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Chevalley–Eilenberg differential
Definition
For define by
This fixes a zero-based sign convention. A hat means omission, and in the second sum the bracket is inserted as the first argument. Empty sums are zero. Thus for , . Alternation of the formula makes it factor through ; the next theorem proves that successive differentials compose to zero.
Depends on
Used by
- Whitehead vanishing does not apply to the nilpotent radical Counterexample
- Classical Lie-action BRST cochain complex Definition
- Central actions on nilradical cohomology factor through the Harish–Chandra projection Lemma
- Gan-Ginzburg and Skryabin inputs for the type-A pyramid good grading Lemma
- The Chevalley–Eilenberg Laplacian is scalar on weight components Lemma
- The extremal weight cochain of a Weyl element is closed and unique Lemma
- The nonequivariance defect of an infinitesimal moment map is a constant Lie-algebra two-cocycle Lemma
- Chevalley–Eilenberg cohomology computes Ext of the trivial module Proposition
- Degree-zero Lie algebra cohomology is the invariant subspace Proposition
- First cohomology is derivations modulo inner derivations Proposition
- The normalizer acts on Lie algebra cohomology Proposition
- Whitehead's second lemma removes the infinitesimal equivariance obstruction for semisimple actions Proposition
- Zeroth Lie algebra cohomology is invariants Proposition
- Second cohomology classifies abelian extensions Theorem
- The Chevalley–Eilenberg differential squares to zero Theorem
Dependency tree · two levels
4 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
- Weibel, Lie Algebra Homology and Cohomology, §7.7 (standard reference, not scraped)