definition 14.74 Coboundary and triviality

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3403 · p. 580

ground object -- no derivation owed

Rests on

depends_on proposition 14.73 The extension datum is a $2$-cocycle ¶
- depends_on definition 14.72 Central extension ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-35 ¶

Supports

depends_on definition 14.75 The classifying group ¶
- depends_on definition A.397 Cochains and the differential ¶
  - depends_on lemma A.398 The differential squares to zero in the degrees used ¶
  - depends_on theorem A.403 Vanishing when the Casimir is invertible ¶
    - depends_on theorem A.410 Whitehead's first and second lemmas ¶
      - depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶
- depends_on example 14.80 The Galilei algebra ¶
- depends_on proposition 14.76 $H^{2}$ classifies the central extensions ¶
  - depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶ ↺
  - depends_on proposition A.408 The two groups vanish for the trivial module ¶
  - depends_on theorem A.414 $H^{2}$ of the Galilei algebra ¶
  - depends_on theorem A.422 The Virasoro extension ¶
- depends_on proposition 14.77 Semisimple algebras admit no nontrivial extension ¶
  - depends_on proposition A.428 Uniqueness in degree zero ¶
  - depends_on proposition 14.84 Properties of a contraction ¶
- depends_on theorem A.414 $H^{2}$ of the Galilei algebra ¶ ↺
- depends_on theorem A.422 The Virasoro extension ¶ ↺
depends_on example 14.79 The Heisenberg algebra ¶
depends_on proposition A.419 The coboundaries ¶
depends_on proposition 14.76 $H^{2}$ classifies the central extensions ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	The extension datum is a $2$-cocycle	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3414
`depends_on`	←	The classifying group	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3425
`depends_on`	←	The Heisenberg algebra	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3535
`depends_on`	←	The coboundaries	declared	appendices/A-long-proofs.tex:20538
`depends_on`	←	$H^{2}$ classifies the central extensions	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3441