/tex], then the projection of z onto the second factor is in [tex]U(\mathfrak g)\mathfrak n^+\cap\mathfrak n^-U(\mathfrak g)[/tex]. This will give zero acting on a highest-weight vector. Defining [tex]\gamma^\prime: Z(\mathfrak g)\rightarrow Z(\mathfrak h)[/tex] to be the projection onto the first factor, the infinitesimal character can be computed by seeing how [tex]\gamma^\prime(z)[/tex] acts on a highest-weight vector.

Remarkably, it turns out that one gets something much simpler if one composes [tex]\gamma^\prime[/tex] with a translation operator

[tex]t_\rho: U(\mathfrak h)\rightarrow U(\mathfrak h)[/tex]

corresponding to the mysterious [tex]\rho\in \mathfrak h^*[/tex], half the sum of the positive roots. To define this, note that since [tex]\mathfrak h[/tex] is commutative, [tex]U(\mathfrak h)=S(\mathfrak h)=\mathbf C[\mathfrak h^*][/tex], the symmetric algebra on [tex]\mathfrak h[/tex], which is isomorphic to the polynomial algebra on [tex]\mathfrak h^*[/tex]. Then one can define

[tex]t_\rho (\phi(\lambda))=\phi(\lambda -\rho)[/tex]

where [tex]\phi\in \mathbf C[\mathfrak h^*][/tex] is a polynomial on [tex]\mathfrak h^*[/tex], and [tex]\lambda\in\mathfrak h^*[/tex].

The composition map

[tex]\gamma=t_\rho\circ\gamma^\prime: Z(\mathfrak g)\rightarrow U(\mathfrak h)=\mathbf C[\mathfrak h^*][/tex]

is a homomorphism, known as the Harish-Chandra homomorphism. One can show that the image is invariant under the action of the Weyl group, and the map is actually an isomorphism

[tex]\gamma: Z(\mathfrak g)\rightarrow \mathbf C[\mathfrak h^*]^W[/tex]

It turns out that the ring [tex]\mathbf C[\mathfrak h^*]^W[/tex] is generated by [tex]dim\ \mathfrak h[/tex] independent homogeneous polynomials. For [tex]\mathfrak g=\mathfrak{sl}(n,\mathbf C)[/tex] these are of degree [tex]2, 3,\cdots,n[/tex] (where the first is the Casimir).

To see how things work in the case of [tex]\mathfrak g=\mathfrak{sl}(2,\mathbf C)[/tex], where there is one generator, the Casimir [tex]\Omega[/tex], recall that

[tex]\Omega=\frac{1}{8}h^2 + \frac{1}{4}(ef +fe)=\frac{1}{8}h^2 + \frac{1}{4}(h +2fe)[/tex]

so one has
[tex]\gamma^\prime(\Omega)= \frac{1}{4}(h +\frac{1}{2}h^2)[/tex]

Here [tex]t_\rho(h)=h-1[/tex], so

[tex]\gamma(\Omega)=\frac{1}{4}((h-1)+\frac{1}{2}(h-1)^2)=\frac{1}{8}(h^2-1)[/tex]

which is invariant under the Weyl group action [tex]h\rightarrow -h[/tex].

Once one has the Harish-Chandra homomorphism [tex]\gamma[/tex], for each[tex] \lambda\in\mathfrak h^*[/tex] one has a homomorphism

[tex]\chi_{\lambda}: z\in Z(\mathfrak g)\rightarrow \chi_\lambda(z)=\gamma(z)(\lambda)\in \mathbf C[/tex]

and the infinitesimal character of an irreducible representation of highest weight [tex]\lambda[/tex] is [tex]\chi_{\lambda + \rho}[/tex].

The Casselman-Osborne Lemma

We have computed the infinitesimal character of a representation of highest weight [tex]\lambda[/tex] by looking at how [tex]Z(\mathfrak g)[/tex] acts on [tex]V^{\mathfrak n^+}=H^0(\mathfrak n^+,V)[/tex]. On [tex]V^{\mathfrak n^+}, z\in Z(\mathfrak g)[/tex] acts by

[tex]z\cdot v = \chi_V(z)v[/tex]

This space has weight [tex]\lambda[/tex], so [tex]U(\mathfrak h)=\mathbf C[\mathfrak h^*][/tex] acts by evaluation at [tex]\lambda[/tex]

[tex]\phi\cdot v=\phi(\lambda)v[/tex]

These two actions are related by the map [tex]\gamma^\prime: Z(\mathfrak g)\rightarrow U(\mathfrak h)[/tex] and we have

[tex]\chi_V(z)=(\gamma^\prime(z))(\lambda)=(\gamma(z))(\lambda + \rho)[/tex]

It turns out that one can consider the same question, but for the higher cohomology groups [tex]H^k(\mathfrak n^+,V)[/tex]. Here one again has an action of [tex]Z(\mathfrak g)[/tex] and an action of [tex]U(\mathfrak h)[/tex]. [tex]Z(\mathfrak g)[/tex] acts on k-cochains [tex]C^k(\mathfrak n^+,V)= Hom_{\mathbf C}(\Lambda^k\mathfrak n^+,V)[/tex] just by acting on [tex]V[/tex], and this action commutes with [tex]d[/tex] so is an action on cohomology. [tex]U(\mathfrak h)[/tex] acts simultaneously on [tex]\mathfrak n^+[/tex] and on [tex]V[/tex], again in a way that descends to cohomology. The content of the Casselman-Osborne lemma is that these two actions are again related in the same way by the Harish-Chandra homomorphism. If [tex]\mu[/tex] is a weight for the [tex]\mathfrak h[/tex] action on [tex]H^k(\mathfrak n^+,V)[/tex], then

[tex]\chi_V(z)=(\gamma^\prime(z))(\mu)=(\gamma(z))(\mu + \rho)[/tex]

Since [tex]\chi_V(z)=(\gamma(z))(\lambda + \rho)[/tex], one can use this equality to show that the weights occurring in [tex]H^k(\mathfrak n^+,V)[/tex] must satisfy

[tex](\mu +\rho)=w(\lambda + \rho)[/tex]

and thus

[tex]\mu=w(\lambda + \rho)-\rho[/tex]

for some element [tex]w\in W[/tex]. Non zero elements of [tex]H^k(\mathfrak n^+,V)[/tex] can be constructed with these weights, and the Casselman-Osborne lemma used to show that these are the only possible weights. This gives the computation of [tex]H^k(\mathfrak n^+,V)[/tex] as an [tex]\mathfrak h[/tex] – module referred to earlier in these notes, which is known as Kostant’s theorem (the algebraic proof was due to Kostant, an earlier one using geometry and sheaf cohomology was due to Bott).

For more details about this and a proof of the Casselman-Osborne lemma, see Knapp’s Lie Groups, Lie Algebras and Cohomology, where things are worked out for the case of [tex]\mathfrak g=\mathfrak{gl}(n,\mathbf C)[/tex] in chapter VI.

Generalizations

So far we have been considering the case of a Cartan subalgebra [tex]\mathfrak h\subset \mathfrak g[/tex], and its orthogonal complement with a choice of splitting into two conjugate subalgebras, [tex]\mathfrak n^+ \oplus \mathfrak n^-[/tex]. Equivalently, we have a choice of Borel subalgebra [tex]\mathfrak b\subset \mathfrak g[/tex], where [tex]\mathfrak b =\mathfrak h \oplus \mathfrak n^+[/tex]. At the group level, this corresponds to a choice of Borel subgroup [tex]B\subset G[/tex], with the space [tex]G/B[/tex] a complex projective variety known as a flag manifold. More generally, much of the same structure appears if we choose larger subgroups [tex]P \subset G[/tex] containing [tex]B[/tex] such that [tex]G/P[/tex] is a complex projective variety of lower dimension. In these cases [tex]Lie\ P=\mathfrak l \oplus \mathfrak u^+[/tex], with [tex]\mathfrak l[/tex] (the Levi subalgebra) a reductive algebra playing the role of the Cartan subalgebra, and [tex]\mathfrak u^+[/tex] playing the role of [tex]\mathfrak n^+[/tex].

In this more general setting, there is a generalization of the Harish-Chandra homomorphism, now taking [tex]Z(\mathfrak g)[/tex] to [tex]Z(\mathfrak l)[/tex]. This acts on the cohomology groups [tex]H^k(\mathfrak u^+,V)[/tex], with a generalization of the Casselman-Osborne lemma determining what representations of [tex]\mathfrak l[/tex] occur in this cohomology. The Dirac cohomology formalism to be discussed later generalizes this even more, to cases of a reductive subalgebra [tex]\mathfrak r[/tex] with orthogonal complement that cannot be given a complex structure and split into conjugate subalgebras. It also provides a compelling explanation for the continual appearance of [tex]\rho[/tex], as the highest weight of the spin representation.

For the case of [tex]G=SU(2)[/tex], it is well-known from the discussion of angular momentum in any quantum mechanics textbook that irreducible representations can be labeled either by j, the highest weight (here, highest eigenvalue of [tex]J_3[/tex] ), or by [tex]j(j+1)[/tex], the eigenvalue of [tex]\mathbf{J\cdot J}[/tex]. The first of these requires making a choice (the z-axis) and looking at a specific vector in the representation, the second doesn’t. It was a physicist (Hendrik Casimir), who first recognized the existence of an analog of [tex]\mathbf{J\cdot J}[/tex] for general semi-simple Lie algebras, and the important role that this plays in representation theory.

The Casimir Operator

Recall that for a semi-simple Lie algebra [tex]\mathfrak g[/tex] one has a non-degenerate, invariant, symmetric bi-linear form [tex](\cdot,\cdot)[/tex], the Killing form, given by

[tex](X,Y)= tr(ad(X)ad(Y))[/tex]

If one starts with [tex]\mathfrak g[/tex] the Lie algebra of a compact group, this bilinear form is defined on [tex]\mathfrak g_{\mathbf C}[/tex], and negative-definite on [tex]\mathfrak g[/tex]. For a simple Lie algebra, taking the trace in a different representation gives the same bilinear form up to a constant. As an example, for the case [tex]\mathfrak g_{\mathbf C}={\mathfrak{sl}(n,\mathbf C)}[/tex], one can show that

([tex]X,Y)=2n\ tr(XY)[/tex]

here taking the trace in the fundamental representation as [tex]n[/tex] by [tex]n[/tex] complex matrices.
One can use the Killing form to define a distinguished quadratic element [tex]\Omega[/tex] of [tex]U(\mathfrak g)[/tex], the Casimir element

[tex]\Omega=\sum_iX_iX^i[/tex]

where [tex]X_i[/tex] is an orthonormal basis with respect to the Killing form and [tex]X^i[/tex] is the dual basis. On any representation [tex]V[/tex], this gives a Casimir operator

[tex]\Omega_V=\sum_i\pi(X_i)\pi(X^i)[/tex]

Note that, taking the representation [tex]V[/tex] to be the space of functions [tex]C^\infty(G)[/tex] on the compact Lie group G, [tex]\Omega_V[/tex] is an invariant second-order differential operator, (minus) the Laplacian.

[tex]\Omega[/tex] is independent of the choice of basis, and belongs to [tex]U(\mathfrak g)^{\mathfrak g}[/tex], the subalgebra of [tex]U(\mathfrak g)[/tex] invariant under the adjoint action. It turns out that [tex]U(\mathfrak g)^{\mathfrak g}=Z(\mathfrak g)[/tex], the center of [tex]U(\mathfrak g)[/tex]. By Schur’s lemma, anything in the center [tex]Z(\mathfrak g)[/tex] must act on an irreducible representation by a scalar. One can compute the scalar for an irreducible representation [tex](\pi,V)[/tex] as follows:

Choose a basis [tex](H_i, X_{\alpha},X_{-\alpha})[/tex] of [tex]\mathfrak g_{\mathbf C}[/tex] with [tex]H_i[/tex] an orthonormal basis of the Cartan subalgebra [tex]\mathfrak t_{\mathbf C}[/tex], and [tex]X_{\pm\alpha}[/tex] elements of [tex]\mathfrak n^{\pm}[/tex] in the [tex]\pm\alpha[/tex] root-spaces of [tex]\mathfrak g_{\mathbf C}[/tex], orthonormal in the sense of satisfying

[tex](X_{\alpha},X_{-\alpha})=1[/tex]

Then one has the following expression for [tex]\Omega[/tex]:

[tex]\Omega=\sum_i H_i^2 + \sum_{+\ roots} (X_{\alpha} X_{-\alpha} +X_{-\alpha}X_{\alpha})[/tex]

To compute the scalar eigenvalue of this on an irreducible representation [tex](\pi,V_{\lambda})[/tex] of highest weight [tex]\lambda[/tex], one can just act on a highest weight vector [tex]v\in V^{\lambda}=V^{\mathfrak n^+}[/tex]. On this vector the raising operators [tex]\pi(X_{\alpha})[/tex] act trivially, and using the commutation relation

[tex][X_{\alpha},X_{-\alpha}]=H_{\alpha}[/tex]

([tex]H_{\alpha}[/tex] is the element of [tex]\mathfrak t_{\mathbf C} [/tex] satisfying [tex](H,H_{\alpha})=\alpha(H)[/tex]) one finds

[tex]\Omega=\sum_i H_i^2 + \sum_{+\ roots}H_{\alpha}= \sum_i H_i^2 +2H_{\rho}[/tex]

where [tex]\rho[/tex] is half the sum of the positive roots, a quantity which keeps appearing in this story. Acting on [tex]v\in V^{\lambda}[/tex] one finds

[tex]\Omega_{V_{\lambda}}v=(\sum_i\lambda(H_i)^2+2\lambda (H_{\rho}))v[/tex]

Using the inner-product [tex]< \cdot,\cdot>[/tex] induced on [tex]\mathfrak t^*[/tex] by the Killing form, this eigenvalue can be written as:

[tex]< \lambda,\lambda>+2< \lambda,\rho>=||\lambda+\rho||^2- ||\rho||^2[/tex]

In the special case [tex]\mathfrak g = \mathfrak {su}(2),\ \mathfrak g_{\mathbf C}=\mathfrak sl(2,\mathbf C)[/tex], there is just one positive root, and one can take

[tex]H_1=h=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\ X_{\alpha}=e=\begin{pmatrix}0&1\\0&0\end{pmatrix},\ X_{-\alpha}=f=\begin{pmatrix}0&0\\1&0\end{pmatrix}[/tex]

Computing the Killing form, one finds

[tex](h,h)=8,\ (e,f)=4[/tex]

and

[tex]\Omega=\frac{1}{8}h^2 + \frac{1}{4}(ef +fe)=\frac{1}{8}h^2 + \frac{1}{4}(h +2fe)[/tex]

On a highest weight vector [tex]\Omega[/tex] acts as

[tex]\Omega=\frac{1}{8}h^2 + \frac{1}{4}h=\frac{1}{8}h(h+2)=\frac{1}{2}(\frac{h}{2}(\frac{h}{2} +1))[/tex]

This is 1/2 times the physicist’s operator [tex]\mathbf{J\cdot J}[/tex], and in the irreducible representation [tex]V_n[/tex] of spin [tex] j=n/2[/tex], it acts with eigenvalue [tex]\frac{1}{2}j(j+1)[/tex].

In the next posting in this series I’ll discuss the Harish-Chandra homomorphism, and the question of how the Casimir acts not just on [tex]V^{\mathfrak n^+}=H^0(\mathfrak n^+,V)[/tex], but on all of the cohomology [tex]H^*(\mathfrak n^+,V)[/tex]. After that, taking note that the Casimir is in some sense a Laplacian, we’ll follow Dirac and introduce Clifford algebras and spinors in order to take its square root.

I should finish writing the next installment of the Notes on BRST series soon, but thought I’d post here about two pieces of BRST-related news, concerning the “B” and the “T”.