Lie algebra cohomology ($H^k(\mathfrak{g},V)$)

The Chevalley-Eilenberg complex consists of alternating maps from ${\mathfrak{g}}^k$ to a Lie algebra representation $V$, with differential built from the bracket and module action. For trivial real coefficients, a two-cocycle is an alternating bilinear form $\kappa$ satisfying $\kappa ([X,Y],Z)+\kappa ([Y,Z],X)+\kappa ([Z,X],Y)=0$. Coboundaries have the form $b([X,Y])$, up to the differential's sign convention. Their quotient is $H^2(\mathfrak{g},\mathbb{R})$ and classifies central extensions of a Lie algebra.

For trivial coefficients in a field $k$, a Lie algebra two-coboundary is a bilinear map $\beta (x,y)=t([x,y])$ for a linear functional $t$ in the dual space ${\mathfrak{g}}^{\ast }$. Some conventions put a minus sign in this formula; the resulting vector subspace $B^2(\mathfrak{g},k)$ is the same. Each such map is a Lie algebra two-cocycle: it is an alternating bilinear form, and its cyclic expression is $t([[x,y],z]+[[y,z],x]+[[z,x],y])=0$ by the Jacobi identity. Replacing a linear section of a central extension of a Lie algebra by $s^{\prime }(x)=s(x)+t(x)z$ changes its Lie algebra two-cocycle by $-\beta$. Thus $H^2(\mathfrak{g},k)=Z^2(\mathfrak{g},k)/B^2(\mathfrak{g},k)$ in Lie algebra cohomology records exactly the obstruction to a section that is a Lie algebra homomorphism.

For trivial real coefficients, a Lie algebra two-cocycle is an alternating bilinear form satisfying $\kappa ([X,Y],Z)+\kappa ([Y,Z],X)+\kappa ([Z,X],Y)=0$. Adding a form $b([X,Y])$ does not change its Lie algebra cohomology class. The bracket $[(X,a),(Y,b)]=([X,Y],\kappa (X,Y))$ defines a central extension of a Lie algebra by $\mathbb{R}$ precisely when this cocycle identity holds. The constant discrepancy in a moment-map equivariance obstruction is such a cocycle.