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.
