Читать книгу Algebra and Applications 2 - Группа авторов - Страница 32

Let us now look at the prototype of algebraic operads: for any vector space V, the operad Endop(V) is given by:

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The right action of the symmetric group Sn on Endop(V)_n is induced by the left action of Sn on V^⊗n given by:

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Elements of Endop(V)_n are conveniently represented as boxes with n inputs and one output: as illustrated by the graphical representation below, the partial composition a ∘_i b is given by:

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The following result is straightforward:

PROPOSITION 1.13.– For any a ∈ Endop(V)_k, b ∈ Endop(V)_l and c ∈ Endop(V)_m, we have:

The identity e: V → V satisfies the following unit property:

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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: