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

denote the canonical inclusion. Then i is a left C-comodule map, in fact, for every∑i (h i ⊗ g i ) ( l i − ε H (l i ) ) = ∑ i h il i(1) ⊗ g i l i(2) − h i ⊗ g i ε H (l i ) ∈ (H ⊗ H) L + , since∆ H (l i ) = l i(1) ⊗ l i(2) ∈ L ⊗ H, we have∑i π ( h i(1) l i(1))⊗ hi(2) l i(2) ⊗ g i l i(3) − π ( h i(1))⊗ hi(2) ⊗ g i ε H (l i )= ∑ i π ( h i(1))⊗ hi(2) l i(1) ⊗ g i l i(2) − π ( h i(1))⊗ hi(2) ⊗ g i ε H (l i )= ∑ i π ( h i(1))⊗(hi(2) ⊗ g i) (li − ε H (l i ) ) ∈ C ⊗ (H ⊗ H) L + .Hence, for every ( X, ρ C X)∈ Comod-C, we can consider the mapX□ C i : X□ C (H ⊗ H) L + → X□ C (H ⊗ H) .so that, for every ( X, ρ C X)∈ Comod-C, we havewhereI X□C L =Let p : H ⊗ H →E ( X, ρ C X)=X□ C H ⊗ HI X□C z= X□ CH ⊗ HI X□C L〈 ∑i xi ⊗ h i y (1) ⊗ gy (2) − ∑ i xi ⊗ h i ⊗ gε H (y)| ∑ i xi ⊗ h i ⊗ g ⊗ y ∈ X□ C H ⊗ H ⊗ L= X□ C[(H ⊗ H) L+ ] .H⊗H(H⊗H)L +〉195be the canonical projection and let us assume that i isleft C-copure i.e. for every ( X, ρ C X)∈ Comod-C, the sequence0 → X□ C (H ⊗ H) L + X□ Ci−→ X□ C (H ⊗ H) X□ Cp−→ X□ Cis exact. In this case we get that, for every ( X, ρX) C ∈ Comod-C,E ( )X, ρ C ∼= H ⊗ HX X□C(H ⊗ H) L = X□ + C (H ⊗ H) LH ⊗ H(H ⊗ H) L → 0 +where (H ⊗ H) Ldenotes the invariants with respect to the algebra L. In the sequel,given h i , k i ∈ H we will use the notation][∑i hi ⊗ k i = ∑ i hi ⊗ k i + (H ⊗ H) L +ELet us denote E := −□ C (H ⊗ H) Land let us consider multiplication and unit ofE. Following Theorem 6.29, they are uniquely determined byi.e.wherêm E :m E ◦ (yy) = y ◦ (P χ) and y ◦ δ D = u E ◦ ε Dm E = −□ Ĉm E and u E = −□ C û EH ⊗ H(H ⊗ H) L □ H ⊗ H+ C(H ⊗ H) L −→ H ⊗ H+ (H ⊗ H) L and û + E : C −→ H ⊗ H(H ⊗ H) L +given by(∑ ∑ ∑ [̂m E k i,j ⊗ h i,j] □ [E C g i,s ⊗ l i,s] [∑E)= k i,j S ( h i,j) ]g i,s ⊗ l i,sE