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The classical Lie-action BRST differential squares to zero

Statement

For every classical Lie-action BRST input of Classical Lie-action BRST cochain complex and every $n \geq 0$ , $d_{B R S T}^{n + 1} d_{B R S T}^{n} = 0.$ Consequently, the quotient $H_{B R S T}^{n} (g, A) : = \ker (d_{B R S T}^{n}) / im (d_{B R S T}^{n - 1})$ is a well-defined vector space, with $C_{B R S T}^{- 1} = 0$ .

Facts & Assumptions

Given: A characteristic-zero field $k$ , a Lie algebra $g$ , a unital commutative $k$ -algebra $A$ , and a Lie algebra representation of $g$ on $A$ by derivations.

[F1]

Classical Lie-action BRST cochain complex: the BRST cochains are ${Hom}_{k} (Λ^{n} g, A)$ and the operator is the zero-based two-sum Chevalley–Eilenberg formula for the representation $ρ$ .

[F2]

The Chevalley–Eilenberg differential squares to zero: for a Lie algebra over a field and a representation on a module, the displayed Chevalley–Eilenberg differential satisfies $d^{n + 1} d^{n} = 0$ for every $n$ .

Proof

1.1givenF1F2

The assumptions in [F1] make $A$ a $g$ -module because $ρ$ is a Lie algebra homomorphism, so $[ρ (x), ρ (y)] = ρ ([x, y])$ . The cochain spaces and both signed sums in $d_{B R S T}$ are term-by-term those in [F2].

2.1givenF1F2step 1.1

Therefore [F2] applies to these exact inputs and gives $d_{B R S T}^{n + 1} d_{B R S T}^{n} = 0$ for every $n \geq 0$ .

3.1givenstep 2.1algebra∎

If $z \in im (d_{B R S T}^{n - 1})$ , write $z = d_{B R S T}^{n - 1} w$ . Step 2.1 gives $d_{B R S T}^{n} z = d_{B R S T}^{n} d_{B R S T}^{n - 1} w = 0$ , so the image is contained in the kernel and the displayed quotient is well-defined.

Depends on

Used by

BRST cohomology for the dilation action Example

Dependency tree · two levels

6 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)