In this posting I’ll work out some examples of Lie algebra cohomology, still for finite dimensional Lie algebras and representations.<\/p>\n
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<\/p>\n
[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]<\/p>\n
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. <\/p>\n
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.<\/p>\n
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<\/p>\n
[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]<\/p>\n
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<\/p>\n
[tex]H^*(\\mathfrak g, \\mathbf C)= H^*_{de Rham}(G)[\/tex]<\/p>\n
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
\n[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]<\/p>\n
where all the differentials are zero, so the cohomology is given by<\/p>\n
[tex]H^*(\\mathfrak g,\\mathbf C)=(\\Lambda^*(\\mathfrak g^*))^G=(\\Lambda^*(\\mathfrak g^*))^{\\mathfrak g}[\/tex]<\/p>\n
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<\/p>\n
[tex]H^*(\\mathbf R^n, \\mathbf C)=\\Lambda^*(\\mathbf C^n)[\/tex]<\/p>\n
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<\/p>\n
[tex]H^*(\\mathfrak g_{\\mathbf C},\\mathbf C)= H^*(\\mathfrak g,\\mathbf C)\\otimes \\mathbf C[\/tex]<\/p>\n
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<\/p>\n
[tex]H^*(\\mathfrak g_{\\mathbf C}, \\mathbf C) = (\\Lambda^*(\\mathfrak g_{\\mathbf C}))^{\\mathfrak g_\\mathbf C}[\/tex]<\/p>\n
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]H^*(\\mathfrak{sl}(n,\\mathbf C))=\\Lambda^*(\\omega_3, \\omega_5,\\cdots,\\omega_{2n+1})[\/tex]<\/p>\n the exterior algebra generated by the [tex]\\omega_{2i+1}[\/tex].<\/p>\n 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<\/p>\n [tex]H^*(\\mathfrak g, V)=H^*(\\mathfrak g, \\mathbf C)\\otimes V^{\\mathfrak g}[\/tex]<\/p>\n 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].<\/p>\n 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.<\/x_1><\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading