6.1 Cyclic Operads
117
with the Koszul sign of the permutation
v
1 , . . . , v
m+1 , v
1 , . . . , v
n+1 −→
v
i , v
j , v
1 , . . . , v
i−1 , v
j +1 , . . . , v
n+1 , v
1 , . . . , v
j −1 , v
i+1 , . . . , v
m+1 .
The “classical” ◦ i operations (6.10)
◦ i : Dne V (m+1) ⊗ Dne V (n+1) → Dne V (m+n), 1 ≤ i ≤ m,
are given by
(v
1 ⊗ · · · ⊗ v
m+1 ) ◦ i (v
1 ⊗ · · · ⊗ v
n+1 )
= B(v
i , v
n+1 )v
1 ⊗ · · · ⊗ v
i−1 ⊗ v
1 ⊗ · · · ⊗ v
n ⊗ v
i+1 ⊗ · · · ⊗ v
m ,
where is the Koszul sign if the permutation
v
1 , . . . , v
m+1 , v
1 , . . . , v
n+1 −→ v
i , v
n+1 , v
1 , . . . , v
i−1 , v
1 , . . . , v
n , v
i+1 , . . . , v
m .
Let us make the intuitive concept of cyclic operads based on cobwebs presented
at the beginning of this section more precise. The mathematical abstraction of a
cobweb will be a graph. In the traditional approach, a graph consists of vertices and
edges connecting these vertices. In the context of operads, one needs to distinguish
between internal edges (those connecting two vertices) and external ones (legs) with
a “free end” along which graphs can be glued together, see Definition 6.9. This needs
a refinement of the classical definition. We use the one suggested by M. Kontsevich:
Definition 6.8 A graph Γ is a finite set Flag(Γ ) (whose elements are called flags
or half-edges) together with an involution σ and a partition λ.
The vertices Vert(Γ ) of a graph Γ are the blocks of the partition λ. The edges
Edg(Γ ) are pairs of flags forming a two-cycle of σ relative to the decomposition of
a permutation into disjoint cycles. The legs Leg(Γ ) are the fixed points of σ .
We also denote by Leg(v) the flags belonging to the block v or, in common
speech, half-edges adjacent to the vertex v. The cardinality of Leg(v) is the valency
of v. We say that two flags x, y ∈ Flag(Γ ) meet if they belong to the same block of
the partition λ. In plain language, this means that they share a common vertex.
One associates to a graph Γ a finite one-dimensional cell complex |Γ |, obtained
by taking one copy of [0,
1
2 ] for each flag and imposing the following equivalence
relation: The points 0 ∈ [0,
1
2 ] are identified for all flags in a block of the
partition λ and the points
1
2 ∈ [0,
1
2 ] are identified for pairs of flags exchanged
by the involution σ . We call |Γ | the geometric realization of the graph. We will
sometimes make no distinction between the graph in the sense of Definition 6.8 and
117
with the Koszul sign of the permutation
v
1 , . . . , v
m+1 , v
1 , . . . , v
n+1 −→
v
i , v
j , v
1 , . . . , v
i−1 , v
j +1 , . . . , v
n+1 , v
1 , . . . , v
j −1 , v
i+1 , . . . , v
m+1 .
The “classical” ◦ i operations (6.10)
◦ i : Dne V (m+1) ⊗ Dne V (n+1) → Dne V (m+n), 1 ≤ i ≤ m,
are given by
(v
1 ⊗ · · · ⊗ v
m+1 ) ◦ i (v
1 ⊗ · · · ⊗ v
n+1 )
= B(v
i , v
n+1 )v
1 ⊗ · · · ⊗ v
i−1 ⊗ v
1 ⊗ · · · ⊗ v
n ⊗ v
i+1 ⊗ · · · ⊗ v
m ,
where is the Koszul sign if the permutation
v
1 , . . . , v
m+1 , v
1 , . . . , v
n+1 −→ v
i , v
n+1 , v
1 , . . . , v
i−1 , v
1 , . . . , v
n , v
i+1 , . . . , v
m .
Let us make the intuitive concept of cyclic operads based on cobwebs presented
at the beginning of this section more precise. The mathematical abstraction of a
cobweb will be a graph. In the traditional approach, a graph consists of vertices and
edges connecting these vertices. In the context of operads, one needs to distinguish
between internal edges (those connecting two vertices) and external ones (legs) with
a “free end” along which graphs can be glued together, see Definition 6.9. This needs
a refinement of the classical definition. We use the one suggested by M. Kontsevich:
Definition 6.8 A graph Γ is a finite set Flag(Γ ) (whose elements are called flags
or half-edges) together with an involution σ and a partition λ.
The vertices Vert(Γ ) of a graph Γ are the blocks of the partition λ. The edges
Edg(Γ ) are pairs of flags forming a two-cycle of σ relative to the decomposition of
a permutation into disjoint cycles. The legs Leg(Γ ) are the fixed points of σ .
We also denote by Leg(v) the flags belonging to the block v or, in common
speech, half-edges adjacent to the vertex v. The cardinality of Leg(v) is the valency
of v. We say that two flags x, y ∈ Flag(Γ ) meet if they belong to the same block of
the partition λ. In plain language, this means that they share a common vertex.
One associates to a graph Γ a finite one-dimensional cell complex |Γ |, obtained
by taking one copy of [0,
1
2 ] for each flag and imposing the following equivalence
relation: The points 0 ∈ [0,
1
2 ] are identified for all flags in a block of the
partition λ and the points
1
2 ∈ [0,
1
2 ] are identified for pairs of flags exchanged
by the involution σ . We call |Γ | the geometric realization of the graph. We will
sometimes make no distinction between the graph in the sense of Definition 6.8 and