116
6 Operads
by the formula
˜
f (v 1 , . . . , v n ) :=
(−1)
|s
i |(|v 1 |+···+|v n |) s
i · f (v 1 , . . . , v n , s
i ), v 1 , . . . , v n ∈ V .
When s is the Casimir element of a non-degenerate bilinear form, the map in (6.30)
is an isomorphism defining another incarnation of the skeletal endomorphism
operad, namely
End V (n) ∼ = Lin(V
⊗n−1 , V ), n ≥ 1.
The ◦ i -operations in (6.10) are, for 1 ≤ i ≤ m,
φ ∈ End V (m+1) ∼ = Lin(V
⊗m , V ), ψ ∈ End V (n+1) ∼ = Lin(V
⊗n , V )
and v 1 , . . . , v m+n−1 ∈ V , given by the formula
(φ ◦ i ψ)(v 1 , . . . , v m+n−1 )
= (−1)
κ φ
v 1 , . . . , v i−1 , ψ(v i , . . . , v i+n−1 ), v i+n , . . . , v m+n−1
with κ = |ψ|(|v 1 | + · · · + |v i−1 |). We recognize the classical form of the ◦ i -
operations in the endomorphism operad [9, Example 12] given by inserting ψ into
the ith input of φ.
Example 6.11 Let us describe the skeletal version of the dual cyclic endomorphism
operad Dne V introduced in Example 6.7. As in (6.24) we have for each n ≥ 1 the
canonical isomorphism
Dne V (n) := Dne V
[n]
∼ = V
⊗n ,
under which the symmetric group Σ n acts by
σ (v 1 ⊗ · · · ⊗ v n ) = v σ −1 (1) ⊗ · · · ⊗ v σ −1 (n) , v 1 , . . . , v n ∈ V .
For v
1 ⊗ · · · ⊗ v
m+1 ∈ Dne V (m+1) and v
1 ⊗ · · · ⊗ v
n+1 ∈ Dne V (n+1) one obtains
(v
1 ⊗ · · · ⊗ v
m+1 ) i ◦ j (v
1 ⊗ · · · ⊗ v
n+1 )
=
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