operad governs commutative associative algebras. COM
n is one-dimensional for any
n ≥ 1, given by
for any
n ≥ 0, whereas COM
0 := {0}. The right action of
Sn on COM
n is trivial. The partial compositions are defined by:
[1.75]
The three axioms of an operad are obviously verified. Let V be an algebra over the operad COM, and let Φ : COM → Endop(V) be the corresponding morphism of operads. Let μ : V ⊗ V → V be the binary operation . We obviously have:
[1.76]
where is the flip. Hence, μ is associative and commutative. Here, any k-ary operation in the image of Φ can be obtained, up to a scalar, by iteratively composing with itself. Hence, an algebra over the operad COM is nothing but a commutative associative algebra. In view of [1.69], the free COM-algebra over a vector space W is the (non-unital) symmetric algebra .
The operad governing unital commutative associative algebras is defined similarly, except that the space of 0-ary operations is , with for any i = 1,…,k. The unit element u : k → V of the algebra V is given by u = Φ(e0). The free unital algebra over a vector space W is the full symmetric algebra .
The map is easily seen to define a morphism of operads Ψ : ASSOC → COM. Hence, any COM-algebra is also an ASSOC-algebra. This expressed the fact that, forgetting commutativity, a commutative associative algebra is also an associative algebra.
1.5.4.3. Associative algebras
Any associative algebra A is some degenerate form of operad: indeed, defining by and for n ≠ 1, the collection is obviously an operad. An algebra over is the same as an A-module.
This point of view leads to a more conceptual definition of operads: an operad is nothing but an associative unital algebra in the category of “S-objects”, that is, collections of vector spaces with a right action of Sn on . There is a suitable “tensor product” ⌧ on S-objects, however not symmetric, such that the global composition γ and the unit (defined by u(1) = e) make the following diagrams commute:
These two diagrams commute if and only if e verifies the unit axiom and the partial compositions verify the two associativity axioms and the equivariance axiom (Loday and Vallette 2012).
1.6. Pre-Lie algebras (continued)
1.6.1. Pre-Lie algebras and augmented operads
1.6.1.1. General construction
We adopt the notations of section 1.5. The sum of the partial compositions yields a right pre-Lie algebra structure on the free -algebra with one generator, more precisely on , namely:
[1.77]
Following Chapoton (2002), we can consider the pro-unipotent group associated with the completion of the pre-Lie algebra for the filtration induced by the grading. More precisely, Chapoton’s group is given by the elements , such that g1 ≠ 0, whereas is the subgroup of formed by elements g, such that g1 = e.
Any element gives rise to an n-ary operation , and for any , we have1 (Manchon and Saidi 2011):
[1.78]
1.6.1.2. The pre-Lie operad
Pre-Lie algebras are algebras over the pre-Lie operad, which has been described in detail by Chapoton and Livernet (2001) as follows: is the vector space of labeled rooted trees, and the partial composition s ∘i t is given by summing all of the possible ways of inserting the tree t inside the tree s at the vertex labeled by i. To be precise, the sum runs over the possible ways of branching on t the edges of s, which arrive on the vertex i.
The free left pre-Lie algebra with one generator is then given by the space of rooted trees, as quotienting with the symmetric group actions amounts to neglect the labels. The pre-Lie operation (s,t) ↦ (s → t) is given by the sum of the graftings of s on t at all vertices of t. As a result of [1.78],
