7.1 Modules and Derivations
161
(vii) For finite sets S 1 , S 2 , symbols a, b, u, v, and genera g 1 , g 2 ∈ A, one has the
equality
a
L
• b (• uv ⊗ 1) = − uv a
L
• b
(7.2)
of maps T
S 1 {a, u, v}; g 1
⊗ M
S 2 {b}; g 2
→ M(S 1 S 2 ; g 1 + g 2 + s).
(viii) For finite sets S 1 , S 2 , symbols a, b, u, v, and genera g 1 , g 2 ∈ A, one has the
equality
a
L
• b (1 ⊗ uv ) = − uv a
L
• b
of maps T
S 1 {a}; g 1
⊗ M
S 2 {b, u, v}; g 2
→ M(S 1 S 2 ; g 1 + g 2 + s).
Remark 7.1 The superscript “L” of a
L
• b reminds us that we apply an element of T
from the left to an element of M. It will be convenient to define also the auxiliary
right composition
a
R
• b : M(S 1 {a}; g 1 ) ⊗ T (S 2 {b}; g 2 ) → M(S 1 S 2 ; g 1 + g 2 )
as a
R
• b := b
L
• a τ , i.e. on elements as
x a
R
• b y := (−1)
|x||y| y b
L
• a x
for x ∈ M
S 1 {a}; g 1
and y ∈ T
S 2 {b}; g 2
. One easily proves the following
properties of this operation.
(i) For arbitrary isomorphisms ρ : S 1 {a} → T 1 and σ : S 2 {b} → T 2 of finite
sets and genera g 1 , g 2 ∈ A, one has the equality
M(ρ| S 1 σ | S 2 ) a
R
• b = ρ(a)
R
• σ (b) (M(ρ) ⊗ T (σ ))
of maps
M
S 1 {a}; g 1
⊗ T
S 2 {b}; g 2
→ M
T 1 T 2 \ {ρ(a), σ (b)}; g 1 + g 2
.
(ii) For finite sets S 1 , S 2 , symbols a, b, c, d and genera g 1 , g 2 ∈ A, one has the
equality
ab c
R
• d = − cd a
R
• b
of maps M
S 1 {a, c}; g 1
⊗ T
S 2 {b, d}; g 2
→ M(S 1 S 2 ; g 1 + g 2 + s).