Читать книгу Algebra and Applications 2 - Группа авторов - Страница 33

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 , a right action of the symmetric group Sn on , 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).