6.1 Cyclic Operads
125
The characteristic property of ideals in P is that the operad structure of P
induces an operad structure on the (componentwise) quotient
P/I :=
P(S)/I (S) | S ∈ cyclic
.
Each operad is in fact a quotient of a free one, as stated in
Proposition 6.3 Each cyclic operad P is isomorphic to the quotient F(E)/I for
some cyclic module E and an ideal I in F(E).
Proof. It is clear that the map Π of (6.36) is a (componentwise) epimorphism, thus
P ∼ = F(P)/Ker(Π).
(6.38)
As ideals, e.g., of algebras, also operadic ideals can be generated by sets of
elements. Given a family S of mutually non-isomorphic finite sets and a system
R of elements r S ∈ P(S), S ∈ S, one defines (R), the ideal generated by R, to be
the smallest ideal (i.e., intersection of all ideals) in P containing R.
Presentation (6.38) is huge, usually much smaller ones are available. As an example we describe a small presentation of the operad Com recalled in Example 6.1.
Consider the family S consisting of a single set {1, 2, 3} and take
G {1,2,3} := Span(μ {1,2,3} ) ∼ = k
with the trivial action of the symmetric groups Σ 3 = Aut
{1, 2, 3}
. Let E μ be the
cyclic module generated, in the sense of (6.32), by the generating system G and (R)
be the ideal in F(E μ ) generated by the single element
r {1,2,5,6} := μ {1,2,3} 3 ◦ 4 μ {4,5,6} − μ {2,5,3} 3 ◦ 4 μ {4,6,1} ∈ F(E μ )
{1, 2, 5, 6}
graphically expressed as
r {1,2,5,6} =
2
5
6
1
3 4
−
2
5
6
1
3
4
.
Proposition 6.4 The cyclic operad Com has the presentation
Com ∼ = F(E μ )/(R).
(6.39)