Manuscript XIII-D@1 · XIII-D.tex · lemma · bab69024

Lemma: Necessary cocycle

D3: the necessary cocycle and its consequence · lines 785–815 · proof included or linked

XIII-D@1#lem:cocycle, XIII-D@1#eq:cocycle

For a representation satisfying (eq:representation), an equation

δ ˉ_{ξ} G ˉ = - B ˉ_{ξ}

for one fixed functional $G$ requires

δ ˉ_{ξ} B ˉ_{ζ} - δ ˉ_{ζ} B ˉ_{ξ} = B ˉ_{[ξ, ζ]} .

For a direct slot proposal, take $G = O$ . For a compensating-functional proposal, take $G = F$ .

Proof. Apply $δ ˉ_{ξ}$ to the equation with parameter $ζ$ , subtract the reversed equation, and use (eq:representation). The result is

- δ ˉ_{ξ} B ˉ_{ζ} + δ ˉ_{ζ} B ˉ_{ξ} = δ ˉ_{[ξ, ζ]} G ˉ = - B ˉ_{[ξ, ζ]} .

This is the degree-one Chevalley–Eilenberg condition with the specified representation convention CE.

Displayed formulas in this passage

XIII-D@1#eq:cocycle · line 793
$δ ˉ_{ξ} B ˉ_{ζ} - δ ˉ_{ζ} B ˉ_{ξ} = B ˉ_{[ξ, ζ]} .$
unlabeled · line 789
$δ ˉ_{ξ} G ˉ = - B ˉ_{ξ}$
unlabeled · line 807
$- δ ˉ_{ξ} B ˉ_{ζ} + δ ˉ_{ζ} B ˉ_{ξ} = δ ˉ_{[ξ, ζ]} G ˉ = - B ˉ_{[ξ, ζ]} .$