154
6 Operads
Remark 6.12 Odd modular operads have their skeletal versions. For T as in (6.77),
n ≥ 0 and g ∈ A, denote T (n; g) := T
[n]; g
. The degree +1 operations
i • j : T (m + 1; g 1 ) ⊗ T (n + 1; g 2 ) → T (m + n; g 1 + g 2 )
and
• ij : T (n + 2; g + s) → T (n; g), 1 ≤ i, j ≤ n + 2, g ∈ A
are defined by obvious formulas analogous to (6.8) resp. (6.70). The axioms for
these operations are the same as the skeletal axioms for modular operads, only the
axioms corresponding to (6.80)–(6.83) acquire the minus sign.
Example 6.32 The skeletal version of the odd endomorphism operad from Example 6.31 is described as follows. One has End V (n; g) ∼ = Lin(V ⊗n , k) for n ≥ 0, g ∈
A as in (6.72), with the skeletal i • j -operations defined by formula (6.25), but this
time with
κ = |g|(|f |+1) + |s
i | + |s
i |(|v i+n | + · · · + |v m+n |)
+ |s
i |(|v n+i−j +1 | + · · · + |v i+n−1 |).
The skeletal contractions • ij are given by formula (6.73) with
κ = |f | + (|v 1 | + · · · + |v i−1 |) + |s
i |(|v i | + · · · + |v j −2 |).
We leave the related straightforward verification to the reader.
Odd endomorphism operads are necessary for the definition of algebras over odd
modular operads.
Definition 6.25 Let V be a graded vector space and s ∈ V ⊗ V a symmetric
degree +1 tensor. An algebra over an odd modular operad T , or a T -algebra,
is a morphism α : T → End V of odd modular operads.
The main source of examples of algebras over odd modular operads will be
provided by algebras over the Feynman transform of a modular operads introduced
in Sect. 7.2.
Definition 6.26 For a finite set X, let ↑ k X be the free k-module with basis
X, considered as a dg-vector space concentrated in degree 1. Define the onedimensional dg-vector space concentrated in degree card(X):
det(X) = det(k
X ) := ∧
|X| (↑ k
X ),
6 Operads
Remark 6.12 Odd modular operads have their skeletal versions. For T as in (6.77),
n ≥ 0 and g ∈ A, denote T (n; g) := T
[n]; g
. The degree +1 operations
i • j : T (m + 1; g 1 ) ⊗ T (n + 1; g 2 ) → T (m + n; g 1 + g 2 )
and
• ij : T (n + 2; g + s) → T (n; g), 1 ≤ i, j ≤ n + 2, g ∈ A
are defined by obvious formulas analogous to (6.8) resp. (6.70). The axioms for
these operations are the same as the skeletal axioms for modular operads, only the
axioms corresponding to (6.80)–(6.83) acquire the minus sign.
Example 6.32 The skeletal version of the odd endomorphism operad from Example 6.31 is described as follows. One has End V (n; g) ∼ = Lin(V ⊗n , k) for n ≥ 0, g ∈
A as in (6.72), with the skeletal i • j -operations defined by formula (6.25), but this
time with
κ = |g|(|f |+1) + |s
i | + |s
i |(|v i+n | + · · · + |v m+n |)
+ |s
i |(|v n+i−j +1 | + · · · + |v i+n−1 |).
The skeletal contractions • ij are given by formula (6.73) with
κ = |f | + (|v 1 | + · · · + |v i−1 |) + |s
i |(|v i | + · · · + |v j −2 |).
We leave the related straightforward verification to the reader.
Odd endomorphism operads are necessary for the definition of algebras over odd
modular operads.
Definition 6.25 Let V be a graded vector space and s ∈ V ⊗ V a symmetric
degree +1 tensor. An algebra over an odd modular operad T , or a T -algebra,
is a morphism α : T → End V of odd modular operads.
The main source of examples of algebras over odd modular operads will be
provided by algebras over the Feynman transform of a modular operads introduced
in Sect. 7.2.
Definition 6.26 For a finite set X, let ↑ k X be the free k-module with basis
X, considered as a dg-vector space concentrated in degree 1. Define the onedimensional dg-vector space concentrated in degree card(X):
det(X) = det(k
X ) := ∧
|X| (↑ k
X ),