192
8 Structures Relevant to Physics
can be, using (6.55) and (6.80), rewritten as
−
b ◦ a (1 ⊗ c ◦ d )(θ 1 ⊗ θ 2 ⊗ θ 3 ) ⊗ b • a (1 ⊗ c • d )(θ 1 ⊗ θ 2 ⊗ θ 3 )
ψ.
This in turn, by (6.54) and its odd version, equals
−
a ◦ b ( c ◦ d ⊗1)(θ 2 ⊗ θ 3 ⊗ θ 1 ) ⊗ a • b ( c • d ⊗1)(θ 2 ⊗ θ 3 ⊗ θ 1 )
ψλ,
which is −B(θ 2 , θ 3 , θ 1 )λ as claimed.
The terms in the Jacobi identity can be expressed via the auxiliary maps as
follows. For f ∈ MT(n 1 ; g 1 ), g ∈ MT(n 2 ; g 2 ), and h ∈ MT(n 3 ; g 3 ) one obtains
{f, g}, h
=
A(θ 11 , θ 22 , θ 33 ) + B(θ 12 , θ 23 , θ 31 )
(f ⊗ g ⊗ h),
where the summation running over all disjoint partitions
S 1 S 2 S 3 = [n 1 + n 2 + n 3 − 4]
with arbitrarily chosen isomorphism
θ 11 : [n 1 ]
∼ =
− → S 1 {b}, θ 22 : [n 2 ]
∼ =
− → S 2 {a, c}, θ 33 : [n 3 ]
∼ =
− → S 3 {d} and
θ 12 : [n 1 ]
∼ =
− → S 2 {a, c}, θ 23 : [n 2 ]
∼ =
− → S 3 {d}, θ 31 : [n 3 ]
∼ =
− → S 1 {b}.
Likewise, for the same f, g, and h one has
{g, h}, f
=
A(θ 21 , θ 32 , θ 13 ) + B(θ 22 , θ 33 , θ 11 )
(g ⊗ h ⊗ f ),
with chosen isomorphism
θ 21 : [n 2 ]
∼ =
− → S 1 {b}, θ 32 : [n 3 ]
∼ =
− → S 2 {a, c}, θ 13 : [n 1 ]
∼ =
− → S 3 {d} and
θ 22 : [n 2 ]
∼ =
− → S 2 {a, c}, θ 33 : [n 3 ]
∼ =
− → S 3 {d}, θ 11 : [n 1 ]
∼ =
− → S 1 {b}.
Finally,
{h, f }, g
=
A(θ 31 , θ 12 , θ 23 ) + B(θ 32 , θ 13 , θ 21 )
(h ⊗ f ⊗ g),
with isomorphism
θ 31 : [n 3 ]
∼ =
− → S 1 {b}, θ 12 : [n 1 ]
∼ =
− → S 2 {a, c}, θ 23 : [n 2 ]
∼ =
− → S 3 {d} and
θ 32 : [n 3 ]
∼ =
− → S 2 {a, c}, θ 13 : [n 1 ]
∼ =
− → S 3 {d}, θ 21 : [n 2 ]
∼ =
− → S 1 {b}.