Classical Lie-action BRST cochain complex

Let $k$ be a characteristic-zero field, let $\mathfrak{g}$ be a Lie algebra over $k$, and let $A$ be a unital commutative $k$-algebra. A classical Lie-action BRST input is a Lie algebra homomorphism $\rho :\mathfrak{g}\to {\mathrm{End}}_k(A)$ whose values are derivations: for $x\in \mathfrak{g}$ and $a,b\in A$, $\rho (x)(ab)=\rho (x)(a)b+a\rho (x)(b),\rho (x)(1)=0.$ In particular, $A$ is a $\mathfrak{g}$-module, so its Chevalley–Eilenberg cochains are $C_{\mathrm{B}\mathrm{R}\mathrm{S}\mathrm{T}}^n(\mathfrak{g},A)={\mathrm{Hom}}_k({\mathrm{\Lambda }}^n\mathfrak{g},A)(n\ge 0),$ and are zero for $n<0$. Define the degree-$+1$ classical Lie-action BRST operator by the fixed zero-based formula $\begin{array}{rl}{\displaystyle (d_{\mathrm{B}\mathrm{R}\mathrm{S}\mathrm{T}}f)(x_0,\dots ,x_n)=}& {\displaystyle \sum _{i=0}^n(-1{)}^i\rho (x_i)f(x_0,\dots ,\widehat{x_i},\dots ,x_n)}\\ {\displaystyle }& {\displaystyle +\sum _{0\le i<j\le n}(-1{)}^{i+j}f([x_i,x_j],x_0,\dots ,\widehat{x_i},\dots ,\widehat{x_j},\dots ,x_n)\mathrm{.}}\end{array}$ A hat denotes omission; the bracket is inserted first. This is exactly the Chevalley–Eilenberg differential for the stated representation, with the BRST name reserved here for this classical Lie-action model.

Its square-zero property and the resulting quotient cohomology are established in The classical Lie-action BRST differential squares to zero. This model does not claim a Koszul–Tate resolution, a BFV charge for arbitrary first-class constraints, reducible or open gauge algebras, field-dependent structure functions, or quantum BRST cohomology.