definition 15.14 Truncated polynomial algebra and Takiff algebra

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:494 · p. 606

ground object -- no derivation owed

Rests on

depends_on definition 15.3 Tensor product Lie algebra ¶

Supports

depends_on corollary 15.69 No positive pairing survives the radical ¶
depends_on definition 15.20 Resonant subalgebra ¶
- 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 $D=4$ ¶ ↺
depends_on example 15.61 Maurer–Cartan expansion ¶ ↺
depends_on proposition 15.15 Takiff brackets ¶
- depends_on example 15.16 The Maxwell tower ¶
- depends_on proposition 15.26 The contraction along a grading ¶ ↺
- depends_on proposition 15.21 Resonance ¶ ↺

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
`cites`	→	Rings of invariant polynomials for a class of Lie algebras	derived	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:508
`depends_on`	→	Tensor product Lie algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:510
`depends_on`	←	No positive pairing survives the radical	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2579
`depends_on`	←	Resonant subalgebra	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`	←	Takiff brackets	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:526