6.1 Cyclic Operads
113
In this situation we write s := B −1 and call s the Casimir element associated with
the non-degenerate bilinear form B. We say that s ∈ V ⊗ V is non-degenerate if
there exists a non-degenerate symmetric bilinear form B such that s = B −1 .
As an exercise we recommend to verify that for B resp. s non-degenerate, (6.21)
and (6.22) are automatically satisfied. The pictorial language used above makes this
statement obvious. One easily proves:
Lemma 6.4 The following conditions are equivalent.
(i) The collection Φ S : End V (S) → Dne V (S) is an isomorphism of operads,
(ii) the collection Ψ S : Dne V (S) → End V (S) is an isomorphism of operads,
(iii) the symmetric bilinear form B : V ⊗ V → k is non-degenerate and s = B −1 .
Example 6.8 We are going to describe the skeletal version of the cyclic endomorphism operad End V introduced in Example 6.6. Notice that for S = [n] =
{1, . . . , n}, n ≥ 1, the unordered tensor product
i∈[n] V i of Definition 6.6 is
canonically isomorphic to the ordinary tensor product V 1 ⊗ · · · ⊗ V n . In particular,
if V i = V for each 1 ≤ i ≤ n, then
i∈[n] V i is canonically isomorphic to V ⊗n ,
therefore
End V (n) := End V
[n]
∼ = Lin(V
⊗n , k).
(6.24)
To shorten the formulas, we will write, for e.g., f ∈ End V (n), f (v 1 , . . . , v n )
instead of f (v 1 ⊗ · · · ⊗ v n ). We will also use a variation of Sweedler’s notation and
write the symmetric element s ∈ V ⊗ V as the formal finite sum s =
s
i ⊗s
i .
Under identification (6.24), a permutation σ ∈ Σ n acts on a function f ∈
End V (n), n ≥ 1, by
(σf )(v 1 , . . . , v n ) = (σ )f
v σ (1) , . . . , v σ (n)
, v 1 , . . . , v n ∈ V ,
with (σ ) the Koszul sign ((1) in Part II). For functions f ∈ End V (m + 1), g ∈
End V (n+1), 1 ≤ i ≤ m + 1 and 1 ≤ j ≤ n + 1, one calculates that
(f i ◦ j g)(v 1 , . . . , v m+n )
(6.25)
=
κ
f (v 1 , ..., v i−1 , s
i , v i+n , ..., v m+n )
g(v n+i−j +1 , ..., v i+n−1 , s
i , v i , ..., v n+i−j )
with
κ =|f ||g| + |s
i |(|v i+n | + · · · + |v m+n |) + |s
i |(|v n+i−j +1 | + · · · + |v i+n−1 |)
(6.26)
113
In this situation we write s := B −1 and call s the Casimir element associated with
the non-degenerate bilinear form B. We say that s ∈ V ⊗ V is non-degenerate if
there exists a non-degenerate symmetric bilinear form B such that s = B −1 .
As an exercise we recommend to verify that for B resp. s non-degenerate, (6.21)
and (6.22) are automatically satisfied. The pictorial language used above makes this
statement obvious. One easily proves:
Lemma 6.4 The following conditions are equivalent.
(i) The collection Φ S : End V (S) → Dne V (S) is an isomorphism of operads,
(ii) the collection Ψ S : Dne V (S) → End V (S) is an isomorphism of operads,
(iii) the symmetric bilinear form B : V ⊗ V → k is non-degenerate and s = B −1 .
Example 6.8 We are going to describe the skeletal version of the cyclic endomorphism operad End V introduced in Example 6.6. Notice that for S = [n] =
{1, . . . , n}, n ≥ 1, the unordered tensor product
i∈[n] V i of Definition 6.6 is
canonically isomorphic to the ordinary tensor product V 1 ⊗ · · · ⊗ V n . In particular,
if V i = V for each 1 ≤ i ≤ n, then
i∈[n] V i is canonically isomorphic to V ⊗n ,
therefore
End V (n) := End V
[n]
∼ = Lin(V
⊗n , k).
(6.24)
To shorten the formulas, we will write, for e.g., f ∈ End V (n), f (v 1 , . . . , v n )
instead of f (v 1 ⊗ · · · ⊗ v n ). We will also use a variation of Sweedler’s notation and
write the symmetric element s ∈ V ⊗ V as the formal finite sum s =
s
i ⊗s
i .
Under identification (6.24), a permutation σ ∈ Σ n acts on a function f ∈
End V (n), n ≥ 1, by
(σf )(v 1 , . . . , v n ) = (σ )f
v σ (1) , . . . , v σ (n)
, v 1 , . . . , v n ∈ V ,
with (σ ) the Koszul sign ((1) in Part II). For functions f ∈ End V (m + 1), g ∈
End V (n+1), 1 ≤ i ≤ m + 1 and 1 ≤ j ≤ n + 1, one calculates that
(f i ◦ j g)(v 1 , . . . , v m+n )
(6.25)
=
κ
f (v 1 , ..., v i−1 , s
i , v i+n , ..., v m+n )
g(v n+i−j +1 , ..., v i+n−1 , s
i , v i , ..., v n+i−j )
with
κ =|f ||g| + |s
i |(|v i+n | + · · · + |v m+n |) + |s
i |(|v n+i−j +1 | + · · · + |v i+n−1 |)
(6.26)