6.1 Cyclic Operads
121
For a graph Γ and a cyclic module E, let
E(Γ ) :=
v∈Vert(Γ )
E
Leg(v)
(6.33)
denote the unordered tensor product over the vertices of Γ . It might help to view
generators of E(Γ ) as structure formulas for chemical substances represented by
decorated graphs, with vertices decorated by specific atoms (elements) and internal
edges of Γ representing chemical bonds.
Notice that each graph isomorphism φ : Γ 0
∼ =
−→ Γ 1 induces a natural isomorphism
E(φ) : E(Γ 0 )
∼ =
−→ E(Γ 1 )
of dg-vector spaces. For a finite set S define
F(E)(S) :=
T E(T )
∼
,
(6.34)
where the direct sum runs over all trees T with Leg(T ) = S, and the equivalence
relation ∼ identifies x ∈ E(T 0 ) with its image E(φ)(x) ∈ E(T 1 ) for any
isomorphism φ : T 0 → T 1 such that Leg(φ) = 1 S . For any tree T with Leg(T ) = S
one has the canonical map
i T : E(T ) → F(E)(S).
(6.35)
Proposition 6.1 The family F(E) = {F(E)(S) | S ∈ Cor} has a natural structure
of a cyclic operad.
Proof. Let ρ : S
∼ =
−→ D be an isomorphism of finite sets and T a tree with Leg(T ) =
S. There clearly exists a tree T ρ obtained by renaming the legs of T according to ρ,
together with an obvious isomorphism φ ρ : T
∼ =
−→ T ρ . The collection of induced
isomorphisms
E(φ ρ ) : E(T )
∼ =
−→ E(T ρ ) | Leg(T ) = S
clearly induces a natural isomorphism of the quotients
F(E)(ρ) : F(E)(S)
∼ =
−→ F(E)(D).