Not Even Wrong

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.

In this posting I’ll work out some examples of Lie algebra cohomology, still for finite dimensional Lie algebras and representations.

If [tex]G[/tex] is a compact, connected Lie group, it can be thought of as a compact manifold, and as such one can define its de Rham cohomology [tex]H^*_{deRham}(G)[/tex] as the cohomology of the complex

[tex]0\longrightarrow \Omega^0(G)\stackrel{d}\longrightarrow \Omega^1(G)\stackrel{d}\longrightarrow\cdots\stackrel{d}\longrightarrow\Omega^{dim\ G}(G)\longrightarrow 0[/tex]

where [tex]\Omega^i(G)[/tex] are the differential i-forms on [tex]G[/tex] (note, we’ll use complex-valued forms), and [tex]d[/tex] is the deRham differential.

For a compact group, one has a bi-invariant Haar measure [tex]\int_G[/tex], and can use this to “average” over an action of the group on a space. For a representation [tex](\pi, V)[/tex], we get a projection operator [tex]\int_g \Pi (g)[/tex] onto the invariant subspace [tex]V^G[/tex]. This projection operator gives explicitly the invariants functor on [tex]\mathcal C_{\mathfrak g}[/tex]. It is an exact functor, taking exact sequences to exact sequences.

The differential forms [tex]\Omega^*(G)[/tex] give a representation of [tex]G[/tex] in two ways, taking the induced action on forms by pullback, using either left or right translation on the group. If [tex](\Pi(g), \Omega^*(G))[/tex] is the representation by left translations, we can use this to apply our “averaging over [tex]G[/tex]” projection operator to the de Rham complex. This action commutes with the de Rham differential, so we get a sub-complex of left-invariant forms

[tex]0\longrightarrow \Omega^0(G)^G\stackrel{d}\longrightarrow \Omega^1(G)^G\stackrel{d}\longrightarrow\cdots\stackrel{d}\longrightarrow\Omega^{dim\ G}(G)^G\longrightarrow 0[/tex]

Since elements of the Lie algebra [tex]\mathfrak g[/tex] are precisely left-invariant 1-forms, it turns out that this complex is nothing but the Chevalley-Eilenberg complex considered last time to represent Lie algebra cohomology, for the case of the trivial representation. This means we have [tex]C^*(\mathfrak g, \mathbf R)= \Lambda^*(\mathfrak g^*)=\Omega^*(G)^G[/tex], and the differentials coincide. So, what we have shown is that

[tex]H^*(\mathfrak g, \mathbf C)= H^*_{de Rham}(G)[/tex]

If one knows the cohomology of [tex]G[/tex], the Lie algebra cohomology is thus known, but this identity is normally used in the other direction, to find the cohomology of [tex]G[/tex] from that of the Lie algebra. To compute the Lie-algebra cohomology, we can exploit the right-action of G on the group, averaging over the induced action on the left-invariant forms [tex]\Lambda^*(\mathfrak g)[/tex], which again commutes with the differential. We end up with a complex
[tex]0\longrightarrow (\Lambda^0(\mathfrak g^*))^G \longrightarrow (\Lambda^1(\mathfrak g^*))^G\longrightarrow\cdots\longrightarrow (\Lambda^{\dim\ \mathfrak g}(\mathfrak g^*))^G\longrightarrow 0[/tex]

where all the differentials are zero, so the cohomology is given by

[tex]H^*(\mathfrak g,\mathbf C)=(\Lambda^*(\mathfrak g^*))^G=(\Lambda^*(\mathfrak g^*))^{\mathfrak g}[/tex]

the adjoint-invariant pieces of the exterior algebra on [tex]\mathfrak g^*[/tex]. Finding the cohomology has now been turned into a purely algebraic problem in invariant theory. For [tex]G=U(1)[/tex], [tex]\mathfrak g=\mathbf R[/tex], and we have shown that [tex]H^*(\mathbf R, \mathbf C)=\Lambda^*(\mathbf C)[/tex], this is [tex]\mathbf C[/tex] in degrees 0, and 1, as expected for the de Rham cohomology of the circle [tex]U(1)=S^1[/tex]. For [tex]G=U(1)^n[/tex], we get

[tex]H^*(\mathbf R^n, \mathbf C)=\Lambda^*(\mathbf C^n)[/tex]

Note that complexifying the Lie algebra and working with [tex]\mathfrak g_{\mathbf C}=\mathfrak g\otimes \mathbf C[/tex] commutes with taking cohomology, so we get

[tex]H^*(\mathfrak g_{\mathbf C},\mathbf C)= H^*(\mathfrak g,\mathbf C)\otimes \mathbf C[/tex]

Complexifying the Lie algebra of a compact semi-simple Lie group gives a complex semi-simple Lie algebra, and we have now computed the cohomology of these as

[tex]H^*(\mathfrak g_{\mathbf C}, \mathbf C) = (\Lambda^*(\mathfrak g_{\mathbf C}))^{\mathfrak g_\mathbf C}[/tex]

Besides [tex]H^0[/tex], one always gets a non-trivial [tex]H^3[/tex], since one can use the Killing form [tex]< \cdot,\cdot>[/tex] to produce an adjoint-invariant 3-form [tex]\omega_3(X_1,X_2,X_3)=[/tex]. For [tex]G=SU(n)[/tex], [tex]\mathfrak g_{\mathbf C}=\mathfrak{sl}(n,\mathbf C})[/tex], and one gets non-trivial cohomology classes [tex]\omega_{2i+1}[/tex] for [tex]i=1,2,\cdots n[/tex], such that

[tex]H^*(\mathfrak{sl}(n,\mathbf C))=\Lambda^*(\omega_3, \omega_5,\cdots,\omega_{2n+1})[/tex]

the exterior algebra generated by the [tex]\omega_{2i+1}[/tex].

To compute Lie algebra cohomology [tex]H^*(\mathfrak g, V)[/tex] with coefficients in a representation [tex]V[/tex], we can go through the same procedure as above, starting with differential forms on [tex]G[/tex] taking values in [tex]V[/tex], or we can just use exactness of the averaging functor that takes [tex]V[/tex] to [tex]V^G[/tex]. Either way, we end up with the result

[tex]H^*(\mathfrak g, V)=H^*(\mathfrak g, \mathbf C)\otimes V^{\mathfrak g}[/tex]

The [tex]H^0[/tex] piece of this is just the [tex]V^{\mathfrak g}[/tex] that we want when we are doing BRST, but we also get quite a bit else: [tex]dim\ V^{\mathfrak g}[/tex] copies of the higher degree pieces of the Lie algebra cohomology [tex]H^*(\mathfrak g, \mathbf C)[/tex]. The Lie algebra cohomology here is quite non-trivial, but doesn’t interact in a non-trivial way with the process of identifying the invariants [tex]V^{\mathfrak g}[/tex] in [tex]V[/tex].

In the next posting I’ll turn to an example where Lie algebra cohomology interacts in a much more interesting way with the representation theory, this will be the highest-weight theory of representations, in a cohomological interpretation first studied by Bott and Kostant.

The Invariants Functor

The last posting discussed one of the simplest incarnations of BRST cohomology, in a formalism familiar to physicists. This fits into a much more abstract mathematical context, and that’s what we’ll turn to now.

Given a Lie algebra $\mathfrak g$, we’ll consider Lie algebra representations as modules over $U(\mathfrak g)$. Such modules form a category $\mathcal C_{\mathfrak g}$: what is interesting is not just the objects of the category (the equivalence classes of modules), but also the morphisms between the objects. For two representations $V_1$ and $V_2$ the set of morphisms between them is a linear space denoted $Hom_{U(\mathfrak g)}(V_1,V_2)$. This is just the set of linear maps from $V_1$ to $V_2$ that commute with the action of $\mathfrak g$:

$$Hom_{U(\mathfrak g)}(V_1,V_2)=\{\phi\in Hom_{\mathbf C}(V_1,V_2): \pi(X)\phi=\phi\pi(X)\ \forall X\in \mathfrak g\}$$

Another conventional name for this is the space of intertwining operators between the two representations.

For any representation $V$, its $\mathfrak g$-invariant subspace $V^{\mathfrak g}$ can be identified with the space $Hom_{U(\mathfrak g)}(\mathbf C, V)$, where here $\mathbf C$ is the trivial one-dimensional representation. Having a way to pick out the invariant piece of a representation also allows one to solve the more general problem of picking out the subspace that transforms like a specific irreducible $W$: just find the invariant subspace of $V\otimes W^*$.

The map $V\rightarrow V^{\mathfrak g}$ that takes a representation to its $\mathfrak g$-invariant subspace is a functor: it takes the category $\mathcal C_{\mathfrak g}$ to $\mathcal C_{\mathbf C}$, the category of vector spaces and linear maps ($\mathbf C$ – modules and $\mathbf C$ – homomorphisms). If, instead of taking

$$V\rightarrow V^{\mathfrak g}$$

one takes

$$V\rightarrow V^{\mathfrak h}$$

where [tex]\mathfrak h[/tex] is a Lie subalgebra of [tex]\mathfrak g[/tex], one again gets a functor. If [tex]\mathfrak h[/tex] is an ideal in [tex]\mathfrak g[/tex] (so that [tex]\mathfrak g/\mathfrak h[/tex] is a Lie algebra), then this functor takes [tex]\mathcal C_{\mathfrak g}[/tex] to [tex]\mathcal C_{\mathfrak g/\mathfrak h}[/tex]. This is a simple version of the situation of interest in the case of gauge theory: if [tex]V[/tex] is a state space with [tex]\mathfrak h[/tex] acting as a gauge symmetry, then [tex]V^{\mathfrak h}[/tex] will be the physical subspace, carrying an action of the algebra of operators [tex]U(\mathfrak g/\mathfrak h)[/tex].

Some Homological Algebra

It turns out that when one has a category of modules like [tex]\mathcal C_{\mathfrak g}[/tex], these can usefully be studied by considering complexes of modules, and this is the subject of homological algebra. A complex of modules is a sequence of modules and homomorphisms

$$\cdots\stackrel{\partial}\longrightarrow U\stackrel{\partial}\longrightarrow V \stackrel{\partial}\longrightarrow W\stackrel{\partial}\longrightarrow\cdots$$

such that [tex]\partial\circ\partial =0[/tex]. If the complex satisfies [tex]im\ \partial=ker\ \partial[/tex] at each module, the complex is said to be an “exact complex”.

To motivate the notion of exact complex, note that

$$0\longrightarrow V_0\longrightarrow V \longrightarrow 0$$

is exact iff [tex]V_0[/tex] is isomorphic to [tex]V[/tex], and an exact sequence

$$0\longrightarrow V_1 \longrightarrow V_0 \longrightarrow V \longrightarrow 0$$

represents the module [tex]V[/tex] as the quotient [tex]V_0/V_1[/tex]. Using longer complexes, one gets the notion of a resolution of a module [tex]V[/tex] by a sequence of n modules [tex]V_i[/tex]. This is an exact complex

$$0\longrightarrow V_n \longrightarrow\cdots\longrightarrow V_1 \longrightarrow V_0\longrightarrow V\longrightarrow 0$$

The deviation of a sequence from being exact is measured by its homology $H^*=\frac{ker\ \partial}{im\ \partial}$. Note that if one deletes [tex]V[/tex] from its resolution, the sequence

$$0\longrightarrow V_n \longrightarrow\cdots\longrightarrow V_1 \longrightarrow V_0\longrightarrow 0$$

is exact except at [tex]V_0[/tex]. Indexing the homology in the obvious way, one has [tex]H^i =0[/tex] for [tex]i>0[/tex], and [tex]H^0=V[/tex]. A sequence like this whose only homology is [tex]V[/tex] at [tex]H^0[/tex] is another manifestation of a resolution of [tex]V[/tex].

The reason this construction is useful is that, for many purposes, it allows us to replace a module whose structure we may not understand by a sequence of modules whose structure we do understand. In particular, we can replace a [tex]U(\mathfrak g)[/tex] module [tex]V[/tex] by a sequence of free modules, i.e. modules that are just sums of copies of [tex]U(\mathfrak g)[/tex] itself. This is called a free resolution, and more generally one can work with projective modules (direct summands of free modules).

A functor that takes exact complexes to exact complexes is called an exact functor. Homological invariants of modules come about in cases where one has a functor on a category of modules that is not exact. Applying such a functor to a free or projective resolution gives the homological invariants.

The Koszul Resolution and Lie Algebra Cohomology

There are many possible choices of a free resolution of a module. For the case of [tex]U(\mathfrak g)[/tex] modules, one convenient choice is known as the Koszul (or Chevalley-Eilenberg) resolution. To construct a resolution of the trivial module [tex]\mathbf C[/tex], one uses the exterior algebra on [tex]\mathfrak g[/tex] to make free modules

$$Y_k=U(\mathfrak g)\otimes_{\mathbf C}\Lambda^k(\mathfrak g)$$

and get a resolution of [tex]\mathbf C[/tex]

$$0\longrightarrow Y_{dim\ \mathfrak g}\stackrel{\partial_{dim\ \mathfrak g -1}}\longrightarrow\cdots\stackrel{\partial_1}\longrightarrow Y_1\stackrel{\partial_0}\longrightarrow Y_0\stackrel{\epsilon}\longrightarrow \mathbf C\longrightarrow 0$$

The maps are given by
$$\epsilon : u\in Y_0=U(\mathfrak g) \rightarrow \epsilon (u) = const.\ term\ of\ u$$

and
$$\partial_{k-1} (u\otimes X_1\wedge\cdots\wedge X_k)=$$
$$\sum_{i=1}^k(-1)^{i+1}(uX_i\otimes X_1\wedge\cdots\wedge\hat X_i\wedge\cdots \wedge X_k)$$
$$+\sum_{i<j} (-1)^{i+j}(u\otimes[X_i,X_j]\wedge X_1\wedge\cdots\wedge \hat X_i\wedge\cdots\wedge \hat X_j\wedge\cdots\wedge X_k)$$

To get Lie algebra cohomology, we apply the invariants functor

$$V\longrightarrow V^{\mathfrak g}=Hom_{U(\mathfrak g)}(\mathbf C, V)$$

replacing the trivial representation by its Koszul resolution. This gives us a complex with terms

$$C^k(\mathfrak g, V)=Hom_{U(\mathfrak g)}(Y_k,V)= Hom_{U(\mathfrak g)}(U(\mathfrak g)\otimes \Lambda^k(\mathfrak g),V)$$
$$=Hom_{U(\mathfrak g)}(U(\mathfrak g),Hom_{\mathbf C}(\Lambda^k(\mathfrak g),V))$$
$$=Hom_{\mathbf C}(\Lambda^k(\mathfrak g),V) =V\otimes\Lambda^k(\mathfrak g^*)$$

and induced maps $d_i$

$$0\longrightarrow C^0(\mathfrak g, V)\stackrel{d_0}\longrightarrow C^1(\mathfrak g, V)\cdots\stackrel{d_{dim\ \mathfrak g -1}}\longrightarrow C^{dim\ \mathfrak g}(\mathfrak g, V)\longrightarrow 0$$

The Lie algebra cohomology $H^*(\mathfrak g, V)$ is just the cohomology of this complex, i.e.

$$H^i(\mathfrak g, V)=\frac{ker\ d_i}{im\ d_{i-1}}|_{C^i(\mathfrak g, V)}$$

This is exactly the same definition as that of the BRST cohomology defined in physicist’s formalism in the last posting with $\mathcal H =C^*(\mathfrak g, V)$.

One has $H^0(\mathfrak g, V)=V^{\mathfrak g}$ and so gets the $\mathfrak g$-invariants as expected, but in general the cohomology will be non-zero also in other degrees.

This is all rather abstract, so in the next posting some examples will be worked out, as well as the relationship of all this to the de Rham cohomology of the group. Anthony Knapp’s book Lie Groups, Lie Algebras, and Cohomology is an excellent reference for details on Lie algebra cohomology.