definition 15.20 Resonant subalgebra

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:623 · p. 608

ground object -- no derivation owed

Rests on

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 ¶

Supports

depends_on example 15.61 Maurer–Cartan expansion ¶
depends_on proposition 15.26 The contraction along a grading ¶
- depends_on corollary 15.44 The eight vertices are the eight degenerations of $A$ ¶
  - depends_on remark 15.45 A correction this forces: the general splitting gives the static algebra, not Carroll ¶
- depends_on corollary 15.27 The flat limit ¶
  - 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 proposition 15.21 Resonance ¶
depends_on theorem 15.58 Nondegeneracy on a resonant subalgebra ¶
- depends_on corollary 15.69 No positive pairing survives the radical ¶
- depends_on remark 15.59 The central charge is what makes the form work ¶ ↺
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`	→	$\Z_{2}$-grading, symmetric coset	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:632
`depends_on`	→	Truncated polynomial algebra and Takiff algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:632
`depends_on`	←	Maurer–Cartan expansion	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2305
`depends_on`	←	The contraction along a grading	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:794
`depends_on`	←	Resonance	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:643
`depends_on`	←	Nondegeneracy on a resonant subalgebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2200
`depends_on`	←	The truncation tower is a tower of central extensions	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1166