Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

so that, since L is a subalgebra of H we get that, for every ∑ ∑i j li,j ⊗ h i,j ⊗ g i ∈(L ⊗ H) □ C H,∑ ∑(217)i j li,j S ( h i,j) g i ∈ L.Let us consider the following mapi∑i∑̂χ : (H ⊗ H) □ C H → H∑j ki,j ⊗ h i,j ⊗ g i ↦→ ∑ ij ki,j S ( h i,j) g iwhich is left C-colinear. In fact, in view of (215), we have that ∑ iS (h i,j ) g i ∈ H ⊗ L and by (212), we get that∑ ( ∑j π [ (k ) i,j(1) S hi,jg i] [ ((1))⊗k ) i,j(2) S hi,jg i] = ∑ (k(2) i∑j π i,j(1)Therefore, we can define the coherd χ = −□ C ̂χ given byχ : QP Q = −□ C H ⊗ H□ C H → Q = −□ C H∑∑ ∑−□ Ci∑j ki,j ⊗ h i,j ⊗ g i ↦→ −□ Ci j ki,j S ( h i,j) g i .Let us prove the properties of χ. We have[χ ◦ (QP χ)](−□ C k ⊗ ∑ )h i ⊗ g i ⊗ l j ⊗ n j= [χ ◦ (χ ⊗ H□ C H)](−□ C k ⊗ ∑ )h i ⊗ g i ⊗ l j ⊗ n j∑ (= χ(−□ ) ) ∑ ( (C kS hig i ⊗ l j ⊗ n j = −□ ) C kS hig i) S ( l j) n jand[χ ◦ (χP Q)](−□ C k ⊗ ∑ )h i ⊗ g i ⊗ l j ⊗ n j= [χ ◦ (−□ C H ⊗ H□ C χ)](k ⊗ ∑ )h i ⊗ g i ⊗ l j ⊗ n j= χ(k ⊗ ∑ h i ⊗ g i S ( l j) ) ∑ (n j = −□ ) ( C kS hig i S ( l j) n j)191∑j ki,j ⊗)⊗k i,j(2) S ( h i,j) g i .so that χ is coassociative. Moreover, we have[χ ◦ (δ C Q)] (k ⊗ h) = [χ ◦ (−□ C H ⊗ δ C )] (−□ C k ⊗ h) = χ ( −□ C k ⊗ h (1) ⊗ h (2))and= −□ C kS ( h (1))h(2) = kε H (h) = ( −□ C H ⊗ ε H) (k ⊗ h) = ( ε C Q ) (k ⊗ h)∑ )[χ ◦ (Qδ D )](−□ C k i ⊗ h i∑ )= [χ ◦ (δ D □ C H)](−□ C k i ⊗ h i)hi∑ ) ∑= χ(−□ C ki(1) ⊗ k(2) i ⊗ h i = −□ C ki(1) S ( k(2)i= ∑ ε H ( k i) 1 H h i = ∑ ε H ( k i) h i = −□ C∑ (ε C ◦ π ) ( k i) h i∑= −□ ( C εCπ ( k i)) ∑ (h i ≃ −□ ) C π ki□ C h i∑ (= −□ ) ( ) ∑ )C ε C H C ki□ C h i = ε C H C □ C H(−□ C k i ⊗ h i