Algebra and Applications 2. Группа авторов

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alt="image"/> 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 COM

      This operad governs commutative associative algebras. COM_n is one-dimensional for any n ≥ 1, given by for any n ≥ 0, whereas COM₀ := {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

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