190
8 Structures Relevant to Physics
which, in turn, equals
−
◦ uv a ◦ b ⊗ • uv a • b
τ (θ 1 ⊗ θ 1 ⊗ θ 2 ⊗ θ 2 )
by the commutativity (6.57) resp. the anti-commutativity (6.82). Comparing it
with (8.11) we conclude that the middle term 2 of (8.10) vanishes.
A straightforward calculation shows that, for f, g as in (8.10),
Δ(f ), g
=
S 1 S 2 =[n 1 +n 2 −4]
a ◦ b (◦ uv ⊗ 1) ⊗ a • b (• uv ⊗ 1)
τ (θ
1 ⊗ θ
1 ⊗ θ
2 ⊗ θ
2 )(f ⊗ g)
for some bijections θ
1 : [n 1 ] → S 1 {a, u, v} and θ
2 : [n 2 ] → S 2 {b}. The sum in
the right-hand side however equals
S 1 S 2 =[n 1 +n 2 −4]
−
◦ uv a ◦ b ⊗ • uv a • b
τ (θ
1 ⊗ θ
1 ⊗ θ
2 ⊗ θ
2 )(f ⊗ g)
by the commutativity (6.58) resp. the anti-commutativity (6.83), which is 1 (f ⊗g)
with the minus sign. By exactly the same method we show that
(−1)
|f |
f, Δ(g)
= − 3 (f ⊗ g).
This finishes the proof of (8.7).
Let us move to the proof of the Jacobi identity (8.8). It will be convenient to
introduce two auxiliary maps. For finite sets S 1 , S 2 , S 3 , integers p, q, r ≥ 0, genera
i, j, k ∈ A and isomorphisms
θ 1 : [p]
∼ =
− → S 1 {b}, θ 2 : [q]
∼ =
− → S 2 {a, c}, θ 3 : [r]
∼ =
− → S 3 {d}
the first map
A(θ 1 , θ 2 , θ 3 ) : MT(p; i) ⊗ MT(q; j) ⊗ MT(r; k) →
→ M (S 1 S 2 S 3 ; i + j + k) ⊗ T (S 1 S 2 S 3 ; i + j + k)
is defined by
A(θ 1 , θ 2 , θ 3 ) :=
c ◦ d ( b ◦ a ⊗1)(θ 1 ⊗ θ 2 ⊗ θ 3 ) ⊗ c • d ( b • a ⊗1)(θ 1 ⊗ θ 2 ⊗ θ 3 )
ψ,
where the isomorphism ψ interchanges the tensor factors of the subspace
MT(p; i) ⊗ MT(q; j ) ⊗ MT(r; k)