By Jean-Louis Loday, Bruno Vallette

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**Example text**

Observe that the identity permutation [1 2 3] is both a (1, 2)-shuffle and a (2, 1)-shuffle. The notion of (i1 , . . , ik )-shuffle and the set Sh(i1 , . . , ik ) are defined analogously. Following Jim Stasheff, we call unshuffle the inverse of a shuffle. For instance the three (1, 2)-unshuffles are [1 2 3], [2 1 3] and [3 1 2]. We denote the set of (p, q)-unshuffles by Sh−1 p,q . 3. For any n = p + q and σ ∈ Sn there exist unique permutations α ∈ Sp , β ∈ Sq and ω ∈ Sh(p, q) such that: σ = ω · (α × β).

Let K[x] = K 1 ⊕ K x ⊕ · · · ⊕ K xn ⊕ · · · . It is a unital commutative algebra for the product xn xm = xn+m . Show that the product xn ∗ xm := n + m n+m x n makes it also a unital commutative algebra, that we denote by Γ(K x). Compute explicitly: a) the dual coalgebra of K[x] and of Γ(K x), b) the coalgebra structure of K[x], resp Γ(K x), which makes it a bialgebra and which is uniquely determined by ∆(x) = x ⊗ 1 + 1 ⊗ x. c) Compare the results of a) and b). 6. Polynomial algebra continued. e.

5. DIFFERENTIAL GRADED ALGEBRA 21 The filtration is bounded below whenever, for each n, there exists k such that Fp Vn = 0 for any p < k. The filtration is exhaustive if Vn = ∪p Fp Vn . e. E 1 ) first and then the homology. The main theorem about spectral sequences compares these two procedures, cf. for instance Chapter 11 of [ML95]. 7 (Classical convergence theorem of spectral sequences). If the filtration F• V of the chain complex V = (V• , d) is exhaustive and bounded below, then the spectral sequence converges.

### Algebraic Operads (version 0.99, draft 2010) by Jean-Louis Loday, Bruno Vallette

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