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

196)û E(ε C H C (h) = û E (π (h)) = [ ]h (1) ⊗ h (2) . ELet us check that ̂m E is a well-defined map. Let us considerf : H ⊗ H → Hh ⊗ k ↦→ hS (k) .For every (h ⊗ k) · l ∈ (H ⊗ H) L + , we havef [( hl (1) ⊗ kl (2))− (h ⊗ k) ε (l)]= hl(1) S ( l (2))S (k) − hS (k) ε (l) = 0so that f induces a morphismH ⊗ Hf :(H ⊗ H) L → H +[h ⊗ k] ↦→ hS (k) .Now, let us consider the compositeH ⊗ H(H ⊗ H) L ⊗ H ⊗ H f⊗Id H⊗Id Hm−→ H ⊗ H ⊗ HH ⊗Id Hp−→ H ⊗ H −→H ⊗ H+ (H ⊗ H) L +where p denotes the canonical projection. Note that[p ◦ (m H ⊗ H)] ( H ⊗ (H ⊗ H) L +) = 0in fact, for every x ∈ H and (h ⊗ k) · l ∈ (H ⊗ H) L + we haveand thusx ⊗ [( hl (1) ⊗ kl (2))− (h ⊗ kε (l))]= x ⊗(hl(1) ⊗ kl (2))− x ⊗ (h ⊗ kε (l))xhl (1) ⊗ kl (2) − xh ⊗ kε (l) = (xh ⊗ k) (l − ε (l)) ∈ (H ⊗ H) L + .Therefore, the above composite map induces the mapH ⊗ H(H ⊗ H) L ⊗ H ⊗ H+ (H ⊗ H) L → H ⊗ H+ (H ⊗ H) L +[k ⊗ h] ⊗ [g ⊗ l] ↦→ [kS (h) g ⊗ l]which is well-defined and hence also the mapH ⊗ Ĥm E :(H ⊗ H) L □ H ⊗ H+ C(H ⊗ H) L → H ⊗ H+ (H ⊗ H) L +∑ [ ] k ⊗ hi ⊗ [ g i ⊗ l ] [∑ ↦→ ( )E E kS hig i ⊗ l].is well defined. Observe that, by using (213) and (212) we have∑ ( ( (π k ) (1) S hig i) ) ( (⊗ k ) (1) (2) S hig i) = ∑ π ( ) ( (k ) (2) (1) ⊗ k(2) S hig i)so that the maps∑ [ ] k ⊗ hi ⊗ [ g i ⊗ l ] [∑ ↦→ ( ) ]E E kS hig i ⊗ l∑k ⊗[h i ⊗ g i] A ↦→ ∑ kS ( h i) g iEE,and∑ [k ⊗ hi ] E ⊗ gi ↦→ ∑ kS ( h i) g i