Читать книгу Algebra and Applications 2 - Группа авторов - Страница 35

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.