170
7 Feynman Transform of a Modular Operad
By Lemma 7.4 again, there is a one-to-one correspondence between degree 0
derivations θ : F(E) → M and morphism of modular operads
Θ :=
1 F(E) , θ
: F(E) → F(E) ⊕ M.
(7.14)
By the freeness of F(E), such a morphism is determined by its restriction
Θ| E =
ι, θ| E
: E → F(E) ⊕ M
which is in turn determined by the restriction θ | E : E → M. It is therefore enough
to show that every modular module morphism ϑ : E → M extends to a modular
operad morphism of as in (7.14), determining a derivation that restricts to it.
Let Θ : F(E) → F(E) ⊕ M be the unique extension of the modular module
morphism (ι, ϑ) : E → F(E) ⊕ M. This extension is necessary of the form Θ =
(Φ, θ ) for some Φ : F(E) → F(E). We must show that Φ = 1 F(E) .
The projection pr 1 : F(E)⊕M → F(E) to the first summand is clearly an operad
morphism, and so is the composition Φ = pr 1 ◦ Θ. By construction, Φ| E = ι. Since
the identity 1 F(E) also restricts to ι, Φ must equal 1 F(E) by the uniqueness of the
extension. The equality θ | E = ϑ is obvious.
7.2
Feynman Transform
In this section we define the cobar construction for modular (co)operads, which
is called in this context the Feynman transform. To understand the idea of this
construction better, we recall very briefly the cobar construction for the classical
non-counital coassociative coalgebras.
If C is such a coalgebra with the comultiplication Δ : C → C ⊗C, then its cobar
construction Ω(C) = (T(↑ C), ∂) is the tensor algebra T(↑ C) on the suspension
of C, with the differential that is the unique extension of the linear degree +1 map
(↑⊗ ↑) ◦ Δ◦ ↓: ↑C → ↑C⊗ ↑C
into a degree +1 derivation. Such an unique extension exists since T(↑ C) is the
free associative algebra generated by ↑ C. A straightforward calculation shows that
the coassociativity of Δ implies ∂ 2 = 0. The cobar construction is therefore a dgassociative algebra.
If C is a dg-coalgebra with a differential d C , then ↑ d C ↓ : ↑ C →↑ C extends
into a degree +1 derivation d : T(↑ C) → T(↑ C). One easily proves that
∂
2
= d
2
= d∂ + ∂d = 0.
(7.15)
There is a modification of the above construction that avoids the use of the
suspension. In Example 6.30 we recalled anti-associative algebras; let us denote
7 Feynman Transform of a Modular Operad
By Lemma 7.4 again, there is a one-to-one correspondence between degree 0
derivations θ : F(E) → M and morphism of modular operads
Θ :=
1 F(E) , θ
: F(E) → F(E) ⊕ M.
(7.14)
By the freeness of F(E), such a morphism is determined by its restriction
Θ| E =
ι, θ| E
: E → F(E) ⊕ M
which is in turn determined by the restriction θ | E : E → M. It is therefore enough
to show that every modular module morphism ϑ : E → M extends to a modular
operad morphism of as in (7.14), determining a derivation that restricts to it.
Let Θ : F(E) → F(E) ⊕ M be the unique extension of the modular module
morphism (ι, ϑ) : E → F(E) ⊕ M. This extension is necessary of the form Θ =
(Φ, θ ) for some Φ : F(E) → F(E). We must show that Φ = 1 F(E) .
The projection pr 1 : F(E)⊕M → F(E) to the first summand is clearly an operad
morphism, and so is the composition Φ = pr 1 ◦ Θ. By construction, Φ| E = ι. Since
the identity 1 F(E) also restricts to ι, Φ must equal 1 F(E) by the uniqueness of the
extension. The equality θ | E = ϑ is obvious.
7.2
Feynman Transform
In this section we define the cobar construction for modular (co)operads, which
is called in this context the Feynman transform. To understand the idea of this
construction better, we recall very briefly the cobar construction for the classical
non-counital coassociative coalgebras.
If C is such a coalgebra with the comultiplication Δ : C → C ⊗C, then its cobar
construction Ω(C) = (T(↑ C), ∂) is the tensor algebra T(↑ C) on the suspension
of C, with the differential that is the unique extension of the linear degree +1 map
(↑⊗ ↑) ◦ Δ◦ ↓: ↑C → ↑C⊗ ↑C
into a degree +1 derivation. Such an unique extension exists since T(↑ C) is the
free associative algebra generated by ↑ C. A straightforward calculation shows that
the coassociativity of Δ implies ∂ 2 = 0. The cobar construction is therefore a dgassociative algebra.
If C is a dg-coalgebra with a differential d C , then ↑ d C ↓ : ↑ C →↑ C extends
into a degree +1 derivation d : T(↑ C) → T(↑ C). One easily proves that
∂
2
= d
2
= d∂ + ∂d = 0.
(7.15)
There is a modification of the above construction that avoids the use of the
suspension. In Example 6.30 we recalled anti-associative algebras; let us denote