Lie algebra axioms (equation [1.60]), involving flip and circular permutations, are clearly rewritten as:
where τ is the flip (21) and σ is the circular permutation (231). The next section will give a precise meaning to these “partial compositions”, and we will end up giving the axioms of an operad, which is the natural framework in which equations like [1.61] and [1.62] make sense.
1.5.2. The operad of multi-linear operations
Let us now look at the prototype of algebraic operads: for any vector space V, the operad Endop(V) is given by:
[1.63]
The right action of the symmetric group Sn on Endop(V)n is induced by the left action of Sn on V⊗n given by:
[1.64]
Elements of Endop(V)n are conveniently represented as boxes with n inputs and one output: as illustrated by the graphical representation below, the partial composition a ∘i b is given by:
[1.65]
The following result is straightforward:
PROPOSITION 1.13.– For any a ∈ Endop(V)k, b ∈ Endop(V)l and c ∈ Endop(V)m, we have:
The identity e: V → V satisfies the following unit property:
[1.66]
[1.67]
and finally, the following equivariance property is satisfied:
where is definedby letting permute the set Ei = {i, i + 1,…, i + l – 1} of cardinality l, and then by letting σ permute the set {1,…,i – 1, Ei, i + l,…, k + l – 1} of cardinality k.
The two associativity properties are graphically represented as follows:
1.5.3. A definition for linear operads
We are now ready to give the precise definition of a linear operad:
DEFINITION 1.1.– An operad (in the symmetric monoidal category of k-vector spaces) is given by a collection of vector spaces
subject to the associativity, unit and equivariance axioms of Proposition 1.13.
The global composition is defined by:
and is graphically represented as follows:
The operad
where we have denoted by the same letter γ the element of
Now let V be any k-vector space. The free -algebra is a
