 By Jungnickel D.H. (ed.), Vedder K. (ed.)

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Sample text

8. Let H = n 0 H n be a connected, graded Hopf algebra over R, such that each homogeneous component H n is finite-dimensional. Define the module H ∗ by H∗= (H n )∗ , n 0 n )∗ where (H denotes the set of all linear maps f : H n → R. ∗ Then H is a Hopf algebra with 1. product m : H ∗ ⊗ H ∗ → H ∗ induced by the convolution product f ∗ g = mR ◦ ( f ⊗ g) ◦ ΔH , where mR is the product of R, and ΔH is the coproduct of H (in Sweedler notation, the convolution product is given by ( f ∗ g)(h) = ∑ f (h1 )g(h2 ) for all h ∈ H ), 2.

Then D(w, γ ) = set(α ), since 1. i, i + 1 ∈ set(α ) implies γ (wi ) > γ (wi+1 ), 2. j, j + 1 ∈ [n] − set(α ) implies γ (w j ) < γ (w j+1 ), 3. i ∈ set(α ) and j ∈ [n] − set(α ) implies γ (wi ) > γ (w j ). 21. If α is the composition (3, 2, 4) 9, then set(α ) = {3, 5} and [n] − set(α ) = {1, 2, 4, 6, 7, 8, 9}. Let w be the chain with order w1 < · · · < w9 and labelling γ that respectively maps w3 , w5 → 9, 8 and w1 , w2 , w4 , w6 , w7 , w8 , w9 → 1, 2, 3, 4, 5, 6, 7. 40 3 Hopf algebras Then γ respectively maps w1 , .

Ik ) of indices i1 < · · · < ik . We define M0/ = 1. 5. We have M(2,1) = x21 x12 + x21 x13 + x21 x14 + x22 x13 + · · · while M(1,2) = x11 x22 + x11 x23 + x11 x24 + x12 x23 + · · · . Since the Mα are independent we have QSymn = span{Mα | α n}. A closely related basis is the basis of fundamental quasisymmetric functions. 6. Let α be a composition. Then the fundamental quasisymmetric function Fα is defined by Fα = ∑ Mβ . β α If α n, then in terms of the variables x1 , x2 , . . we have Fα = ∑ xi1 · · · xin , where the sum is over all n-tuples (i1 , .