Manuscript I-B-VP-1@1 · I-B-VP-1-v1.tex · corollary · 6d700be9

Corollary: Simultaneous residues

Dilation residues and compatibility of presentations › The unavoidable scalar residue · lines 1118–1148 · proof included or linked

I-B-VP-1@1#cor:residues, I-B-VP-1@1#eq:allresidues

Back to manuscript

For the canonical prescription,

(E + 2) F_{L} (E + 4) (T_{L})^{A}_{B} (E + 4) (T_{L})^{i}_{j} (E + 3) (J_{L})^{i} = s σ_{α} δ_{W}, = \frac{a σ _{α}}{4} δ^{A}_{B} Δ δ_{W}, = \frac{b σ _{α}}{2} K_{ij} δ_{W}, = - \frac{j σ _{α}}{2 α ^{2}} ϵ^{i}_{k} \partial_{k} δ_{W} .

These residues are unchanged by the contacts in Theorem thm:classification. The $L$ -derivative of each canonical distribution is the negative of its displayed residue.

Proof. Use $[E, \partial_{i}] = - \partial_{i}$ in (eq:EulerU), then apply the defining differential operators. Each permitted contact is exactly homogeneous at the relevant degree and is therefore killed by the corresponding Euler operator.

Displayed formulas in this passage

I-B-VP-1@1#eq:allresidues · line 1121
$(E + 2) F_{L} (E + 4) (T_{L})^{A}_{B} (E + 4) (T_{L})^{i}_{j} (E + 3) (J_{L})^{i} = s σ_{α} δ_{W}, = \frac{a σ _{α}}{4} δ^{A}_{B} Δ δ_{W}, = \frac{b σ _{α}}{2} K_{ij} δ_{W}, = - \frac{j σ _{α}}{2 α ^{2}} ϵ^{i}_{k} \partial_{k} δ_{W} .$

Corollary: Simultaneous residues

Displayed formulas in this passage

Nearby in this section