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

216Definition 9.3<strong>2.</strong> Let H = ( )H, ∆ H , ε H , m H , u H be a k-bialgebra, letA = ( ) (((A, m A , u A ) , ρ H A be a right H-comodule algebra, let D = D, ∆ D , ε D) ), µ H Dbe a right H-module coalgebra and g ∈ D be a grouplike element. We definethe category of (D, A)-Hopf modules (or Doi-Koppinen Hopf modules) denoted byM D A (H) , as follows:• M ∈ Ob ( M D A (H)) is a right D-comodule via ρ D M , a right A-module via µA Msuch that for every m ∈ M we have( ) ∑(230) ρDM ◦ µ A M (m ⊗ a) = µAM (m 0 ⊗ a 0 ) ⊗ µ H D (m 1 ⊗ a 1 )where ρ D M (m) = ∑ m 0 ⊗ m 1 ∈ M ⊗ D and ρ H A (a) = ∑ a 0 ⊗ a 1 ∈ A ⊗ H, i.e.ρ D M is a morphism of right A-modules or equivalently, µA M is a morphism ofright D-comodules• f ∈ Hom M DA (H) (M, N) is both a morphism of right D-comodules and amorphism of right A-modules.Lemma 9.33. Let H = ( ) (H, ∆ H , ε H , m H , u H be a k-bialgebra, let A = (A, mA , u A ) , ρ H Abe a right H-comodule algebra, let D = (( D, ∆ D , ε D) , µ D) H be a right H-modulecoalgebra and g ∈ D be a grouplike element. Then A ∈ M D A (H) and A is a rightD-comodule algebra.Proof. We denote µ H D (d ⊗ h) = d · h. First of all we want to prove that A is a rightD-comodule. In fact we can considerdefined by settingρ D A : A → A ⊗ Dρ D A (a) = a 0 ⊗ g · a 1 .Let us compute, for every a ∈ A,[( ) ] (A ⊗ ∆D◦ ρ ) D A (a) = A ⊗ ∆D(a 0 ⊗ g · a 1 ) = a 0 ⊗ (g · a 1 ) (1)⊗ (g · a 1 ) (2)so thatDisHmodcoalg= a 0 ⊗ g (1) · a 1(1) ⊗ g (2) · a 1(2)A is Hcom= a 00 ⊗ g (1) · a 01 ⊗ g (2) · a 1ggrouplike= a 00 ⊗ g · a 01 ⊗ g · a 1 = ( ρ D A ⊗ D ) (a 0 ⊗ g · a 1 ) = [( ρ D A ⊗ D ) ◦ ρ D A](a)(A ⊗ ∆D ) ◦ ρ D A = ( ρ D A ⊗ D ) ◦ ρ D A.We compute, for every a ∈ A,[rA ◦ ( A ⊗ ε D) ] [◦ ρ D A (a) = rA ◦ ( A ⊗ ε D)] (a 0 ⊗ g · a 1 )(= r A a0 ⊗ ε D (ga 1 ) ) DisHmodcoalg (= r A a0 ⊗ ε D (g) ε H (a 1 ) )so that= a 0 ε D (g) ε H (a 1 ) ggrouplike= a1 k = ar A ◦ ( A ⊗ ε D) ◦ ρ D A = Id A .Note that A is a right A-module via m A . It remains to prove (230) . Recall thatρ D A (a) = ∑ a 0 ⊗ g · a 1 ∈ A ⊗ D)