7.2 Feynman Transform
171
by T(C) the free anti-associative algebra generated by a graded vector space C and
ι : C C→ T(C) the canonical inclusion. The algebra T(C) can be realized as the
tensor algebra T(C) with suitably redefined degrees and the tensor product taken
with an appropriate sign; we leave the details to the interested reader.
If C = (C, Δ) is a coalgebra as above, we define its “odd” cobar construction
Ω(C) as the couple ( T(C), ∂), where ∂ is the unique degree +1 derivation extending
the degree +1 map
C
Δ
−→ C ⊗ C
ι⊗ι
−→ T(C) ⊗ T(C)
−→ T(C).
We leave as an exercise to prove that ∂ 2 = 0, so
Ω(C) is a dg-anti-associative
algebra. As before, a differential d C of C induces a second differential d :
Ω(C) →
Ω(C) such that (7.15) is satisfied.
Notice that if A is an dg-associative algebra, then its desuspension ↓ A has a
natural structure of an anti-associative algebra. Likewise, suspensions of dg-antiassociative algebras are ordinary associative algebras. This correspondence defines
an equivalence between the category of dg-associative algebras and the category
of dg-anti-associative algebras. Since the ordinary and the odd cobar constructions
correspond to each other under this correspondence, both constructions are essentially equivalent.
As the classical cobar construction recalled above acts on coassociative coalgebras, the Feynman transform acts on modular cooperads. The underlying objects of
modular cooperads are modular comodules:
Definition 7.5 A modular comodule is a contravariant functor
E : Cor × A → Chain.
Definition 7.6 Each modular comodule E has its associated modular module
˚
E : Cor × A → Chain
with ˚
E(S; g) := E(S; g) and ˚
E(σ ) : ˚
E(S; g) → ˚
E(T ; g) defined, for an
isomorphism σ : S
∼ =
−→ T , by ˚
E(σ ) := E(σ −1 ).
The assignment E → ˚
E defines an isomorphism between the category of
modular comodules and the category of modular modules.
Definition 7.7 A modular cooperad with step s consists of a modular comodule
C =
C (S; g) ∈ Chain; (S; g) ∈ Cor × A
171
by T(C) the free anti-associative algebra generated by a graded vector space C and
ι : C C→ T(C) the canonical inclusion. The algebra T(C) can be realized as the
tensor algebra T(C) with suitably redefined degrees and the tensor product taken
with an appropriate sign; we leave the details to the interested reader.
If C = (C, Δ) is a coalgebra as above, we define its “odd” cobar construction
Ω(C) as the couple ( T(C), ∂), where ∂ is the unique degree +1 derivation extending
the degree +1 map
C
Δ
−→ C ⊗ C
ι⊗ι
−→ T(C) ⊗ T(C)
−→ T(C).
We leave as an exercise to prove that ∂ 2 = 0, so
Ω(C) is a dg-anti-associative
algebra. As before, a differential d C of C induces a second differential d :
Ω(C) →
Ω(C) such that (7.15) is satisfied.
Notice that if A is an dg-associative algebra, then its desuspension ↓ A has a
natural structure of an anti-associative algebra. Likewise, suspensions of dg-antiassociative algebras are ordinary associative algebras. This correspondence defines
an equivalence between the category of dg-associative algebras and the category
of dg-anti-associative algebras. Since the ordinary and the odd cobar constructions
correspond to each other under this correspondence, both constructions are essentially equivalent.
As the classical cobar construction recalled above acts on coassociative coalgebras, the Feynman transform acts on modular cooperads. The underlying objects of
modular cooperads are modular comodules:
Definition 7.5 A modular comodule is a contravariant functor
E : Cor × A → Chain.
Definition 7.6 Each modular comodule E has its associated modular module
˚
E : Cor × A → Chain
with ˚
E(S; g) := E(S; g) and ˚
E(σ ) : ˚
E(S; g) → ˚
E(T ; g) defined, for an
isomorphism σ : S
∼ =
−→ T , by ˚
E(σ ) := E(σ −1 ).
The assignment E → ˚
E defines an isomorphism between the category of
modular comodules and the category of modular modules.
Definition 7.7 A modular cooperad with step s consists of a modular comodule
C =
C (S; g) ∈ Chain; (S; g) ∈ Cor × A