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A nonrepresentation does not give a BRST differential

Statement refuted

The two-sum Chevalley–Eilenberg formula squares to zero for every linear map $ρ : g \to {End}_{k} (A)$ , even when $ρ$ is not a Lie algebra representation.

Facts & Assumptions

Given: The two-sum formula from Classical Lie-action BRST cochain complex, evaluated algebraically even though the proposed action below is not a representation.

[F1]

Classical Lie-action BRST cochain complex: on a $0$ -cochain $f \in A$ , $(d f) (z) = ρ (z) f$ ; on a $1$ -cochain $g$ , $(d g) (z_{0}, z_{1}) = ρ (z_{0}) g (z_{1}) - ρ (z_{1}) g (z_{0}) - g ([z_{0}, z_{1}])$ .

Counterexample

Take $k = C$ , the two-dimensional abelian Lie algebra $g = C u \oplus C v$ , and the commutative algebra $A = C [x, y]$ . Let $ρ (u) = \partial_{x}, ρ (v) = x \partial_{y} .$ Both maps are derivations, but their commutator is not the zero map.

1.1givenF1algebra

The maps $\partial_{x}$ and $x \partial_{y}$ are derivations by the polynomial product rule, and $[\partial_{x}, x \partial_{y}] f = \partial_{x} (x \partial_{y} f) - x \partial_{y} (\partial_{x} f) = \partial_{y} f$ . Since $[u, v] = 0$ but $\partial_{y} \neq 0$ , $ρ$ is not a Lie algebra homomorphism and is not an admissible input in Classical Lie-action BRST cochain complex.

2.1givenF1step 1.1algebra∎

For the $0$ -cochain $f = y$ , [F1] gives $(d f) (u) = 0$ and $(d f) (v) = x$ . Applying [F1] again gives $d^{2} f (u, v) = ρ (u) (x) - ρ (v) (0) - 0 = \partial_{x} x = 1$ . Thus the formula fails to square to zero when the representation hypothesis is dropped, refuting the statement.

Depends on

Classical Lie-action BRST cochain complex

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, An Introduction to Homological Algebra, §7.7, Lie Algebra Homology and Cohomology (standard reference, not scraped)