theorem 15.34 The truncation tower is a tower of central extensions

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1129 · p. 613

Rests on

depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶
depends_on definition 15.20 Resonant subalgebra ¶
- depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶ ↺
- depends_on definition 15.14 Truncated polynomial algebra and Takiff algebra ¶
  - depends_on definition 15.3 Tensor product Lie algebra ¶
depends_on definition 14.72 Central extension ¶
depends_on proposition 15.6 Functoriality ¶
- depends_on definition 15.3 Tensor product Lie algebra ¶ ↺
- depends_on theorem 15.4 The construction produces a Lie algebra ¶
  - depends_on definition 15.3 Tensor product Lie algebra ¶ ↺
  - proves proof ch:13-lie-algebra-expansions@proof-1 ¶
- proves proof ch:13-lie-algebra-expansions@proof-2 ¶
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 ¶
proves proof ch:13-lie-algebra-expansions@proof-20 ¶

Supports

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 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`	→	$\Z_{2}$-grading, symmetric coset	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1166
`depends_on`	→	Resonant subalgebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1166
`depends_on`	→	Central extension	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1166
`depends_on`	→	Functoriality	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1166
`depends_on`	→	The extension datum is a $2$-cocycle	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1166
`depends_on`	←	Bargmann in $3{+}1$, extended Bargmann in $2{+}1$	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1284
`depends_on`	←	Where the central charges are	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1235
`proves`	←	ch:13-lie-algebra-expansions@proof-20	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1169