⊂
[1.57]
for any a, b ∈ A.
1.5. Algebraic operads
An operad is a combinatorial device which appeared in algebraic topology (May 1972), coined for coding “types of algebras”. Hence, for example, a Lie algebra is an algebra over some operad denoted by LIE, an associative algebra is an algebra over some operad denoted by ASSOC, a commutative algebra is an algebra over some operad denoted by COM and so on.
1.5.1. Manipulating algebraic operations
Algebra starts, in most cases, with some set E and some binary operation * : E × E → E. The set E shows some extra structure most of the time. Here, we will stick to the linear setting, where E is replaced by a vector space V (over some base field k), and * is bilinear, that is, a linear map from V ⊗ V into V. A second bilinear map is deduced from the first by permuting the entries:
[1.58]
It also makes sense to look at tri-, quadri- and multi-linear operations, that is, linear maps from V⊗n to V for any V. For example, it is very easy to produce 12 tri-linear maps starting with the bilinear map * by considering:
and the others deduced by permuting the three entries a, b and c. We could also introduce some tri- or multi-linear operations from scratch, that is, without deriving them from the bilinear operation *. We can even consider 1-ary and 0-ary operations, the latter being just distinguished elements of V. Note that there is a canonical 1-ary operation, namely, the identity map e : V → V. At this stage note that the symmetric group Sn obviously acts on the n-ary operations on the right by permuting the entries before composing them.
The bilinear operation * is not arbitrary in general: its properties determine the “type of algebra” considered. For example, V will be an associative or a Lie algebra if for any a, b, c ∈ V, we have respectively:
The concept of operad emerges when we try to rewrite such relations in terms of the operation * only, discarding the entries a, b, c . For example, the associativity axiom equation [1.59] informally expresses itself as follows: composing the operation * twice in two different ways gives the same result. Otherwise said:
The 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