Algebra and Applications 2. Группа авторов

a distinguished element and a collection of partial compositions:

      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 is augmented if and . For any operad , a -algebra structure on the vector space V is a morphism of operads from to Endop(V). For any two -algebras V and W, a morphism of -algebras is a linear map f : V → W, such that for any n ≥ 0 and for any , the following diagram commutes,

      where we have denoted by the same letter γ the element of and its images in Endop(V)_n and Endop(W)_n.

      Now let V be any k-vector space. The free -algebra is a -algebra endowed with a linear map , such that for any -algebra A and for any linear map f : V → A, there is a unique -algebra morphism , such that . The free -algebra is unique up to isomorphism, and we can prove that a concrete presentation of it is given by:

[1.69]

      with the map ι being obviously defined. When V is of finite dimension d, the corresponding free -algebra is often called the free -algebra with d generators.

      There are several other equivalent definitions for an operad. For more details about operads, see, for example, Loday (1996) and Loday and Vallette (2012).

      1.5.4. A few examples of operads

      1.5.4.1. The operad ASSOC

      This operad governs associative algebras. ASSOC_n is given by k[Sn] (the algebra of the symmetric group Sn) for any n ≥ 0, whereas ASSOC₀ := {0}. The right action of Sn on ASSOC_n is given by linear extension of right multiplication:

      [1.70]

      Let σ ∈ ASSOC_k and . The partial compositions are given for any i = 1,…,k by:

      [1.71]

with the notations of equation [1.68]. The reader is invited to check the two associativity axioms, as well as the equivariance axiom which reads:

      [1.72]

      for any σ, σ′ ∈ ASSOC_k and . Let us denote by ek the unit element in the symmetric group Sk. We obviously have ek ∘_i el = e_{k + l – 1} for any i = 1,…, k. In particular,

[1.73]

Now let V be an algebra over the operad ASSOC, and let Φ : ASSOC → Endop(V) be the corresponding morphism of operads. Let μ : V ⊗ V → V be the binary operation Φ(e2). In view of equation [1.73] we have:

      [1.74]

In other words, μ is associative. As ek can be obtained, for any k ≥ 3, by iteratively composing k – 2 times the element e₂, we see that any element of ASSOC_k can be obtained from e₂, partial compositions, symmetric group actions and linear combinations. As a result, any k-ary operation on V, which is in the image of Φ, can be obtained in terms of the associative product μ, partial compositions, symmetric group actions and linear combinations. Summing up, an algebra over the operad ASSOC is nothing but an associative algebra. In view of equation [1.69], the free ASSOC-algebra over a vector space W is the (non-unital) tensor algebra .

      In the same line of thoughts, the operad governing unital associative algebras is defined similarly, except that the space of 0-ary operations is k.e₀, with ek ∘_i e₀ = e_{k – 1} 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 tensor algebra .

      1.5.4.2. The operad