proposition 14.76 $H^{2}$ classifies the central extensions
open in the book ·
parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3431
· p. 580
Rests on
- depends_on definition 14.72 Central extension ¶
-
depends_on
definition 14.74
Coboundary and triviality
¶
-
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 ¶
-
depends_on
proposition 14.73
The extension datum is a $2$-cocycle
¶
-
depends_on
definition 14.75
The classifying group
¶
- depends_on definition 14.74 Coboundary and triviality ¶ ↺
- depends_on proposition 14.73 The extension datum is a $2$-cocycle ¶ ↺
- depends_on proposition 14.73 The extension datum is a $2$-cocycle ¶ ↺
- proves proof ch:12-lie-groups-fibre-bundles@proof-36 ¶
Supports
- 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 ¶
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:3441 |
depends_on |
→ | Coboundary and triviality | declared | parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3441 |
depends_on |
→ | The classifying group | declared | parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3441 |
depends_on |
→ | The extension datum is a $2$-cocycle | declared | parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3441 |
depends_on |
← | Semisimple algebras admit no nontrivial extension | declared | appendices/A-long-proofs.tex:20107 |
depends_on |
← | The two groups vanish for the trivial module | declared | appendices/A-long-proofs.tex:19902 |
depends_on |
← | $H^{2}$ of the Galilei algebra | declared | appendices/A-long-proofs.tex:20243 |
depends_on |
← | The Virasoro extension | declared | appendices/A-long-proofs.tex:20724 |
proves |
← | ch:12-lie-groups-fibre-bundles@proof-36 | declared | parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3444 |