Algebra and Applications 2

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and finally, the following equivariance property is satisfied :

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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:

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 ) nand 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:

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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).

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

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