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

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

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[( ) ( ) ( )= A U ˜CλA ˜CA F ◦ ˜CA F Cu A ◦ λ A ˜CA F C ◦ ( A F Cu A C) ◦ ( AF ∆ C)][( ) (λ=AA U λ A ˜C ˜CA F ◦ AF A U ˜Cλ)A ˜CA F ◦(AF A U ˜C)A F Cu A ◦ ( A F Cu A C) ◦ ( AF ∆ C)][( ) ()= A U λ A ˜C ˜CA F ◦ AF C A Uλ A ˜CA F ◦ ( A F C A U A F Cu A ) ◦ ( A F Cu A C) ◦ ( AF ∆ C)][( ) ()u=AA U λ A ˜C ˜CA F ◦ AF C A Uλ A ˜CA F ◦ ( A F Cu A CA) ◦ ( A F CCu A ) ◦ ( AF ∆ C)][( ) () (= A U λ A ˜C ˜CA F ◦ AF C A Uλ A ˜CA F ◦ AF Cu AA U ˜C)A F ◦ ( A F CCu A ) ◦ ( AF ∆ C)][( )(λ A ,u A )adj,∆= CAU λ A ˜C ˜CA F ◦ ( AF ∆ C A ) ]◦ ( A F Cu A )[( ) (= A U λ A ˜C ˜CA F ◦ AF A U∆ e ) ]CA F ◦ ( A F Cu A )[(λ=AA U ∆ e ) ( ) ]CA F ◦ λ A ˜CA F ◦ ( A F Cu A )(= AU∆ e ) ( )CA F ◦ AUλ A ˜CA F ◦ ( A U A F Cu A ) = ( ∆ C A ) ◦ (Φ)85so thatMoreoverso that(CΦ) ◦ (ΦC) ◦ ( A∆ C) = ( ∆ C A ) ◦ (Φ) .(ε C A ) ◦ (Φ) = ( ε C A ) ( )◦ AUλ A ˜CA F ◦ ( A U A F Cu A )(= AUε e ) ( )CA F ◦ AUλ A ˜CA F ◦ ( A U A F Cu A )[(= A U ε e ) ( ) ]CA F ◦ λ A ˜CA F ◦ ( A F Cu A )[ (λ=AA U (λ AA F ) ◦ AF A Uε e ) ]CA F ◦ ( A F Cu A )= A U [ (λ AA F ) ◦ ( AF ε C AU A F ) ◦ ( A F Cu A ) ]ε C = A U [ (λ AA F ) ◦ ( A F u A ) ◦ ( AF ε C)](λ A ,u A )adj= AU A F ε C = Aε C(ε C A ) ◦ (Φ) = Aε C .Therefore Φ is a mixed distributive law.C<strong>on</strong>versely let Φ ∈ D. Then we know that b (Φ) = ˜C is a functor ˜C : A A → A Athat is a lifting of C (i.e. AU ˜C = C A U). We have to prove that such a ˜C givesrise to a com<strong>on</strong>ad <strong>on</strong> the category A A. Let us prove that ∆ C and ε C are A-modulesmorphisms. Indeed, for every ( X, A µ X)∈ A A, by Lemma 5.3 we haveA µ CX = ( C A µ X)◦ (ΦX)and alsoA µ CCX = ( C A µ CX)◦ (ΦCX) =(CC A µ X)◦ (CΦX) ◦ (ΦCX) .

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