Manuscript VII-I@1 · VII-I.tex · proposition · 907e96ab

Proposition: Associativity and normalization interfaces

The triangle obstruction and what VII-D does supply › Necessary compatibility beyond one finite graph · lines 1491–1532 · proof included or linked

VII-I@1#prop:associativity, VII-I@1#eq:associativity-equations, VII-I@1#eq:comparison-equation

Back to manuscript

Suppose a proposed sharp product has defined bilinear maps

F ⋆ G = j = 0 \sum^{3} ℏ^{j} μ_{j} (F, G) (mod ℏ^{4}), μ_{0} (F, G) = F G .

Then associativity is equivalent to

δ μ_{1} = 0, δ μ_{n} (F, G, K) = i + j = n i, j \geq 1 \sum [μ_{i} (μ_{j} (F, G), K) - μ_{i} (F, μ_{j} (G, K))], n = 2, 3,

where

δ μ (F, G, K) = F μ (G, K) - μ (F G, K) + μ (F, G K) - μ (F, G) K .

If two products agree below order $p$ , a comparison $Z = id + ℏ^{p} D (mod ℏ^{p + 1})$ must satisfy

μ_{p}^{b} (F, G) - μ_{p}^{a} (F, G) = D (F G) - D (F) G - F D (G) .

Proof. Expand both associations coefficientwise and collect all terms with one occurrence of $μ_{n}$ ; the remaining terms are the right-hand side of (eq:associativity-equations). Expanding $Z (F ⋆_{a} G) = Z F ⋆_{b} Z G$ at order $p$ gives (eq:comparison-equation).

Displayed formulas in this passage

unlabeled · line 1494
$F ⋆ G = j = 0 \sum^{3} ℏ^{j} μ_{j} (F, G) (mod ℏ^{4}), μ_{0} (F, G) = F G .$
VII-I@1#eq:associativity-equations · line 1499
$δ μ_{1} = 0, δ μ_{n} (F, G, K) = i + j = n i, j \geq 1 \sum [μ_{i} (μ_{j} (F, G), K) - μ_{i} (F, μ_{j} (G, K))], n = 2, 3,$
unlabeled · line 1512
$δ μ (F, G, K) = F μ (G, K) - μ (F G, K) + μ (F, G K) - μ (F, G) K .$
VII-I@1#eq:comparison-equation · line 1518
$μ_{p}^{b} (F, G) - μ_{p}^{a} (F, G) = D (F G) - D (F) G - F D (G) .$

Nearby in this section

section: Necessary compatibility beyond one finite graph