108
6 Operads
Proof. Each ω : {1, . . . , n}
∼ =
→ S and ω : {1, . . . , m}
∼ =
→ S determine an
isomorphism
ω
ω
: {1, . . . , n + m}
∼ =
−→ S
S
by the formula
(ω
ω
)(i) :=
ω (i),
if 1 ≤ i ≤ n, and
ω (i − n), if n < i ≤ n + m .
The isomorphism of the lemma is then given by the assignment
[v ω (1) ⊗· · ·⊗v ω (n) ]⊗[v ω (1) ⊗· · ·⊗v ω (m) ] → [v (ω ω )(1) ⊗· · ·⊗v (ω ω )(n+m) ].
Example 6.4 Let S = {c 1 , . . . , c n }. By iterating Lemma 6.2 one obtains a canonical
isomorphism
c∈S
V c ∼ = V c 1 ⊗ · · · ⊗ V c n
which, crucially, depends on the order of elements of S.
Example 6.5 Let ↑ k be the one-dimensional graded vector space concentrated in
degree +1, S = {1, . . . , n} and ↑ k i :=↑ k for each i ∈ S. Then
i∈S ↑ k i is
the one-dimensional vector space concentrated in degree n. Action (6.14) applied
to isomorphisms σ : {1, . . . , n} → {1, . . . , n}, i.e., to elements of the symmetric
group Σ n , is the signum representation.
We are ready to define the endomorphism operad.
Example 6.6 Let V be a graded vector space and s ∈ V ⊗ V a symmetric degree
0 tensor. Its symmetry means that τ (s) = s, where τ is the flip (3). For a finite set
S put
End V (S) := Lin
c∈S V c , k
= (
c∈S V c ) # ,
where V c := V for each c ∈ S. Given an isomorphism σ : S → D of finite sets,
define
End V (σ ) : End V (S) → End V (D)
by End V (σ )(f ) := f σ −1 for f :
c∈S V c → k ∈ End V (S) and σ as in (6.14).