proposition 14.73 The extension datum is a $2$-cocycle

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3377 · p. 580

Rests on

depends_on definition 14.72 Central extension ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-35 ¶

Supports

depends_on definition 14.74 Coboundary and triviality ¶
- 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 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 ¶ ↺
depends_on definition 14.75 The classifying group ¶ ↺
depends_on example 14.79 The Heisenberg algebra ¶ ↺
depends_on proposition 14.76 $H^{2}$ classifies the central extensions ¶ ↺
depends_on proposition 25.14 The Galilei cocycle is not a coboundary ¶
- depends_on corollary 15.36 Bargmann in $3{+}1$, extended Bargmann in $2{+}1$ ¶
  - depends_on example 15.64 The whole chapter at $D=4$ ¶
  - depends_on remark 15.59 The central charge is what makes the form work ¶
- depends_on example 14.80 The Galilei algebra ¶ ↺
depends_on theorem 15.34 The truncation tower is a tower of central extensions ¶
- depends_on corollary 15.36 Bargmann in $3{+}1$, extended Bargmann in $2{+}1$ ¶ ↺
- depends_on corollary 15.35 Where the central charges are ¶
  - depends_on corollary 15.36 Bargmann in $3{+}1$, extended Bargmann in $2{+}1$ ¶ ↺
  - depends_on example 15.64 The whole chapter at $D=4$ ¶ ↺

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`	→	Central extension	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3388
`depends_on`	←	Coboundary and triviality	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`	←	$H^{2}$ classifies the central extensions	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3441
`depends_on`	←	The Galilei cocycle is not a coboundary	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:608
`depends_on`	←	The truncation tower is a tower of central extensions	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1166
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-35	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3391