7.1 Modules and Derivations
165
with the obvious diagonal action of isomorphisms of finite sets. The structure
operations (6.78) are given by
(t 1 , m 1 ) a • b (t 2 , m 2 ) := (t 1 a • b t 2 , t 1 a
L
• b m 2 + m 1 a
R
• b t 2 ),
where
t 1 ∈ T
S 1 {a}; g 1
, t 2 ∈ T
S 2 {b}; g 2
and
m 1 ∈ M
S 1 {a}; g 1
, m 2 ∈ M
S 2 {b}; g 2
.
The contractions (6.79) are defined diagonally,
• uv (t, m) := (• uv t, uv m).
for t ∈ T
S {u, v}; g
and m ∈ M
S {u, v}; g
.
Lemma 7.2 The modular module T ⊕ M with the operations defined above is
an odd modular operad.
Proof. The axioms of modular operads can be verified directly. Let us, for instance,
verify axiom (6.80) of Definition 6.24. For
t 1 ∈ T
S 1 {a}; g 1
, t 2 ∈ T
S 2 {b, c}; g 2
, t 3 ∈ T
S 3 {d}; g 3
and
m 1 ∈ M
S 1 {a}; g 1
, m 2 ∈ M
S 2 {b, c}; g 2
, m 3 ∈ M
S 3 {d}; g 3
one has
a • b (1 ⊗ c • d )
(t 1 , m 1 ) ⊗ (t 2 , m 2 ) ⊗ (t 3 , m 3 )
= (t 1 , m 1 ) a • b (t 2 c • d t 3 , t 2 c
L
• d m 3 + m 2 c
R
• d t 3 )
=
t 1 a • b (t 2 c • d t 3 ), t 1 a
L
• b (t 2 c
L
• d m 3 ) + t 1 a
L
• b (m 2 c
R
• d t 3 ) + m 1 a
R
• b (t 2 c • d t 3 )
,
while
− c • d ( a • b ⊗1)
(t 1 , m 1 ) ⊗ (t 2 , m 2 ) ⊗ (t 3 , m 3 )
= −(t 1 a • b t 2 , t 1 a
L
• b m 2 + m 1 a
R
• b t 2 ) c
L
• d (t 3 , m 3 )
= −
(t 1 a • b t 2 ) c • d t 3 , (t 1 a • b t 2 ) c
L
• d m 3 + (t 1 a
L
• b m 2 ) c
R
• d t 3 + (m 1 a
R
• b t 2 ) c
R
• d t 3
.