Algebraic operads by Loday J.-L., Vallette B.
By Loday J.-L., Vallette B.
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When A is augmented and C coaugmented, a twisting morphism between C and A ¯ is supposed to send K to 0 and C to A. 9. For every augmented dga algebra A and every conilpotent dga coalgebra C there exist natural bijections Homdga alg (ΩC, A) ∼ = Tw(C, A) ∼ = Homdga coalg (C, BA) . Proof. Let us make the first bijection explicit. Since ΩC = T (s−1 C) is a free algebra, any morphism of algebras from ΩC to A is characterized by its restriction to C (cf. 4). Let ϕ be a map from C to A of degree −1. Define the map ϕ¯ : s−1 C → A of degree 0 by the formula ϕ(s ¯ −1 c) := ϕ(c).
For instance Sh1 is one-dimensional spanned by [1 2] − [2 1]. 9 ([Ree58]). Let Lie(n) the functorial inclusion Lie(n) ⊗Sn V ⊗n Lie(V ) = K[Sn ] be the inclusion deduced from K[Sn ] ⊗Sn V ⊗n . T (V ) = n n Under the isomorphism K[Sn ] ∼ = K[Sn ]∗ obtained by taking the dual basis, the kernel ∗ ∗ Ker K[Sn ] → Lie(n) is the subspace Shn of K[Sn ] spanned by the nontrivial shuffles: Ker K[Sn ]∗ → Lie(n)∗ = Shn . Proof. Since the graded dual of T (V ) is T (V )∗ = T c (V ∗ ), and since the dual of the primitives are the indecomposables (cf.
4. Small coalgebras. Make explicit the coalgebra which is the linear dual of the dual numbers algebra K[t]/(t2 = 0), resp. the group algebra K[t]/(t2 = 1). Are they conilpotent ? 26 1. 5. Polynomial algebra. 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.