7.1 Modules and Derivations
163
Definition 7.2 Let M be a module over an odd operad T . A degree k derivation
θ : T → M is a degree k morphism
θ =
θ(S; g) : T (S; g) → M(S; g); (S; g) ∈ Cor × A
of modular modules 1 satisfying, for finite S 1 , S 2 , symbols a, b, and genera g 1 , g 2 ∈
A the equality
θ a • b = (−1)
k
a
R
• b (θ ⊗ 1) + a
L
• b (1 ⊗ θ)
,
(7.5)
of maps T
S 1 {a}; g 1
⊗ T
S 2 {b}; g 2
→ M(S 1 S 2 ; g 1 + g 2 ), and for a finite
set S, symbols u, v, and a genus g ∈ A the equality
θ • ab = (−1)
k ab θ
(7.6)
of maps T
S {u, v}; g
→ T (S; g + s).
One can of course replace a
R
• b (θ ⊗1) in (7.5) by b
L
• a τ (θ ⊗1) = b
L
• a (1⊗θ)τ , but
using a
R
• b makes the analogy between (7.5) and, e.g. the standard Leibniz property
of derivations of associative algebras manifest. The auxiliary a
R
• b will also simplify
some formulas in the proofs that follow.
Example 7.2 The case when M = T , considered as a module over itself as in
Example 7.1, is particularly important. With this convention, an example of a
degree +1 derivation is the internal differential d T : T → T of an odd modular
operad.
The following lemma in which an odd operad T is considered as a module over
itself will be used in the proof of Theorem 7.2.
Lemma 7.1 Let α, β : T → T be derivations of an odd modular operad T such
that α is of degree k and β of degree l. Then the linear map
[α, β] := αβ − (−1)
kl βα : T → T
is a degree k +l derivation. In particular, if k = l = 1, then both α 2 , β 2 and αβ +βα
are degree 2 derivations.
1 See Remark 6.10.