Manuscript II@2 · II.tex · section · 50d64def

Associator coordinate

Ward reduction and coupling factorization › Associator coordinate · lines 687–742

II@2#eq:q

On the internal Euclidean space with basis $e_{1}, e_{2}$ , set

e_{I} \circ e_{J} = K \sum d_{I J K} e_{K} .

The left multiplication matrices are

L_{1} = (a b b c), L_{2} = (b c c e) .

Define

q (d) = b^{2} + c^{2} - a c - b e .

Then

[L_{2}, L_{1}] = (0 - q q 0) .

For

A_{I J K L} = ⟨ (e_{I} \circ e_{J}) \circ e_{K} - e_{I} \circ (e_{J} \circ e_{K}), e_{L} ⟩,

the nonzero possible components are

A_{1122} = A_{2211} = - q, A_{1221} = A_{2112} = q .

Hence

A = 0 ⟺ q = 0, ∥ A ∥_{F} = 2∣ q ∣.

When $q = 0$ , the symmetric matrices $L_{1}, L_{2}$ commute. In a common orthonormal eigenbasis $f_{1}, f_{2}$ , the product $f_{1} \circ f_{2}$ is simultaneously proportional to $f_{1}$ and to $f_{2}$ , and is therefore zero. Inner-product invariance then gives $f_{i} \circ f_{i} = λ_{i} f_{i}$ . Consequently

d_{I J K} = i = 1 \sum^{2} λ_{i} O_{I i} O_{J i} O_{K i}

for an orthogonal matrix $O$ . The cubic and reference quartic in that basis are

i \sum λ_{i} E_{3} [H^{i}], i \sum λ_{i}^{2} E_{4} [H^{i}] .

This elementary decomposition is used as a control, not as an imported multi-graviton classification.

Displayed formulas in this passage

II@2#eq:q · line 700
$q (d) = b^{2} + c^{2} - a c - b e .$
unlabeled · line 690
$e_{I} \circ e_{J} = K \sum d_{I J K} e_{K} .$
unlabeled · line 694
$L_{1} = (a b b c), L_{2} = (b c c e) .$
unlabeled · line 705
$[L_{2}, L_{1}] = (0 - q q 0) .$
unlabeled · line 709
$A_{I J K L} = ⟨ (e_{I} \circ e_{J}) \circ e_{K} - e_{I} \circ (e_{J} \circ e_{K}), e_{L} ⟩,$
unlabeled · line 714
$A_{1122} = A_{2211} = - q, A_{1221} = A_{2112} = q .$
unlabeled · line 719
$A = 0 ⟺ q = 0, ∥ A ∥_{F} = 2∣ q ∣.$
unlabeled · line 730
$d_{I J K} = i = 1 \sum^{2} λ_{i} O_{I i} O_{J i} O_{K i}$
unlabeled · line 735
$i \sum λ_{i} E_{3} [H^{i}], i \sum λ_{i}^{2} E_{4} [H^{i}] .$