6.5 Odd Modular Operads
153
To finish the proof of (6.86), we observe that
1 S 1 ⊗ ¯
s ⊗ 1 S 2 ⊗ ¯ ¯
s ⊗ 1 S 3
# =
1 S 1 ⊗ ¯
s ⊗ 1 S 2 3
#
1 S 1 {a,b}}S 2 ⊗ ¯ ¯
s ⊗ 1 S 3
#
= −
1 S 1 2 ⊗ ¯ ¯
s ⊗ 1 S 3
#
1 S 1 ⊗ ¯
s ⊗ 1 S 2 {c,d}}S 3
# ,
the minus sign coming from commuting ¯
s over ¯ ¯
s.
Let us also verify that the a • b -operations defined by the odd version of
composition (6.15) satisfy (6.87). The related calculation is of course obtained from
the above one by removing duals and inverting the order of compositions but, very
crucially, without inserting Koszul signs. We obtain
f a • b (g c • d h) =
f ⊗ (g c • d h)
1 S 1 ⊗ ¯
s ⊗ 1 S 2 3
=
f ⊗ (g ⊗ h)(1 S 2 ⊗ ¯ ¯
s ⊗ 1 S 3 )
1 S 1 ⊗ ¯
s ⊗ 1 S 2 3
= (f ⊗ g ⊗ h)
1 S 1 {a,b}}S 2 ⊗ ¯ ¯
s ⊗ 1 S 3
1 S 1 ⊗ ¯
s ⊗ 1 S 2 3
on the one hand and
(f a • b g) c • d h =
(f a • b g) ⊗ h
1 S 1 2 ⊗ ¯ ¯
s ⊗ 1 S 3
=
(f ⊗ g)(1 S 1 ⊗ ¯
s ⊗ 1 S 2 ) ⊗ h
1 S 1 2 ⊗ ¯ ¯
s ⊗ 1 S 3
= (−1)
|h| (f ⊗ g ⊗ h)
1 S 1 ⊗ ¯
s ⊗ 1 S 2 {c,d}}S 3
1 S 1 2 ⊗ ¯ ¯
s ⊗ 1 S 3
on the other hand. Axiom (6.87) now follows from the equality
1 S 1 ⊗ ¯
s ⊗ 1 S 2 ⊗ ¯ ¯
s ⊗ 1 S 3
=
1 S 1 ⊗ ¯
s ⊗ 1 S 2 {c,d}}S 3
1 S 1 2 ⊗ ¯ ¯
s ⊗ 1 S 3
= −
1 S 1 {a,b}}S 2 ⊗ ¯ ¯
s ⊗ 1 S 3
1 S 1 ⊗ ¯
s ⊗ 1 S 2 3
.
Notice that the sign difference between the results of the above two computations
is (−1) |x| versus (−1) |z| as it should be. The verification of axioms (6.88)
resp. (6.89) is similar.
We will call the family (6.61) with the a • b - and • uv -operations defined via
compositions (6.16) and (6.64) with |s| = 1 the odd modular endomorphism operad
and denote it End V ; whether End V means the odd modular endomorphism operad
or the ordinary one (with |s| = 0) will always be clear from the context. For a
degree +1 bilinear form B : V ⊗ V → k we also have the odd modular version of
the operad Dne V of Example 6.7. We leave the details to the reader.
153
To finish the proof of (6.86), we observe that
1 S 1 ⊗ ¯
s ⊗ 1 S 2 ⊗ ¯ ¯
s ⊗ 1 S 3
# =
1 S 1 ⊗ ¯
s ⊗ 1 S 2 3
#
1 S 1 {a,b}}S 2 ⊗ ¯ ¯
s ⊗ 1 S 3
#
= −
1 S 1 2 ⊗ ¯ ¯
s ⊗ 1 S 3
#
1 S 1 ⊗ ¯
s ⊗ 1 S 2 {c,d}}S 3
# ,
the minus sign coming from commuting ¯
s over ¯ ¯
s.
Let us also verify that the a • b -operations defined by the odd version of
composition (6.15) satisfy (6.87). The related calculation is of course obtained from
the above one by removing duals and inverting the order of compositions but, very
crucially, without inserting Koszul signs. We obtain
f a • b (g c • d h) =
f ⊗ (g c • d h)
1 S 1 ⊗ ¯
s ⊗ 1 S 2 3
=
f ⊗ (g ⊗ h)(1 S 2 ⊗ ¯ ¯
s ⊗ 1 S 3 )
1 S 1 ⊗ ¯
s ⊗ 1 S 2 3
= (f ⊗ g ⊗ h)
1 S 1 {a,b}}S 2 ⊗ ¯ ¯
s ⊗ 1 S 3
1 S 1 ⊗ ¯
s ⊗ 1 S 2 3
on the one hand and
(f a • b g) c • d h =
(f a • b g) ⊗ h
1 S 1 2 ⊗ ¯ ¯
s ⊗ 1 S 3
=
(f ⊗ g)(1 S 1 ⊗ ¯
s ⊗ 1 S 2 ) ⊗ h
1 S 1 2 ⊗ ¯ ¯
s ⊗ 1 S 3
= (−1)
|h| (f ⊗ g ⊗ h)
1 S 1 ⊗ ¯
s ⊗ 1 S 2 {c,d}}S 3
1 S 1 2 ⊗ ¯ ¯
s ⊗ 1 S 3
on the other hand. Axiom (6.87) now follows from the equality
1 S 1 ⊗ ¯
s ⊗ 1 S 2 ⊗ ¯ ¯
s ⊗ 1 S 3
=
1 S 1 ⊗ ¯
s ⊗ 1 S 2 {c,d}}S 3
1 S 1 2 ⊗ ¯ ¯
s ⊗ 1 S 3
= −
1 S 1 {a,b}}S 2 ⊗ ¯ ¯
s ⊗ 1 S 3
1 S 1 ⊗ ¯
s ⊗ 1 S 2 3
.
Notice that the sign difference between the results of the above two computations
is (−1) |x| versus (−1) |z| as it should be. The verification of axioms (6.88)
resp. (6.89) is similar.
We will call the family (6.61) with the a • b - and • uv -operations defined via
compositions (6.16) and (6.64) with |s| = 1 the odd modular endomorphism operad
and denote it End V ; whether End V means the odd modular endomorphism operad
or the ordinary one (with |s| = 0) will always be clear from the context. For a
degree +1 bilinear form B : V ⊗ V → k we also have the odd modular version of
the operad Dne V of Example 6.7. We leave the details to the reader.