aa r X i v : . [ m a t h . QA ] M a y HOM-ENTWINING STRUCTURES AND HOM-HOPF-TYPE MODULES
SERKAN KARAC¸ UHA
Abstract.
The notions of Hom-coring, Hom-entwining structure and associated entwined Hom-module are introduced. A theorem regarding base ring extension of a Hom-coring is provenand then is used to acquire the Hom-version of Sweedler coring [36]. Motivated by [4], aHom-coring associated to an entwining Hom-structure is constructed and an identification ofentwined Hom-modules with Hom-comodules of this Hom-coring is shown. The dual algebraof this Hom-coring is proven to be a ψ -twisted convolution algebra. By a construction, itis shown that a Hom-Doi-Koppinen datum comes from a Hom-entwining structure and thatthe Doi-Koppinen Hom-Hopf modules are the same as the associated entwined Hom-modules,following [3]. A similar construction regarding an alternative Hom-Doi-Koppinen datum isalso given. A collection of Hom-Hopf-type modules are gathered as special examples of Hom-entwining structures and corresponding entwined Hom-modules, and structures of all relevantHom-corings are also considered. Introduction
Motivated by the study of symmetry properties of noncommutative principal bundles, entwin-ing structures (over a commutative ring k with unit) were introduced in [5] as a triple ( A, C ) ψ consisting of a k -algebra A , a k -coalgebra C and a k -module map ψ : C ⊗ A → A ⊗ C satisfying,for all a, a ′ ∈ A and c ∈ C ,( aa ′ ) κ ⊗ c κ = a κ a ′ λ ⊗ c κλ , κ ⊗ c κ = 1 ⊗ c,a κ ⊗ c κ ⊗ c κ = a λκ ⊗ c κ ⊗ c λ , a κ ε ( c κ ) = aε ( c ) , where the notation ψ ( c ⊗ a ) = a κ ⊗ c κ (summation over κ is understood) is used. Given an entwiningstructure ( A, C ) ψ , the notion of ( A, C ) ψ -entwined module M was first defined in [3] as a right A -module with action m ⊗ a m · a and a right C -comodule with coaction ρ M : m m (0) ⊗ m (1) (summation understood) such that the following compatibility condition holds: ρ M ( m · a ) = m (0) · a κ ⊗ m κ (1) , ∀ a ∈ A, m ∈ M. Hopf-type modules are typically the objects with an action of an algebra and a coaction of a coal-gebra which satisfy some compatibility condition. The family of Hopf-type modules includes wellknown examples such as Hopf modules of Sweedler [35], relative Hopf modules of Doi and Takeuchi[14], [37], Long dimodules [27], Yetter-Drinfeld modules [33], [42], Doi-Koppinen Hopf modules[15], [25] and alternative Doi-Koppinen Hopf modules of Schauenburg [34]. All these modulesexcept alternative Doi-Koppinen modules are special cases of Doi-Koppinen modules. As newerspecial cases of them, the family of Hopf-type modules also includes anti-Yetter-Drinfeld moduleswhich were obtained as coefficients for the cyclic cohomology of Hopf algebras [19], [20], [22], andtheir generalizations termed ( α, β )-Yetter-Drinfeld modules [32] (also called ( α, β )-equivariant C -comodules in [24]). Basically, entwining structures and modules associated to them enable us tounify several categories of Hopf modules in the sense that the compatibility conditions for all ofthem can be restated in the form of the above condition required for entwined modules. Onecan refer to [6] and [9] for more information on the relationship between entwining structures andHopf-type modules.Hom-type algebras have been introduced in the form of Hom-Lie algebras in [21], where theJacobi identity was twisted along a linear endomorphism. Meanwhile, Hom-associative algebrashave been suggested in [28] to give rise to a Hom-Lie algebra using the commutator bracket. Key words and phrases.
Hom-coring, Hom-entwining structure, entwined Hom-module, (alternative) Hom-Doi-Koppinen data, unifying Hom-Hopf module, generalized Hom-Yetter-Drinfeld module.
Other Hom-type structures such as Hom-coalgebras, Hom-bialgebras, Hom-Hopf algebras and theirproperties have been considered in [29, 30, 38]. The so-called twisting principle has been introducedin [39] to provide Hom-type generalization of algebras, and it has been used to obtain many moreproperties of Hom-bialgebras and Hom-Hopf algebras; for instance see [1, 16, 17, 31, 40, 41].The authors of [8] investigated the counterparts of Hom-bialgebras and Hom-Hopf algebras inthe context of tensor categories, and termed them monoidal
Hom-bialgebras and monoidal Hom-Hopf algebras with slight variations in their definitions. Further properties of monoidal Hom-Hopfalgebras and many structures on them have been lately studied [10],[11],[12], [13], [18], [23], [26].Entwining structures have been generalized to weak entwining structures in [7] to define Doi-Koppinen data for a weak Hopf algebra, motivated by [2]. Thereafter, it has been proven in [4]that both entwined modules and weak entwined modules are comodules of certain type of coringswhich built on a tensor product of an algebra and a coalgebra, and shown that various propertiesof entwined modules can be obtained from properties of comodules of a coring. Here we recall from[36] that for an associative algebra A with unit, an A - coring is an A -bimodule C with A -bilinearmaps ∆ C : C → C ⊗ A C , c c ⊗ c called coproduct and ε C : C → A called counit, such that∆ C ( c ) ⊗ c = c ⊗ ∆ C ( c ) , ε C ( c ) c = c = c ε C ( c ) , ∀ c ∈ C . Given an A -coring C , a right C - comodule is a right A -module M with a right A -linear map ρ M : M → M ⊗ C , m m (0) ⊗ m (1) called coaction, such that ρ M ( m (0) ) ⊗ m (1) = m (0) ⊗ ∆ C ( m (1) ) , m = m (0) ε C ( m (1) ) , ∀ m ∈ M. The main aim of the present paper is to generalize the entwining structures, entwined modulesand the associated corings within the framework of monoidal Hom-structures and then to studyHopf-type modules in the Hom-setting. The idea is to replace algebra and coalgebra in a classicalentwining structure with a monoidal Hom-algebra and a monoidal Hom-coalgebra to make adefinition of Hom-entwining structures and associated entwined Hom-modules. Following [4],these entwined Hom-modules are identified with Hom-comodules of the associated Hom-coring.The dual algebra of this Hom-coring is proven to be the Koppinen smash. Furthermore, we give aconstruction regarding Hom-Doi-Kopinen datum and Doi-Koppinen Hom-Hopf modules as specialcases of Hom-entwining structures and associated entwined Hom-modules. Besides, we introducealternative Hom-Doi-Koppinen datum. By using these constructions, we get Hom-versions of theaforementioned Hopf-type modules as special cases of entwined Hom-modules, and give examplesof Hom-corings in addition to trivial Hom-coring and canonical Hom-coring.Throughout the paper k will be a commutative ring with unit. Unadorned tensor product isover k . 2. Preliminaries
Let M k = ( M k , ⊗ , k, a, l, r ) be the monoidal category of k -modules. We associate to M k a new monoidal category H ( M k ) whose objects are ordered pairs ( M, µ ), with M ∈ M k and µ ∈ Aut k ( M ), and morphisms f : ( M, µ ) → ( N, ν ) are morphisms f : A → B in M k satisfying ν ◦ f = f ◦ µ. The monoidal structure is given by (
M, µ ) ⊗ ( N, ν ) = ( M ⊗ N, µ ⊗ ν ) and ( k, e H ( M k ) =( H ( M k ) , ⊗ , ( k, id ) , ˜ a, ˜ l, ˜ r ) introduced in ([8]), with the associativity constraint ˜ a defined by(2.1) ˜ a A,B,C = a A,B,C ◦ (( α ⊗ id ) ⊗ γ − ) = ( α ⊗ ( id ⊗ γ − )) ◦ a A,B,C , for ( A, α ), (
B, β ), (
C, γ ) ∈ H ( M k ), and the right and left unit constraints ˜ r , ˜ l given by(2.2) ˜ r A = α ◦ r A = r A ◦ ( α ⊗ id ); ˜ l A = α ◦ l A = l A ◦ ( id ⊗ α ) , which we write elementwise:˜ a A,B,C (( a ⊗ b ) ⊗ c ) = α ( a ) ⊗ ( b ⊗ γ − ( c )) , ˜ l A ( x ⊗ a ) = xα ( a ) = ˜ r A ( a ⊗ x ) . The category e H ( M k ) is termed Hom-category associated to M k , where a k -submodule N ⊂ M iscalled a subobject of ( M, µ ) if (
N, µ | N ) ∈ e H ( M k ), that is µ restricts to an automorphism of N . OM-ENTWINING STRUCTURES AND HOM-HOPF-TYPE MODULES 3
We now recall some definitions of monoidal Hom-structures from [8]. An algebra in e H ( M k )is called a monoidal Hom-algebra, i.e., an object ( A, α ) ∈ e H ( M k ) together with a k -linear map m : A ⊗ A → A, a ⊗ b ab and an element 1 A ∈ A such that(2.3) α ( a )( bc ) = ( ab ) α ( c ) ; a A = 1 A a = α ( a ) , (2.4) α ( ab ) = α ( a ) α ( b ) ; α (1 A ) = 1 A for all a, b, c ∈ A . A right ( A, α )-Hom-module consists of an object (
M, µ ) ∈ e H ( M k ) togetherwith a k -linear map ψ : M ⊗ A → M, ψ ( m ⊗ a ) = ma , in e H ( M k ) satisfying the following(2.5) µ ( m )( ab ) = ( ma ) α ( b ) ; m A = µ ( m )for all m ∈ M and a, b ∈ A . For ψ to be a morphism in e H ( M k ) means(2.6) µ ( ma ) = µ ( m ) α ( a ) . The map ψ is termed a right Hom-action of ( A, α ) on (
M, µ ). Let (
M, µ ) and (
N, ν ) be two right(
A, α )-Hom-modules. We call a morphism f : M → N right ( A, α )-linear if f ◦ µ = ν ◦ f and f ( ma ) = f ( m ) a for all m ∈ M and a ∈ A .Let ( A, α ) and (
B, β ) be two monoidal Hom-algebras. A left ( A, α ), right ( B, β ) Hom-bimodule (for short [(
A, α ) , ( B, β )]-Hom-bimodule), consists of an object (
M, µ ) ∈ e H ( M k ) together witha left ( A, α )-Hom-action φ : A ⊗ M → M , φ ( a ⊗ m ) = am and a right ( B, β )-Hom-action ϕ : M ⊗ B → M , ϕ ( m ⊗ b ) = mb fulfilling the compatibility condition, for all a ∈ A , b ∈ B and m ∈ M ,(2.7) ( am ) β ( b ) = α ( a )( mb ) . Let (
M, µ ) and (
N, ν ) be two [(
A, α ) , ( B, β )]-Hom-bimodules. A morphism f : M → N is called amorphism of [( A, α ) , ( B, β )]-Hom-bimodules if it is both left (
A, α )-linear and right (
B, β )-linear,and satisfies the following property(2.8) ( af ( m )) β ( b ) = α ( a )( f ( m ) b ) , for all a ∈ A , b ∈ B and m ∈ M . Of course, any monoidal Hom-algebra ( A, α ) is a [(
A, α ) , ( A, α )]-Hom-bimodule ((
A, α )-Hom-bimodule, for short). Moreover, if f : A → B is a morphism ofmonoidal Hom-algebras ( A, α ) and (
B, β ), then naturally (
B, β ) is a [(
A, α ) , ( A, α )]-Hom-bimodulealong f , i.e., the left and right ( A, α )-Hom-action on (
B, β ) is given via f .Let ( M, µ ) be a right (
A, α )-Hom-module and (
N, ν ) be a left (
A, α )-Hom-module. The tensorproduct ( M ⊗ A N, µ ⊗ ν ) of ( M, µ ) and (
N, ν ) over (
A, α ) is the coequalizer of ρ ⊗ id N , ( id M ⊗ ¯ ρ ) ◦ ˜ a M,A,N : ( M ⊗ A ) ⊗ N → M ⊗ N , where ρ and ¯ ρ are the right and left Hom-actions of ( A, α )on (
M, µ ) and (
N, ν ) respectively. That is,(2.9) m ⊗ A n = { m ⊗ n ∈ M ⊗ N | ma ⊗ n = µ ( m ) ⊗ aν − ( n ) , ∀ a ∈ A } . Hom-corings and Hom-Entwining structures
Analogous with monoidal Hom-algebras, one defines monoidal Hom-coalgebras as coalgebras in e H ( M k ). More generally we define the notion of a Hom-coring. Definition 3.1. (1) Let (
A, α ) be a monoidal Hom-algebra. An (
A, α )- Hom-coring consists ofan (
A, α )-Hom-bimodule ( C , χ ) together with A -bilinear maps ∆ C : C → C ⊗ A C , c c ⊗ c and ε C : C → A called comultiplication and counit such that(3.10) χ − ( c ) ⊗ ∆ C ( c ) = c ⊗ ( c ⊗ χ − ( c )); ε C ( c ) c = c = c ε C ( c ) , (3.11) ∆ C ( χ ( c )) = χ ( c ) ⊗ χ ( c ); ε C ( χ ( c )) = α ( ε C ( c )) . SERKAN KARAC¸ UHA (2) A right ( C , χ ) -Hom-comodule ( M, µ ) is defined as a right (
A, α )-Hom-module with a right A -linear map ρ : M → M ⊗ A C , m m (0) ⊗ m (1) satisfying(3.12) µ − ( m (0) ) ⊗ ∆ C ( m (1) ) = m (0)(0) ⊗ ( m (0)(1) ⊗ χ − ( m (1) )); m = m (0) ε C ( m (1) ) , (3.13) µ ( m ) (0) ⊗ µ ( m ) (1) = µ ( m (0) ) ⊗ χ ( m (1) ) . A monoidal Hom-coalgebra (
C, γ ) is simply a coalgebra in the category e H ( M k ). It is nothard to see that ( C, γ ) is a monoidal Hom-coalgebra if and only if it is a ( k, id )-Hom-coring, i.e.an object (
C, γ ) ∈ e H ( M k ) together with k -linear maps ∆ : C → C ⊗ C, ∆( c ) = c ⊗ c and ε : C → k satisfying equations (3.10) and (3.11). A right Hom-comodule ( M, µ ) over a monoidalHom-coalgebra (
C, γ ) is a right (
C, γ )-Hom-comodule over the ( k, id )-Hom-coring (
C, γ ), i.e., anobject (
M, µ ) ∈ e H ( M k ) together with a k -linear map ρ : M → M ⊗ C, ρ ( m ) = m [0] ⊗ m [1] , in e H ( M k ) such that (3.12) and (3.13) hold. The map ρ is called a right Hom-coaction of ( C, γ )on (
M, µ ). Let (
M, µ ) and (
N, ν ) be two right (
C, γ )-Hom-comodules, then we call a morphism f : M → N right ( C, γ )-colinear if f ◦ µ = ν ◦ f and f ( m [0] ) ⊗ m [1] = f ( m ) [0] ⊗ f ( m ) [1] for all m ∈ M . Theorem 3.2.
Let φ : ( A, α ) → ( B, β ) be a morphism of monoidal Hom-algebras. Then, for an ( A, α ) -Hom-coring ( C , χ ) , ( B C ) B = (( B ⊗ A C ) ⊗ A B, ( β ⊗ χ ) ⊗ β ) is a ( B, β ) -Hom-coring, calleda base ring extension of the ( A, α ) -Hom-coring ( C , χ ) , with a comultiplication and a counit, (3.14) ∆ ( B C ) B (( b ⊗ A c ) ⊗ A b ′ ) = (( β − ( b ) ⊗ A c ) ⊗ A B ) ⊗ B ((1 B ⊗ A c ) ⊗ A β − ( b ′ )) , (3.15) ε ( B C ) B (( b ⊗ A c ) ⊗ A b ′ ) = ( bφ ( ε C ( c ))) b ′ . Proof.
For b, b ′ , b ′′ ∈ B and c ∈ C ,∆ ( B C ) B ((( b ′ ⊗ A c ) ⊗ A b ′′ ) b )= ∆ ( B C ) B (( β ( b ′ ) ⊗ A χ ( c )) ⊗ A b ′′ β − ( b ))= (( b ′ ⊗ A χ ( c ) ) ⊗ A B ) ⊗ B ((1 B ⊗ A χ ( c ) ) ⊗ A β − ( b ′′ β − ( b ))) ( . ) = (( b ′ ⊗ A χ ( c )) ⊗ A B ) ⊗ B ((1 B ⊗ A χ ( c )) ⊗ A β − ( b ′′ ) β − ( b ))= (( b ′ ⊗ A χ ( c )) ⊗ A B ) ⊗ B (( β ⊗ χ )(1 B ⊗ A c ) ⊗ A β − ( b ′′ ) β − ( β − ( b )))= (( β ⊗ χ ) ⊗ β )(( β − ( b ′ ) ⊗ A c ) ⊗ A B ) ⊗ B ((1 B ⊗ A c ) ⊗ A β − ( b ′′ )) β − ( b )= ∆ ( B C ) B (( b ′ ⊗ A c ) ⊗ A b ′′ ) b, which proves the right ( B, β )-linearity of ∆ ( B C ) B . It can also be shown that ∆ ( B C ) B ◦ ¯ χ =( ¯ χ ⊗ ¯ χ ) ◦ ∆ ( B C ) B , where ¯ χ = ( β ⊗ χ ) ⊗ β . And as well, the left ( B, β )-linearity of ∆ ( B C ) B and thefact that it preserves the compatibility condition between the left and right ( B, β )-Hom-actionson ( B C ) B can be checked similarly, that is,∆ ( B C ) B ( b (( b ′ ⊗ A c ) ⊗ A b ′′ )) = b ∆ ( B C ) B (( b ′ ⊗ A c ) ⊗ A b ′′ ) , ( b ∆ ( B C ) B (( b ′′ ⊗ A c ) ⊗ A b ′′′ )) β ( b ′ ) = β ( b )(∆ ( B C ) B (( b ′′ ⊗ A c ) ⊗ A b ′′′ ) b ′ ) . Next we prove the Hom-coassociativity of ∆ ( B C ) B : OM-ENTWINING STRUCTURES AND HOM-HOPF-TYPE MODULES 5 (( β − ⊗ χ − ) ⊗ β − )((( b ⊗ A c ) ⊗ A b ′ ) ) ⊗ B ∆ ( B C ) B ((( b ⊗ A c ) ⊗ A b ′ ) )= (( β − ( b ) ⊗ A χ − ( c )) ⊗ A B ) ⊗ B (((1 B ⊗ A c ) ⊗ A B ) ⊗ B ((1 B ⊗ A c ) ⊗ A β − ( b ′ ))) ( . ) = (( β − ( b ) ⊗ A c ) ⊗ A B ) ⊗ B (((1 B ⊗ A c ) ⊗ A B ) ⊗ B ((1 B ⊗ A χ − ( c )) ⊗ A β − ( b ′ )))= (( β − ( b ) ⊗ A c ) ⊗ A B ) ⊗ B ((( β − ( b ) ⊗ A c ) ⊗ A B ) ⊗ B ((1 B ⊗ A χ − ( c )) ⊗ A β − ( b ′ )))= (( b ⊗ A c ) ⊗ A b ′ ) ⊗ B ((( b ⊗ A c ) ⊗ A b ′ ) ⊗ B (( β − ⊗ χ − ) ⊗ β − )(( b ⊗ A c ) ⊗ A b ′ ) ) . Now we demonstrate that ε ( B C ) B is left ( B, β )-linear: ε ( B C ) B ( b (( b ′ ⊗ A c ) ⊗ A b ′′ ))= ε ( B C ) B (( β − ( b ) b ′ ⊗ A χ ( c )) ⊗ A β ( b ′′ ))= (( β − ( b ) b ′ ) φ ( ε C ( χ ( c )))) β ( b ′′ ) ( . ) = (( β − ( b ) b ′ ) φ ( α ( ε C ( c )))) β ( b ′′ )= ( β − ( b )( b ′ β − ( φ ( α ( ε C ( c )))))) β ( b ′′ ) = ( β − ( b )( b ′ φ ( ε C ( c )))) β ( b ′′ )= b (( b ′ φ ( ε C ( c ))) b ′′ ) = bε ( B C ) B (( b ′ ⊗ A c ) ⊗ A b ′′ ) , where φ ◦ α = β ◦ φ was used in the fifth equality. Additionally, we have( ε ( B C ) B ◦ ¯ χ )(( b ⊗ A c ) ⊗ A b ′ ) = ( β ( b ) φ ( ε C ( χ ( c )))) β ( b ′ )= β (( bφ ( ε C ( c ))) b ′ ) = ( β ◦ ε C )(( b ⊗ A c ) ⊗ A b ′ ) , meaning ε ( B C ) B ∈ e H ( M k ). In the same manner, one can show that ε ( B C ) B is right ( B, β )-linearand it preserves the compatibility condition between the left and right (
B, β )-Hom-actions on( B C ) B , i.e., ε ( B C ) B ((( b ′ ⊗ A c ) ⊗ A b ′′ ) b ) = ε ( B C ) B (( b ′ ⊗ A c ) ⊗ A b ′′ ) b, ( bε ( B C ) B (( b ′′ ⊗ A c ) ⊗ A b ′′′ )) β ( b ′ ) = β ( b )( ε ( B C ) B (( b ′′ ⊗ A c ) ⊗ A b ′′′ ) b ′ ) . Below, we prove the counity condition:(( β − ( b ) ⊗ A c ) ⊗ A B ) ε ( B C ) B ((1 B ⊗ A c ) ⊗ A β − ( b ′ ))= (( β − ( b ) ⊗ A c ) ⊗ A B )((1 B φ ( ε C ( c ))) β − ( b ′ ))= (( β − ( b ) ⊗ A c ) ⊗ A B )( β ( φ ( ε C ( c ))) β − ( b ′ ))= ( b ⊗ A χ ( c )) ⊗ A β ( φ ( ε C ( c ))) β − ( b ′ )= ( b ⊗ A χ ( c )) ⊗ A φ ( α ( ε C ( c ))) β − ( b ′ ) ( . ) = ( β − ( b ) ⊗ c ) α ( ε C ( c )) ⊗ A b ′ = ( b ⊗ A c ε C ( c )) ⊗ A b ′ ( . ) = ( b ⊗ A c ) ⊗ A b ′ ( . ) = ( b ⊗ A ε C ( c ) c ) ⊗ A b ′ ( . ) = ( β − ( b ) φ ( ε C ( c )) ⊗ A χ ( c )) ⊗ A b ′ = ( β − ( bφ ( α ( ε C ( c ))))1 B ⊗ A χ ( c )) ⊗ A β ( β − ( b ′ ))= ( bφ ( α ( ε C ( c ))))((1 B ⊗ A c ) ⊗ A β − ( b ′ ))= (( β − ( b ) φ ( ε C ( c )))1 B )((1 B ⊗ A c ) ⊗ A β − ( b ′ ))= ε ( B C ) B (( β − ( b ) ⊗ A c ) ⊗ A B )((1 B ⊗ A c ) ⊗ A β − ( b ′ )) , SERKAN KARAC¸ UHA which completes the proof that given a morphism of monoidal Hom-algebras φ : ( A, α ) → ( B, β ),(( B ⊗ A C ) ⊗ A B, ( β ⊗ χ ) ⊗ β ) is a ( B, β )-Hom-coring. (cid:3)
Example 3.3.
A monoidal Hom-algebra (
A, α ) has a natural (
A, α )-Hom-bimodule structure withits Hom-multiplication. (
A, α ) is an (
A, α )-Hom-coring by the canonical isomorphism A → A ⊗ A A , a α − ( a ) ⊗ A , in e H ( M k ), as a comultiplication and the identity A → A as a counit. ThisHom-coring is called a trivial ( A, α )-Hom-coring.
Example 3.4.
Let φ : ( B, β ) → ( A, α ) be a morphism of monoidal Hom-algebras. Then ( C , χ ) =( A ⊗ B A, α ⊗ α ) is an ( A, α )-Hom-coring with comultiplication∆ C ( a ⊗ B a ′ ) = ( α − ( a ) ⊗ B A ) ⊗ A (1 A ⊗ B α − ( a ′ )) = ( α − ( a ) ⊗ B A ) ⊗ B a ′ and counit ε C ( a ⊗ B a ′ ) = aa ′ for all a, a ′ ∈ A . Proof.
By Theorem (3.2), for φ : ( B, β ) → ( A, α ) and the trivial (
B, β )-Hom-coring (
B, β ) with∆ B ( b ) = β − ( b ) ⊗ B B and ε B ( b ) = b , we have the base ring extension of the trivial ( B, β )-Hom-coring (
B, β ) to (
A, α )-Hom-coring ( AB ) A = (( A ⊗ B B ) ⊗ B A, ( α ⊗ β ) ⊗ α ) with∆ ( AB ) A (( a ⊗ B b ) ⊗ B a ′ ) = (( α − ( a ) ⊗ B β − ( b )) ⊗ B A ) ⊗ A ((1 A ⊗ B B ) ⊗ B α − ( a ′ )) ,ε ( AB ) A (( a ⊗ B b ) ⊗ B a ′ ) = ( aφ ( b )) a ′ . On the other hand we have the isomorphism ϕ : A → A ⊗ B B, a α − ( a ) ⊗ B B , in e H ( M k ),with the inverse ψ : A ⊗ B B → A, a ⊗ B b aφ ( b ): For a ∈ A and b ∈ B , ψ ( ϕ ( a )) = α − ( a ) φ (1 B ) = α − ( a )1 A = a,ϕ ( ψ ( a ⊗ B b )) = ϕ ( aφ ( b )) = α − ( aφ ( b )) ⊗ B B = α − ( a ) α − ( φ ( b )) ⊗ B B = α − ( a ) φ ( β − ( b )) ⊗ B B = a ⊗ B β − ( b )1 B = a ⊗ B b, in addition one can check that α ◦ ψ = ψ ◦ ( α ⊗ β ) and ( α ⊗ β ) ◦ ϕ = ϕ ◦ α . Thus, ( AB ) A ψ ⊗ ≃ A ⊗ B A = C and∆ C ( a ⊗ B b ) = (( ψ ⊗ id ) ⊗ ( ψ ⊗ id )) ◦ ∆ ( AB ) A ◦ ( ϕ ⊗ id )( a ⊗ B b ) = ( α − ( a ) ⊗ B A ) ⊗ A (1 A ⊗ B α − ( a ′ )) ,ε C ( a ⊗ B a ′ ) = ε ( AB ) A ◦ ( ϕ ⊗ id )( a ⊗ B a ′ ) = aa ′ . ( A ⊗ B A, α ⊗ α ) is called the Sweedler or canonical ( A, α )- Hom-coring associated to a monoidalHom-algebra extension φ : ( B, β ) → ( A, α ). (cid:3) For the monoidal Hom-algebra (
A, α ) and the (
A, α )-Hom-coring ( C , χ ), let us put ∗ C = A Hom H ( C , A ), consisting of left ( A, α )-linear morphisms f : ( C , χ ) → ( A, α ), that is, f ( ac ) = af ( c )for a ∈ A , c ∈ C and f ◦ χ = α ◦ f . Similarly, C ∗ = Hom H A ( C , A ) and ∗ C ∗ = A Hom H A ( C , A ) consistof right ( A, α )-Hom-module maps and (
A, α )-Hom-bimodule maps, respectively. Now we provethat these modules of (
A, α )-linear morphisms
C → A have ring structures. Proposition 3.5. (1) ∗ C is an associative algebra with unit ε C and multiplication ( f ∗ l g )( c ) = f ( c g ( c )) for f, g ∈ ∗ C and c ∈ C . (2) C ∗ is an associative algebra with unit ε C and multiplication ( f ∗ r g )( c ) = g ( f ( c ) c ) for f, g ∈ C ∗ and c ∈ C . OM-ENTWINING STRUCTURES AND HOM-HOPF-TYPE MODULES 7 (3) ∗ C ∗ is an associative algebra with unit ε C and multiplication ( f ∗ g )( c ) = f ( c ) g ( c ) for f, g ∈ ∗ C ∗ and c ∈ C .Proof. (1) For f, g, h ∈ ∗ C and c ∈ C ,(( f ∗ l g ) ∗ l h )( c ) = f (( c h ( c )) g (( c h ( c )) )) = f ( χ ( c ) g ( c α − ( h ( c ))))= f ( χ ( c ) g ( c h ( χ − ( c )))) ( . ) = f ( c g ( c h ( c )))= ( f ∗ l ( g ∗ l h ))( c ) , where the second equality comes from the fact that ∆ C is right ( A, α )-linear, i.e., ∆ C ( ca ) =( ca ) ⊗ A ( ca ) = ∆ C ( c ) a = ( c ⊗ A c ) a = χ ( c ) ⊗ A c α − ( a ), ∀ c ∈ C , a ∈ A .( f ∗ l ε C )( c ) = f ( c ε C ( c )) = f ( c ) , ( ε C ∗ l f )( c ) = ε C ( c f ( c )) = ε C ( c ) f ( c ) = f ( ε C ( c ) c ) = f ( c ) . By similar computations one can prove (2) and (3). (cid:3)
Definition 3.6.
A (right-right)
Hom-entwining structure is a triple [(
A, α ) , ( C, γ )] ψ consisting ofa monoidal Hom-algebra ( A, α ), a monoidal Hom-coalgebra (
C, γ ) and a k -linear map ψ : C ⊗ A → A ⊗ C in e H ( M k ) satisfying the following conditions for all a, a ′ ∈ A , c ∈ C :(3.16) ( aa ′ ) κ ⊗ γ ( c ) κ = a κ a ′ λ ⊗ γ ( c κλ ) , (3.17) α − ( a κ ) ⊗ c κ ⊗ c κ = α − ( a ) κλ ⊗ c λ ⊗ c κ , (3.18) 1 κ ⊗ c κ = 1 ⊗ c, (3.19) a κ ε ( c κ ) = aε ( c ) , where we have used the notation ψ ( c ⊗ a ) = a κ ⊗ c κ , a ∈ A , c ∈ C , for the so-called entwiningmap ψ . It is said that ( C, γ ) and (
A, α ) are entwined by ψ . ψ ∈ e H ( M k ) means that the relation(3.20) α ( a ) κ ⊗ γ ( c ) κ = α ( a κ ) ⊗ γ ( c κ )holds. Definition 3.7.
A [(
A, α ) , ( C, γ )] ψ - entwined Hom-module is an object ( M, µ ) ∈ e H ( M k ) whichis a right ( A, α )-Hom-module with action ρ M : M ⊗ A → M , m ⊗ a ma and a right ( C, γ )-Hom-comodule with coaction ρ M : M → M ⊗ C , m m (0) ⊗ m (1) fulfilling the condition, for all m ∈ M , a ∈ A ,(3.21) ρ M ( ma ) = m (0) α − ( a ) κ ⊗ γ ( m κ (1) ) . By f M CA ( ψ ), we denote the category of [( A, α ) , ( C, γ )] ψ -entwined Hom-modules together withthe morphisms in which are both right ( A, α )-linear and right (
C, γ )-colinear.With the following theorem, we construct a Hom-coring associated to an entwining Hom-structure and show an identification of entwined Hom-modules with Hom-comodules of this Hom-coring, pursuing the Proposition 2.2 in [4].
Theorem 3.8.
Let ( A, α ) be a monoidal Hom-algebra and ( C, γ ) be a monoidal Hom-coalgebra. SERKAN KARAC¸ UHA (1)
For a Hom-entwining structure [( A, α ) , ( C, γ )] ψ , ( A ⊗ C, α ⊗ γ ) is an ( A, α ) -Hom-bimodulewith a left Hom-module structure a ( a ′ ⊗ c ) = α − ( a ) a ′ ⊗ γ ( c ) and a right Hom-modulestructure ( a ′ ⊗ c ) a = a ′ α − ( a ) κ ⊗ γ ( c κ ) , for all a, a ′ ∈ A , c ∈ C . Furthermore, ( C , χ ) =( A ⊗ C, α ⊗ γ ) is an ( A, α ) -Hom-coring with the comultiplication and counit (3.22) ∆ C : C → C ⊗ A C , a ⊗ c ( α − ( a ) ⊗ c ) ⊗ A (1 ⊗ c ) , (3.23) ε C : C →
A, a ⊗ c α ( a ) ε ( c ) . (2) If C = ( A ⊗ C, α ⊗ γ ) is an ( A, α ) -Hom-coring with the comultiplication and counit givenabove, then [( A, α ) , ( C, γ )] ψ is a Hom-entwining structure, where ψ : C ⊗ A → A ⊗ C, c ⊗ a (1 ⊗ γ − ( c )) a. (3) Let ( C , χ ) = ( A ⊗ C, α ⊗ γ ) be the ( A, α ) -Hom-coring associated to [( A, α ) , ( C, γ )] ψ asin (1). Then the category of [( A, α ) , ( C, γ )] ψ -entwined Hom-modules is isomorphic to thecategory of right ( C , χ ) -Hom-comodules.Proof. (1) We first show that the right Hom-action of ( A, α ) on ( A ⊗ C, α ⊗ γ ) is Hom-associative and Hom-unital, for all a, d, e ∈ A and c ∈ C :( α ( a ) ⊗ γ ( c ))( de ) = α ( a ) α − ( de ) κ ⊗ γ ( γ ( c ) κ )= α ( a )( α − ( d ) α − ( e )) κ ⊗ γ ( γ ( c ) κ ) ( . ) = α ( a )( α − ( d ) κ α − ( e ) λ ) ⊗ γ ( c κλ )= ( aα − ( d ) κ ) α ( α − ( e ) λ ) ⊗ γ ( γ ( c κλ )) ( . ) = ( aα − ( d ) κ ) α ( α − ( e )) λ ⊗ γ ( γ ( c κ ) λ )= ( aα − ( d ) κ ⊗ γ ( c κ )) α ( e )= (( a ⊗ c ) d ) α ( e ) , ( a ⊗ c )1 = aα − (1) κ ⊗ γ ( c κ ) = a κ ⊗ γ ( c κ )= α − ( α ( a ))1 κ ⊗ γ ( c κ ) = α ( a )(1 κ ⊗ c κ ) ( . ) = α ( a )(1 ⊗ c ) = a ⊗ γ ( c )= ( α ⊗ γ )( a ⊗ c ) . One can also show that the left Hom-action, too, satisfies the Hom-associativity and Hom-unity. For any a, b, d ∈ A and c ∈ C ,( b ( a ⊗ c )) α ( d ) = ( α − ( b ) a ⊗ γ ( c )) α ( d ) = ( α − ( b ) a ) α − ( α ( d )) κ ⊗ γ ( γ ( c ) κ )= ( α − ( b ) a ) α ( α − ( d )) κ ⊗ γ ( γ ( c ) κ ) ( . ) = ( α − ( b ) a ) α ( α − ( d ) κ ) ⊗ γ ( c κ )= b ( aα − ( d ) κ ) ⊗ γ ( c κ ) = α − ( α ( b ))( aα − ( d ) κ ) ⊗ γ ( γ ( c κ ))= α ( b )( aα − ( d ) κ ⊗ γ ( c κ )) = α ( b )(( a ⊗ c ) d ) , proves the compatibility condition between left and right ( A, α )-Hom-actions.First, it can easily be proven that the morphisms A ⊗ ( C ⊗ A C ) → C ⊗ A C ,(3.24) a ⊗ (( a ′ ⊗ c ) ⊗ A ( a ′′ ⊗ c ′ )) α − ( a )( a ′ ⊗ c ) ⊗ A ( α ( a ′′ ) ⊗ γ ( c ′ ))and ( C ⊗ A C ) ⊗ A → C ⊗ A C , (3.25) (( a ′ ⊗ c ) ⊗ A ( a ′′ ⊗ c ′ )) a ( α ( a ′ ) ⊗ γ ( c )) ⊗ A ( a ′′ ⊗ c ′ ) α − ( a )define a left Hom-action and a right Hom-action of ( A, α ) on (
C ⊗ A C , χ ⊗ χ ), respectively.Next it is shown that the comultiplication ∆ C is ( A, α )-bilinear, that is, ∆ C preserves OM-ENTWINING STRUCTURES AND HOM-HOPF-TYPE MODULES 9 the left and right (
A, α )-Hom-actions and the compatibility condition between them asfollows: Let a, a ′ , b, d ∈ A and c ∈ C , then we have the following computations∆ C ( a ( a ′ ⊗ c )) = ( α − ( α − ( a ) a ′ ) ⊗ γ ( c ) ) ⊗ A (1 ⊗ γ ( c ) ) ( . ) = ( α − ( a ) α − ( a ′ ) ⊗ γ ( c )) ⊗ A (1 ⊗ γ ( c ))= α − ( a )( α − ( a ′ ) ⊗ c ) ⊗ A ( α (1) ⊗ γ ( c )) ( . ) = a (( α − ( a ′ ) ⊗ c ) ⊗ A (1 ⊗ c )) = a ∆ C ( a ′ ⊗ c ) , ∆ C (( a ′ ⊗ c ) a ) = ∆ C ( a ′ α − ( a ) κ ⊗ γ ( c κ ))= ( α − ( a ′ α − ( a ) κ ) ⊗ γ ( c κ ) ) ⊗ A (1 ⊗ γ ( c κ ) ) ( . ) = ( α − ( a ′ ) α − ( α − ( a ) κ ) ⊗ γ ( c κ )) ⊗ A (1 ⊗ γ ( c κ )) ( . ) = ( α − ( a ′ ) α − ( a ) κλ ⊗ γ ( c λ )) ⊗ A (1 ⊗ γ ( c κ ))= ( α − ( a ′ ) ⊗ c ) α ( α − ( a ) κ ) ⊗ A (1 ⊗ γ ( c κ )) ( . ) = ( a ′ ⊗ γ ( c )) ⊗ A α ( α − ( a ) κ )(1 ⊗ c κ )= ( a ′ ⊗ γ ( c )) ⊗ A ( α ( α − ( a ) κ ) ⊗ γ ( c κ ))= ( a ′ ⊗ γ ( c )) ⊗ A (1 α − ( α − ( a )) κ ⊗ γ ( c κ ))= ( a ′ ⊗ γ ( c )) ⊗ A (1 ⊗ c ) α − ( a ) ( . ) = (( α − ( a ′ ) ⊗ c ) ⊗ A (1 ⊗ c )) a = ∆ C ( a ′ ⊗ c ) a,α ( b )(∆ C ( a ⊗ c ) d ) = α ( b )((( α − ( a ) ⊗ c ) ⊗ A (1 ⊗ c )) d )= α ( b )(( a ⊗ γ ( c )) ⊗ A (1 ⊗ c ) α − ( d )) ( . ) = α ( b )(( a ⊗ γ ( c )) ⊗ A (1 α − ( α − ( d )) κ ⊗ γ ( c κ )))= α ( b )(( a ⊗ γ ( c )) ⊗ A ( α ( α − ( d ) κ ) ⊗ γ ( c κ ))) ( . ) = b ( a ⊗ γ ( c )) ⊗ A ( α ( α − ( d ) κ ) ⊗ γ ( c κ ))= ( α − ( b ) a ⊗ γ ( c )) ⊗ A α ( α − ( d ) κ )(1 ⊗ γ ( c κ )) ( . ) = ( α − ( α − ( b ) a ) ⊗ γ ( c )) α ( α − ( d ) κ ) ⊗ A (1 ⊗ γ ( c κ ))= (( α − ( b ) α − ( a )) α ( α − ( d ) κ ) λ ⊗ γ ( γ ( c ) λ )) ⊗ A (1 ⊗ γ ( c κ )) ( . ) = (( α − ( b ) α − ( a )) α ( α − ( d ) κλ ) ⊗ γ ( γ ( c λ ))) ⊗ A (1 ⊗ γ ( c κ ))= ( α − ( b )( α − ( a ) α − ( d ) κλ ) ⊗ γ ( c λ )) ⊗ A (1 ⊗ γ ( c κ ))= ( α − ( b )( α − ( a ) α − ( α − ( d )) κλ ) ⊗ γ ( c λ )) ⊗ A (1 ⊗ γ ( c κ )) ( . ) = ( α − ( b )( α − ( a ) α − ( α − ( d ) κ )) ⊗ γ ( c κ )) ⊗ A (1 ⊗ γ ( c κ ))= (( α − ( b ) α − ( a )) α − ( d ) κ ⊗ γ ( c κ )) ⊗ A (1 ⊗ γ ( c κ )) ( . ) = ( α − (( α − ( b ) a ) α ( α − ( d ) κ )) ⊗ γ ( c κ ) ) ⊗ A (1 ⊗ γ ( c κ ) )= ∆ C (( α − ( b ) a ) α ( α − ( d ) κ ) ⊗ γ ( c κ )) = ∆ C ( b ( a ⊗ c )) α ( d ) . One easily checks that the counit ε C is both left and right ( A, α )-linear. For any a, b, d ∈ A and c ∈ C we have ε C (( b ( a ⊗ c )) α ( d )) = ε C ( b ( aα − ( d ) κ ) ⊗ γ ( c κ ))= α ( b ( aα − ( d ) κ )) ε ( γ ( c κ )) ( . ) = α ( b )( α ( a ) α ( α − ( d ) κ )) ε ( c κ )= α ( b )( α ( a ) α ( α − ( d ) κ ε ( c κ ))) ( . ) = α ( b )( α ( a ) α ( α − ( d ) ε ( c )))= α ( b )( α ( a ) ε ( c ) d ) = α ( b )( ε C ( a ⊗ c ) d ) . This finishes the proof that ε C is ( A, α )-bilinear. Let us put∆ C ( a ⊗ c ) = ( a ⊗ c ) ⊗ A ( a ⊗ c ) = ( α − ( a ) ⊗ c ) ⊗ A (1 ⊗ c ) . Then we get the following( α − ⊗ γ − )(( a ⊗ c ) ) ⊗ A ∆ C (( a ⊗ c ) )= ( α − ( a ) ⊗ γ − ( c )) ⊗ A ((1 ⊗ c ) ⊗ A (1 ⊗ c ))= ( α − ( a ) ⊗ c ) ⊗ A ((1 ⊗ c ) ⊗ A (1 ⊗ γ − ( c )))= ( α − ( a ) ⊗ c ) ⊗ A (( α − ( a ) ⊗ c ) ⊗ A (1 ⊗ γ − ( c )))= ( a ⊗ c ) ⊗ A (( a ⊗ c ) ⊗ A ( α − ⊗ γ − )(( a ⊗ c ) )) , where in the second step the Hom-coassociativity of ( C, γ ) is used. ε C (( a ⊗ c ) )( a ⊗ c ) = ε C (( α − ( a ) ⊗ c ))(1 ⊗ c )= α ( α − ( a ) ε ( c ))(1 ⊗ c ) = a (1 ⊗ ε ( c ) c )= a (1 ⊗ γ − ( c )) = a ⊗ c, on the other hand we have( a ⊗ c ) ε C (( a ⊗ c ) ) = ( α − ( a ) ⊗ c ) α (1) ε ( c )= ( α − ( a ) ⊗ c ε ( c ))1 = ( α − ( a ) ⊗ γ − ( c ))1 = a ⊗ c. We also show that the following relations∆ C ( α ( a ) ⊗ γ ( c )) = ( α − ( α ( a )) ⊗ γ ( c ) ) ⊗ A (1 ⊗ γ ( c ) )= ( α ( α − ( a )) ⊗ γ ( c )) ⊗ A ( α (1) ⊗ γ ( c )= (( α ⊗ γ ) ⊗ ( α ⊗ γ ))(∆ C ( a ⊗ c )) ,ε C ( α ( a ) ⊗ γ ( c )) = α ( α ( a )) ε ( γ ( c )) = α ( α ( a )) ε ( c ) = α ( ε C ( a ⊗ c ))hold, which completes the proof that ( A ⊗ C, α ⊗ γ ) is an ( A, α )-Hom-coring.(2) Let us denote ψ ( c ⊗ a ) = (1 ⊗ γ − ( c )) a = a κ ⊗ c κ . ψ is in e H ( M k ):( α ⊗ γ )( ψ ( c ⊗ a )) = α ( a κ ) ⊗ γ ( c κ ) = ( α ⊗ γ )((1 ⊗ γ − ( c )) a )= ( α (1) ⊗ γ ( γ − ( c ))) α ( a ) = (1 ⊗ c ) α ( a )= (1 ⊗ γ − ( γ ( c ))) α ( a ) = α ( a ) κ ⊗ γ ( c ) κ = ψ ( γ ( c ) ⊗ α ( a )) , OM-ENTWINING STRUCTURES AND HOM-HOPF-TYPE MODULES 11 where in the third equality the fact that the right Hom-action of (
A, α ) on ( A ⊗ C, α ⊗ γ )is a morphism in e H ( M k ) was used. Now, let a, a ′ ∈ A and c ∈ C , then ψ ( c ⊗ aa ′ ) = ( aa ′ ) κ ⊗ c κ = (1 ⊗ γ − ( c ))( aa ′ )= (( α − (1) ⊗ γ − ( γ − ( c ))) a ) α ( a ′ ) = ((1 ⊗ γ − ( γ − ( c ))) a ) α ( a ′ )= ( a κ ⊗ γ − ( c ) κ ) α ( a ′ ) = ( α − ( a κ )1 ⊗ γ ( γ − ( γ − ( c ) κ ))) α ( a ′ )= ( a κ (1 ⊗ γ − ( γ − ( c ) κ ))) α ( a ′ )= α ( a κ )((1 ⊗ γ − ( γ − ( c ) κ )) a ′ ) = α ( a κ ) ψ ( γ − ( c ) κ ⊗ a ′ )= α ( a κ )( a ′ λ ⊗ γ − ( c ) κλ ) = α − ( α ( a κ )) a ′ λ ⊗ γ ( γ − ( c ) κλ )= a κ a ′ λ ⊗ γ ( γ − ( c ) κλ ) . In the above equality, if we replace c by γ ( c ) we obtain ( aa ′ ) κ ⊗ γ ( c ) κ = a κ a ′ λ ⊗ γ ( c κλ ).Next, by using the right ( A, α )-linearity of ∆ C we prove the following α − ( a ) κλ ⊗ c λ ⊗ c κ = ψ ( c ⊗ α − ( a ) κ ) ⊗ c κ = (1 ⊗ γ − ( c )) α − ( a ) κ ⊗ c κ = (1 ⊗ γ − ( c )) α − ( a ) κ ⊗ A (1 ⊗ γ − ( c κ )) ( . ) = (1 ⊗ c ) ⊗ A α − ( a ) κ (1 ⊗ γ − ( c κ ))= (1 ⊗ c ) ⊗ A ( α − ( a ) κ ⊗ γ − ( c κ ))= ( id A ⊗ C ⊗ id A ⊗ γ − )((1 ⊗ c ) ⊗ A ψ ( c ⊗ α − ( a )))= ( id A ⊗ C ⊗ id A ⊗ γ − )((1 ⊗ c ) ⊗ A ((1 ⊗ γ − ( c )) α − ( a ))) ( . ) = ( id A ⊗ C ⊗ id A ⊗ γ − )((( α − (1) ⊗ γ − ( c )) ⊗ A (1 ⊗ γ − ( c ))) a ) ( . ) = ( id A ⊗ C ⊗ id A ⊗ γ − )(((1 ⊗ γ − ( c ) ) ⊗ A (1 ⊗ γ − ( c ) )) a )= ( id A ⊗ C ⊗ id A ⊗ γ − )(∆ C (1 ⊗ γ − ( c )) a )= ( id A ⊗ C ⊗ id A ⊗ γ − )(∆ C ((1 ⊗ γ − ( c )) a ))= ( id A ⊗ C ⊗ id A ⊗ γ − )(∆ C ( a κ ⊗ c κ ))= ( id A ⊗ C ⊗ id A ⊗ γ − )(( α − ( a κ ) ⊗ c κ ) ⊗ A (1 ⊗ c κ ))= ( α − ( a κ ) ⊗ c κ ) ⊗ A (1 ⊗ γ − ( c κ ))= ( α − ( α − ( a κ )) ⊗ γ − ( c κ ))1 ⊗ γ ( γ − ( c κ )) = α − ( a κ ) ⊗ c κ ⊗ c κ . We also find ψ ( c ⊗
1) = 1 κ ⊗ c κ = (1 ⊗ γ − ( c ))1 = 1 ⊗ c. Finally, the fact of ε C being right( A, α )-linear gives α ( a κ ) ε ( c κ ) = ε C ( a κ ⊗ c κ ) = ε C ((1 ⊗ γ − ( c )) a )= ε C (1 ⊗ γ − ( c )) a = α (1) ε ( γ − ( c )) a = 1 aε ( c ) = α ( a ) ε ( c ) , which means that a κ ε ( c κ ) = aε ( c ). Therefore [( A, α ) , ( C, γ )] ψ is a Hom-entwining struc-ture.(3) The essential point is that if ( M, µ ) is a right (
A, α )-Hom-module, then ( M ⊗ C, µ ⊗ γ )is a right ( A, α )-Hom-module with the Hom-action ρ M ⊗ C : ( M ⊗ C ) ⊗ A → M ⊗ C ,( m ⊗ c ) ⊗ a ( m ⊗ c ) a = mα − ( a ) κ ⊗ γ ( c κ ). ρ M ⊗ C indeed satisfies Hom-associativityand Hom-unity as follows. For all m ∈ M , a, a ′ ∈ A and c ∈ C ,( µ ( m ) ⊗ γ ( c ))( aa ′ ) = µ ( m ) α − ( aa ′ ) κ ⊗ γ ( γ ( c ) κ ) ( . ) = µ ( m )( α − ( a ) κ α − ( a ′ ) λ ) ⊗ γ ( γ ( c κλ ))= ( mα − ( a ) κ ) α ( α − ( a ′ ) λ ) ⊗ γ ( γ ( c κλ )) ( . ) = ( mα − ( a ) κ ) α ( α − ( a ′ )) λ ⊗ γ ( γ ( c κ ) λ )= ( mα − ( a ) κ ) α − ( α ( a ′ )) λ ⊗ γ ( γ ( c κ ) λ )= ( mα − ( a ) κ ⊗ γ ( c κ )) α ( a ′ ) = (( m ⊗ c ) a ) α ( a ′ ) , ( m ⊗ c )1 = mα − (1) κ ⊗ γ ( c κ ) = m κ ⊗ γ ( c κ ) ( . ) = m ⊗ γ ( c ) = µ ( m ) ⊗ γ ( c ) . With respect to this Hom-action of (
A, α ) on ( M ⊗ C, µ ⊗ γ ), becoming an [( A, α ) , ( C, γ )] ψ -entwined Hom-module is equivalent to the fact that the Hom-coaction of ( C, γ ) on (
M, µ )is right (
A, α )-linear. Let (
M, µ ) ∈ f M CA ( ψ ) with the right ( C, γ )-Hom-comodule structure m m (0) ⊗ m (1) . Then ( M, µ ) ∈ f M C with the Hom-coaction ρ M : M → M ⊗ A C , m m (0) ⊗ A (1 ⊗ γ − ( m (1) )), which actually is ρ M ( m ) = m (0) ⊗ A (1 ⊗ γ − ( m (1) )) = µ − ( m )1 ⊗ γ ( γ − ( m (1) ))= m (0) ⊗ m (1) , where in the second equality we have used the canonical identification φ : M ⊗ A ( A ⊗ C ) ≃ M ⊗ C, m ⊗ A ( a ⊗ c ) µ − ( m ) a ⊗ γ ( c ) , and ρ M is ( A, α )-linear since ρ M ( ma ) = ( ma ) (0) ⊗ ( ma ) (1) = m (0) α − ( a ) κ ⊗ γ ( m κ (1) ) = ( m (0) ⊗ m (1) ) a. Conversely, if (
M, µ ) is a right ( A ⊗ C, α ⊗ γ )-Hom-comodule with the coaction ρ M : M → M ⊗ A ( A ⊗ C ), by using the canonical identification above, one gets the ( C, γ )-Hom-comodule structure ¯ ρ M = φ ◦ ρ M : M → M ⊗ C on ( M, µ ). One can also check that φ isright ( A, α )-linear once the following (
A, α )-Hom-module structure on M ⊗ A C is given: ρ M ⊗ A C : ( M ⊗ A C ) ⊗ A → M ⊗ A C , ( m ⊗ A ( a ⊗ c )) ⊗ a ′ µ ( m ) ⊗ A ( a ⊗ c ) α − ( a ′ ) , thus ¯ ρ M is ( A, α )-linear since by definition ρ M is ( A, α )-linear. Therefore (
M, µ ) hasan [(
A, α ) , ( C, γ )] ψ -entwined Hom-module structure. (cid:3) One should refer to both [9, Proposition 25] and [6, Item 32.9] for the classical version of thefollowing theorem.
Theorem 3.9.
Let [( A, α ) , ( C, γ )] ψ be an entwining Hom-structure and ( C , χ ) = ( A ⊗ C, α ⊗ γ ) bethe associated ( A, α ) -Hom-coring. Then the so-called Koppinen smash or ψ -twisted convolutionalgebra Hom H ψ ( C, A ) = (
Hom H ( C, A ) , ∗ ψ , η A ◦ ε C ) , where ( f ∗ ψ g )( c ) = f ( c ) κ g ( c κ ) for any f, g ∈ Hom H ( C, A ) , is anti-isomorphic to the algebra ( ∗ C , ∗ l , ε C ) in Proposition (3.5).Proof. For f, g, h ∈ Hom H ( C, A ) and c ∈ C ,(( f ∗ ψ g ) ∗ ψ h )( c )= ( f ∗ ψ g )( c ) κ h ( c κ ) = ( f ( c ) λ g ( c λ )) h ( c κ ) ( . ) = ( f ( c ) λκ g ( c λ ) σ ) h ( γ ( γ − ( c ) κσ )) = ( f ( c ) λκ g ( c λ ) σ ) α ( h ( γ − ( c ) κσ ))= α ( f ( c ) λκ )( g ( c λ ) σ h ( γ − ( c ) κσ )) κ ↔ λ = α ( f ( c ) κλ )( g ( c κ ) σ h ( γ − ( c ) λσ )) ( . ) = α ( f ( γ − ( c )) κλ )( g ( c κ ) σ h ( c λσ )) = α ( α − ( f ( c )) κλ )( g ( c κ ) σ h ( c λσ )) ( . ) = f ( c ) κ ( g ( c κ ) σ h ( c κ σ ))= f ( c ) κ ( g ∗ ψ h )( c κ )= ( f ∗ ψ ( g ∗ ψ h ))( c ) , proving that ∗ ψ is associative. Now we show that ηε is the unit for ∗ ψ : OM-ENTWINING STRUCTURES AND HOM-HOPF-TYPE MODULES 13 ( ηε ∗ ψ f )( c ) = ηε ( c ) κ f ( c κ ) = ε ( c )1 κ f ( c κ )= 1 κ f ( γ − ( c ) κ ) ( . ) = 1 f ( γ − ( c ))= f ( c )= f ( γ − ( c ))1 = f ( c ) ε ( c )1 ( . ) = f ( c ) κ ε ( c κ )1 = f ( c ) κ ηε ( c κ )= ( f ∗ ψ ηε )( c ) . The map φ : ∗ C = A Hom H ( A ⊗ C, A ) → Hom H ( C, A ) given by(3.26) φ ( ξ )( c ) = ξ (1 ⊗ γ − ( c ))for any ξ ∈ ∗ C and c ∈ C , is a k -module isomorphism with the inverse ϕ : Hom H ( C, A ) → ∗ C given by ϕ ( f )( a ⊗ c ) = af ( c ) for all f ∈ Hom H ( C, A ) and a ⊗ c ∈ A ⊗ C : Let a ∈ A , a ′ ⊗ c ∈ A ⊗ C and f ∈ Hom H ( C, A ). Then ϕ ( f )( a ( a ′ ⊗ c )) = ϕ ( f )( α − ( a ) a ′ ⊗ γ ( c )) = ( α − ( a ) a ′ ) f ( γ ( c ))= ( α − ( a ) a ′ ) α ( f ( c )) = a ( a ′ f ( c )) = aϕ ( f )( a ′ ⊗ c )and ϕ ( f )( α ( a ) ⊗ γ ( c )) = α ( a ) f ( γ ( c )) = α ( af ( c )) = α ( ϕ ( f )( a ⊗ c )) , showing that ϕ ( f ) is ( A, α )-linear. On the other hand, ϕ ( φ ( ξ ))( a ⊗ c ) = aφ ( ξ )( c ) = aξ (1 ⊗ γ − ( c )) = ξ ( a (1 ⊗ γ − ( c ))) = ξ ( a ⊗ c ) ,φ ( ϕ ( f ))( c ) = ϕ ( f )(1 ⊗ γ − ( c )) = 1 f ( γ − ( c )) = f ( c ) . Now if we put φ ( ξ ) = f and φ ( ξ ′ ) = f ′ , we have f ( c ) = ξ (1 ⊗ γ − ( c )), f ′ ( c ) = ξ ′ (1 ⊗ γ − ( c )) for c ∈ C , and then( ξ ∗ l ξ ′ )( a ⊗ c ) = ξ (( a ⊗ c ) ξ ′ (( a ⊗ c ) )) ( . ) = ξ (( α − ( a ) ⊗ c ) ξ ′ (1 ⊗ c ))= ξ (( α − ( a ) ⊗ c ) f ′ ( γ ( c ))) = ξ (( α − ( a ) ⊗ c ) α ( f ′ ( c )))= ξ ( α − ( a ) α − ( α ( f ′ ( c ))) κ ⊗ γ ( c κ )) = ξ ( α − ( a ) f ′ ( c ) κ ⊗ γ ( c κ ))= ( α − ( a ) f ′ ( c ) κ ) f ( γ ( c κ )) = ( α − ( a ) f ′ ( c ) κ ) α ( f ( c κ ))= a ( f ′ ( c ) κ f ( c κ )) = a ( f ′ ∗ ψf )( c ) , which induces the following φ ( ξ ∗ l ξ ′ )( c ) = ( ξ ∗ l ξ ′ )(1 ⊗ γ − ( c )) = 1( f ′ ∗ ψ f )( γ − ( c ))= α (( f ′ ∗ ψ f )( γ − ( c ))) = ( f ′ ∗ ψ f )( γ ( γ − ( c )))= ( f ′ ∗ ψ f )( c ) = ( φ ( ξ ′ ) ∗ ψ φ ( ξ ))( c ) . Moreover, φ ( ε C )( c ) = ε C (1 ⊗ γ − ( c )) = α (1) ε ( γ − ( c )) = ηε ( c ) . Therefore φ is the anti-isomorphismof the algebras ∗ C and Hom H ψ ( C, A ). (cid:3) Entwinings and Hom-Hopf-type Modules
A bialgebra in e H ( M k ) is called a monoidal Hom-bialgebra (see [8]), i.e. a monoidal Hom-bialgebra ( H, α ) is a sextuple (
H, α, m, η, ∆ , ε ) where ( H, α, m, η ) is a monoidal Hom-algebra and(
H, α, ∆ , ε ) is a monoidal Hom-coalgebra such that(4.27) ∆( hh ′ ) = ∆( h )∆( h ′ ) ; ∆(1 H ) = 1 H ⊗ H , (4.28) ε ( hh ′ ) = ε ( h ) ε ( h ′ ) ; ε (1 H ) = 1 , for any h, h ′ ∈ H . Definition 4.1. [13] Let (
B, β ) be a monoidal Hom-bialgebra. A right ( B, β )- Hom-comodulealgebra ( A, α ) is a monoidal Hom-algebra and a right (
B, β )-Hom-comodule with a Hom-coaction ρ A : A → A ⊗ B, a a (0) ⊗ a (1) such that ρ A is a Hom-algebra morphism, i.e., for any a, a ′ ∈ A (4.29) ( aa ′ ) (0) ⊗ ( aa ′ ) (1) = a (0) a ′ (0) ⊗ a (1) a ′ (1) , ρ A (1 A ) = 1 A ⊗ B ,ρ A ◦ α = ( α ⊗ β ) ◦ ρ A . Definition 4.2.
Let (
B, β ) be a monoidal Hom-bialgebra. A right ( B, β )- Hom-module coalgebra ( C, γ ) is a monoidal Hom-coalgebra and a right (
B, β )-Hom-module with the Hom-action ρ C : C ⊗ B → C, c ⊗ b cb such that ρ C is a Hom-coalgebra morphism, that is, for any c ∈ C and b ∈ B (4.30) ( cb ) ⊗ ( cb ) = c b ⊗ c b , ε C ( cb ) = ε C ( c ) ε B ( b ) ,ρ C ◦ ( γ ⊗ β ) = γ ◦ ρ C . By the following construction, we show that a Hom-Doi-Koppinen datum comes from a Hom-entwining structure and that the Doi-Koppinen Hom-Hopf modules are the same as the associatedentwined Hom-modules, and give the structure of Hom-coring corresponding to the relevant Hom-entwining structure.
Proposition 4.3.
Let ( B, β ) be a monoidal Hom-bialgebra. Let ( A, α ) be a right ( B, β ) -Hom-comodule algebra with Hom-coaction ρ A : A → A ⊗ B, a a (0) ⊗ a (1) and ( C, γ ) be a right ( B, β ) -Hom-module coalgebra with Hom-action ρ C : C ⊗ B → C, c ⊗ b cb . Define the morphism (4.31) ψ : C ⊗ A → A ⊗ C, c ⊗ a α ( a (0) ) ⊗ γ − ( c ) a (1) = a κ ⊗ c κ . Then the following assertions hold. (1) [(
A, α ) , ( C, γ )] ψ is an Hom-entwining structure. (2) ( M, µ ) is an [( A, α ) , ( C, γ )] ψ -entwined Hom-module if and only if it is a right ( A, α ) -Hom-module with ρ M : M ⊗ A → M, m ⊗ a ma and a right ( C, γ ) -Hom-comodule with ρ M : M → M ⊗ C, m m (0) ⊗ m (1) such that (4.32) ρ M ( ma ) = m (0) a (0) ⊗ m (1) a (1) for any m ∈ M and a ∈ A . (3) ( C , χ ) = ( A ⊗ C, α ⊗ γ ) is an ( A, α ) -Hom-coring with comultiplication and counit given by(3.22) and (3.23), respectively, and it has the ( A, α ) -Hom-bimodule structure a ( a ′ ⊗ c ) = α − ( a ) a ′ ⊗ γ ( c ) , ( a ′ ⊗ c ) a = a ′ a (0) ⊗ ca (1) for a, a ′ ∈ A and c ∈ C . (4) Hom H ( C, A ) is an associative algebra with the unit ηε and the multiplication ∗ ψ definedby (4.33) ( f ∗ ψ g )( c ) = α ( f ( c ) (0) ) g ( γ − ( c ) f ( c ) (1) ) = α ( f ( c )) (0) α − ( g ( c α ( f ( c ) (1) ))) , for all f, g ∈ Hom H ( C, A ) and c ∈ C . OM-ENTWINING STRUCTURES AND HOM-HOPF-TYPE MODULES 15
Proof. (1) By (4.31) we have a κ ⊗ γ ( c ) κ = α ( a (0) ) ⊗ ca (1) , and thus( aa ′ ) κ ⊗ γ ( c ) κ = α (( aa ′ ) (0) ) ⊗ c (( aa ′ ) (1) ) ( . ) = α ( a (0) a ′ (0) ) ⊗ c ( a (1) a ′ (1) ) = α ( a (0) ) α ( a ′ (0) ) ⊗ ( γ − ( c ) a (1) ) β ( a ′ (1) ) ( . ) = a κ α ( a ′ (0) ) ⊗ c κ β ( a ′ (1) )= a κ α ( a ′ (0) ) ⊗ γ ( γ − ( c κ ) a ′ (1) ) ( . ) = a κ a ′ λ ⊗ γ ( c κλ ) , which shows that ψ satisfies (3.16). To prove that ψ fulfills (3.17) we have the compu-tation α − ( a κ ) ⊗ c κ ⊗ c κ = α − ( α ( a (0) )) ⊗ ( γ − ( c ) a (1) ) ⊗ ( γ − ( c ) a (1) ) . ) = a (0) ⊗ γ − ( c ) a (1)1 ⊗ γ − ( c ) a (1)2 = a (0) ⊗ γ − ( c ) a (1)1 ⊗ γ − ( c ) a (1)2( . ) = α ( a (0)(0) ) ⊗ γ − ( c ) a (0)(1) ⊗ γ − ( c ) β − ( a (1) ) ( . ) = a (0) κ ⊗ c κ ⊗ γ − ( c ) β − ( a (1) )= α ( α − ( a (0) )) κ ⊗ c κ ⊗ γ − ( c ) β − ( a (1) ) ( . ) = α ( α − ( a ) (0) ) κ ⊗ c κ ⊗ γ − ( c ) α − ( a ) (1)( . ) = α − ( a ) λκ ⊗ c κ ⊗ c λ . To finish the proof of (1) we finally verify that ψ satisfies (3.18) and (3.19) as follows,1 κ ⊗ c κ = α (1 (0) ) ⊗ γ − ( c )1 (1) = α (1 A ) ⊗ γ − ( c )1 B = 1 ⊗ c,a κ ε ( c κ ) = α ( a (0) ) ε ( γ − ( c ) a (1) ) = α ( a (0) ) ε ( γ − ( cβ ( a (1) ))) ( . ) = α ( a (0) ) ε ( cβ ( a (1) )) ( . ) = α ( a (0) ) ε ( c ) ε B ( β ( a (1) ))= α ( a (0) ε B ( a (1) )) ε ( c ) ( . ) = α ( α − ( a )) ε ( c )= aε ( c ) . (2) We see that the condition for entwined Hom-modules,i.e., ρ M ( ma ) = m (0) α − ( a ) κ ⊗ γ ( m κ (1) ) and the condition in (4.32) are equivalent by the following, for m ∈ M and a ∈ A , m (0) α − ( a ) κ ⊗ γ ( m κ (1) ) = m (0) α ( α − ( a ) (0) ) ⊗ γ ( γ − ( m (1) ) α − ( a ) (1) )= m (0) α ( α − ( a (0) )) ⊗ γ ( γ − ( m (1) ) β − ( a (1) ))= m (0) a (0) ⊗ γ ( γ − ( m (1) a (1) ))= m (0) a (0) ⊗ m (1) a (1) . (3) We only prove that the right ( A, α )-Hom-module structure holds as is given in the asser-tion. The rest of the structure of the corresponding Hom-coring can be seen at once fromTheorem (3.8). For a, a ′ ∈ A and c ∈ C , ( a ′ ⊗ c ) a = a ′ α − ( a ) κ ⊗ γ ( c κ )= a ′ α ( α − ( a ) (0) ) ⊗ γ ( γ − ( c ) α − ( a ) (1) ) = a ′ a (0) ) ⊗ γ ( γ − ( c ) β − ( a (1) ))= a ′ a (0) ⊗ ca (1) . (4) By the definition of product ∗ ψ given in Theorem (3.9) and the definition of ψ given in(4.31) we have, for f, g ∈ Hom H ( C, A ) and c ∈ C ,( f ∗ ψ g )( c ) = f ( c ) κ g ( c κ )= α ( f ( c ) (0) ) g ( γ − ( c ) f ( c ) (1) ) = α ( f ( c ) (0) ) g ( γ − ( c β ( f ( c ) (1) )))= α ( f ( c ) (0) ) α − ( g ( c β ( f ( c ) (1) ))) = α ( f ( c )) (0) α − ( g ( c α ( f ( c )) (1) )) . (cid:3) Definition 4.4.
A triple [(
A, α ) , ( B, β ) , ( C, γ )] is called a (right-right) Hom-Doi-Koppinen datum if it satisfies the conditions of Proposition (4.3), that is, if (
A, α ) is a right (
B, β )-Hom-comodulealgebra and (
C, γ ) is a right (
B, β )-Hom-module coalgebra for a monoidal Hom-bialgebra (
B, β ).[(
A, α ) , ( C, γ )] ψ in Proposition (4.3) is called a Hom-entwining structure associated to a Hom-Doi-Koppinen datum .A Doi-Koppinen Hom-Hopf module or a unifying Hom-Hopf module is a Hom-module satisfyingthe condition (4.32).Now we give the following collection of examples. Each of them is a special case of the con-struction given above.
Example 4.5. Relative entwinings and relative Hom-Hopf modules
Let (
B, β ) be amonoidal Hom-bialgebra and let (
A, α ) be a (
B, β )-Hom-comodule algebra with Hom-coaction ρ A : A → A ⊗ B, a a (0) ⊗ a (1) .(1) [( A, α ) , ( B, β )] ψ , with ψ : B ⊗ A → A ⊗ B, b ⊗ a α ( a (0) ) ⊗ β − ( b ) a (1) , is an Hom-entwiningstructure.(2) ( M, µ ) is an [(
A, α ) , ( B, β )] ψ -entwined Hom-module if and only if it is a right ( A, α )-Hom-module with ρ M : M ⊗ A → M, m ⊗ a ma and a right ( B, β )-Hom-comodule with ρ M : M → M ⊗ B, m m [0] ⊗ m [1] such that(4.34) ρ M ( ma ) = m [0] a (0) ⊗ m [1] a (1) for all m ∈ M and a ∈ A . Hom-modules fulfilling the above condition are called relativeHom-Hopf modules (see [18]).(3) ( C , χ ) = ( A ⊗ B, α ⊗ β ) is a ( A, α )-Hom-coring with comultiplication ∆ C ( a ⊗ b ) = ( α − ( a ) ⊗ b ) ⊗ A (1 A ⊗ b ) and counit ε C ( a ⊗ b ) = α ( a ) ε B ( b ), and ( A, α )-Hom-bimodule structure a ( a ′ ⊗ b ) = α − ( a ) a ′ ⊗ β ( b ) , ( a ′ ⊗ b ) a = a ′ a (0) ⊗ ba (1) for all a, a ′ ∈ A and b ∈ B . Proof.
The relevant Hom-Doi-Koppinen datum is [(
A, α ) , ( B, β ) , ( B, β )], where the first object(
A, α ) is assumed to be a right (
B, β )-Hom-comodule algebra with the Hom-coaction ρ A : a a (0) ⊗ a (1) and the third object ( B, β ) is a right (
B, β )-Hom-module coalgebra with Hom-actiongiven by its Hom-multiplication. Hence, [(
A, α ) , ( B, β )] ψ is the associated Hom-entwining struc-ture, where ψ ( b ⊗ a ) = α ( a (0) ) ⊗ β − ( b ) a (1) . Assertions (2) and (3) can be seen at once fromProposition (4.3). (cid:3) Remark . ( A, α ) itself is a relative Hom-Hopf-module by its Hom-multiplication and the (
B, β )-Hom-coaction ρ A . For ( A, α ) = (
B, β ) one gets the notion of
Hom-Hopf modules (see [8]).
Example 4.7. Dual-relative entwinings and [( C, γ ) , ( A, α )] -Hom-Hopf modules Let (
A, α )be a monoidal Hom-bialgebra and let (
C, γ ) be a right (
A, α )-Hom-module coalgebra with Hom-action ρ C : C ⊗ A → C, c ⊗ a ca . OM-ENTWINING STRUCTURES AND HOM-HOPF-TYPE MODULES 17 (1) [(
A, α ) , ( C, γ )] ψ , with ψ : C ⊗ A → A ⊗ C, c ⊗ a α ( a ) ⊗ β − ( c ) a , is an Hom-entwiningstructure.(2) ( M, µ ) is an [(
A, α ) , ( C, γ )] ψ -entwined Hom-module if and only if it is a right ( A, α )-Hom-module with ρ M : M ⊗ A → M, m ⊗ a ma and a right ( C, γ )-Hom-comodule with ρ M : M → M ⊗ B, m m (0) ⊗ m (1) such that(4.35) ρ M ( ma ) = m (0) a ⊗ m (1) a for all m ∈ M and a ∈ A . Such a Hom-module is called [( C, γ ) , ( A, α )] -Hom-Hopf module .(3) ( C , χ ) = ( A ⊗ C, α ⊗ γ ) is a ( A, α )-Hom-coring with comultiplication ∆ C ( a ⊗ c ) = ( α − ( a ) ⊗ c ) ⊗ A (1 A ⊗ c ) and counit ε C ( a ⊗ b ) = α ( a ) ε C ( c ), and ( A, α )-Hom-bimodule structure a ( a ′ ⊗ b ) = α − ( a ) a ′ ⊗ γ ( c ) , ( a ′ ⊗ c ) a = a ′ a ⊗ ca for all a, a ′ ∈ A and c ∈ C . Proof. ( A, α ) is a right (
A, α )-Hom-comodule algebra with Hom-coaction given by the Hom-comultiplication ρ A = ∆ A : A → A ⊗ A, a a (0) ⊗ a (1) = a ⊗ a , since ∆ A is a Hom-algebra morphism. Besides ( C, γ ) is assumed to be a right (
A, α )-Hom-modulecoalgebra with Hom-action ρ C ( c ⊗ a ) = ca . Thus, the related Hom-Doi-Koppinen datum is[( A, α ) , ( A, α ) , ( C, γ )]. Then [(
A, α ) , ( C, γ )] ψ is the Hom-entwining structure associated to thedatum, where ψ ( c ⊗ a ) = α ( a (0) ) ⊗ γ − ( c ) a (1) = α ( a ) ⊗ γ − ( c ) a . The assertions (2) and (3) are also immediate by Proposition (4.3). (cid:3)
Remark . ( C, γ ) itself is a [(
C, γ ) , ( A, α )]-Hom-Hopf-module by the (
A, α )-Hom-action ρ C andits Hom-comultiplication.The example below gives a Hom-generalization of the so-called ( α, β )-Yetter-Drinfeld modulesintroduced in [32] as an entwined Hom-module.Following [8], a monoidal Hom-Hopf algebra ( H, α ) is a Hopf algebra in in e H ( M k ), i.e. itconsists of a septuple ( H, α, m, η, ∆ , ε, S ) where ( H, α, m, η, ∆ , ε ) is a monoidal Hom-bialgebraand S : H → H is a morphism in e H ( M k ) such that S ∗ id H = id H ∗ S = η ◦ ε . S is called antipodeand it has the following properties S ( gh ) = S ( h ) S ( g ) ; S (1 H ) = 1 H ; ∆( S ( h )) = S ( h ) ⊗ S ( h ) ; ε ◦ S = ε, for any elements g, h ∈ H . Example 4.9. Generalized Yetter-Drinfeld entwinings and ( φ, ϕ ) -Hom-Yetter-Drinfeldmodules Let (
H, α ) be a monoidal Hom-Hopf algebra and let φ, ϕ : H → H be two monoidalHom-Hopf algebra automorphisms. Define the map, for all h, g ∈ H (4.36) ψ : H ⊗ H → H ⊗ H, g ⊗ h α ( h ) ⊗ ϕ ( S ( h ))( α − ( g ) φ ( h )) , where S is the antipode of H .(1) [( H, α ) , ( H, α )] ψ is an Hom-entwining structure.(2) ( M, µ ) is an [(
H, α ) , ( H, α )] ψ -entwined Hom-module if and only if it is a right ( H, α )-Hom-module with ρ M : M ⊗ H → M, m ⊗ h mh and a right ( H, α )-Hom-comodule with ρ M : M → M ⊗ H, m m (0) ⊗ m (1) such that(4.37) ρ M ( mh ) = m (0) α ( h ) ⊗ ϕ ( S ( h ))( α − ( m (1) ) φ ( h ))for all m ∈ M and h ∈ H . A Hom-module ( M, µ ) satisfying this condition is called( φ, ϕ )- Hom-Yetter-Drinfeld module .(3) ( C , χ ) = ( H ⊗ H, α ⊗ α ) is an ( H, α )-Hom-coring with comultiplication ∆ C ( h ⊗ h ′ ) =( α − ( h ) ⊗ h ′ ) ⊗ H (1 H ⊗ h ′ ) and counit ε C ( h ⊗ h ′ ) = α ( h ) ε H ( h ′ ), and ( H, α )-Hom-bimodulestructure g ( h ⊗ h ′ ) = α − ( g ) h ⊗ α ( h ′ ) , ( h ⊗ h ′ ) g = hα ( g ) ⊗ ϕ ( S ( g ))( α − ( h ′ ) φ ( g ))for all h, h ′ , g ∈ H . Proof.
In the first place, we prove that the map ρ H : H → H ⊗ ( H op ⊗ H ) , h h (0) ⊗ h (1) := α ( h ) ⊗ ( α − ( ϕ ( S ( h ))) ⊗ h )defines a ( H op ⊗ H, α ⊗ α )-Hom-comodule algebra structure on ( H, α ). Let us put ( H op ⊗ H, α ⊗ α ) =( e H, ˜ α ). Then h (0)(0) ⊗ ( h (0)(1) ) ⊗ ˜ α − ( h (1) )= α ( α ( h ) ) ⊗ (( α − ( ϕ ( S ( α ( h ) ))) ⊗ α ( h ) ) ⊗ ( α − ( ϕ ( S ( h ))) ⊗ α − ( h )))= α ( h ) ⊗ (( α − ( ϕ ( S ( α ( h )))) ⊗ α ( h )) ⊗ ( α − ( ϕ ( S ( h ))) ⊗ α − ( h )))= α ( h ) ⊗ (( ϕ ( S ( h )) ⊗ α ( h )) ⊗ ( α − ( ϕ ( S ( h ))) ⊗ α − ( h )))= h ⊗ (( α − ( ϕ ( S ( h ))) ⊗ h ) ⊗ ( α − ( ϕ ( S ( h ))) ⊗ h ))= h ⊗ (( α − ( ϕ ( S ( h ))) ⊗ h ) ⊗ ( α − ( ϕ ( S ( h ))) ⊗ h ))= α − ( h (0) ) ⊗ ∆ e H ( h (1) ) , where in the fourth step we used α ( h ) ⊗ α − ( h ) ⊗ α − ( h ) ⊗ α − ( h ) ⊗ α ( h ) = h ⊗ h ⊗ h ⊗ h ⊗ h , which can be obtained by applying the Hom-coassociativity of ∆ H three times. We also have h (0) ε e H ( h (0) ) = α ( h ) ε ( α − ( ϕ ( S ( h )))) ε ( h )= α ( h ε ( h )) ε ( α − ( ϕ ( S ( h )))) = α ( α − ( h )) ε ( h )= α − ( h ) , where in the third equality we used the relations ε ◦ α − = ε , ε ◦ ϕ = ε and ε ◦ S = ε . One caneasily check that the relations ρ H ◦ α = ( α ⊗ ˜ α ) ◦ ρ H and ρ H (1 H ) = 1 H ⊗ e H hold. For g, h ∈ H , ρ H ( g ) ρ H ( g ) = ( α ( g ) ⊗ ( α − ( ϕ ( S ( g ))) ⊗ g ))( α ( h ) ⊗ ( α − ( ϕ ( S ( h ))) ⊗ h ))= α ( g ) α ( h ) ⊗ ( α − ( ϕ ( S ( h ))) α − ( ϕ ( S ( g ))) ⊗ g h )= α ( g h ) ⊗ ( α − ( ϕ ( S ( h ) S ( g ))) ⊗ g h )= α (( gh ) ) ⊗ ( α − ( ϕ ( S (( gh ) ))) ⊗ ( gh ) )= ρ H ( gh ) , which completes the proof of the statement that ρ H makes ( H, α ) an ( e H, ˜ α )-Hom-comodule alge-bra. We next consider the map, for all g, h, k ∈ Hρ H : H ⊗ e H → H, g · ( h ⊗ k ) := ( hα − ( g )) φ ( α ( k ))and we claim that it defines an ( e H, ˜ α )-Hom-module coalgebra structure on ( H, α ): Indeed,( g · ( h ⊗ k )) · ( α ( h ′ ) ⊗ α ( k ′ )) = (( hα − ( g )) φ ( α ( k ))) · ( α ( h ′ ) ⊗ α ( k ′ ))= ( α ( h ′ )(( α − ( h ) α − ( g )) α − ( φ ( α ( k ))))) φ ( α ( k ′ ))= ( α ( h ′ )(( α − ( h ) α − ( g )) φ ( k ))) φ ( α ( k ′ ))= (( h ′ ( α − ( h ) α − ( g ))) α ( φ ( k ))) φ ( α ( k ′ ))= ( α − (( h ′ h ) g ) α ( φ ( k ))) φ ( α ( k ′ ))= (( h ′ h ) g )( α ( φ ( k )) α − ( φ ( α ( k ′ )))) = (( h ′ h ) g )( φ ( α ( k )) φ ( α ( k ′ )))= (( h ′ h ) g ) φ ( α (( kk ′ ))) = (( h ′ h ) α − ( α ( g ))) φ ( α (( kk ′ )))= α ( g ) · ( h ′ h ⊗ kk ′ ) = α ( g ) · (( h ⊗ k )( h ′ ⊗ k ′ )) ,h · (1 H ⊗ H ) = (1 H α − ( h )) φ ( α (1 H )) = α ( h ) , OM-ENTWINING STRUCTURES AND HOM-HOPF-TYPE MODULES 19 ( g · ( h ⊗ k )) ⊗ ( g · ( h ⊗ k )) = (( hα − ( g )) φ ( α ( k ))) ⊗ (( hα − ( g )) φ ( α ( k ))) = ( hα − ( g )) φ ( α ( k )) ⊗ ( hα − ( g )) φ ( α ( k )) = ( h α − ( g )) φ ( α ( k )) ⊗ ( h α − ( g )) φ ( α ( k ))= g · ( h ⊗ k ) ⊗ g · ( h ⊗ k )= g · ( h ⊗ k ) ⊗ g · ( h ⊗ k ) ,ε ( g · ( h ⊗ k )) = ε (( hα − ( g )) φ ( α ( k ))) = ε ( h ) ε ( α − ( g )) ε ( φ ( α ( k ))) = ε ( h ) ε ( g ) ε ( k ) = ε ( h ) ε e H ( g ⊗ k ) , proving that ( H, α ) is an ( e H, ˜ α )-Hom-module coalgebra with the Hom-action ρ H . Hence, the Hom-Doi-Koppinen datum is given by [( H, α ) , ( H op ⊗ H, α ⊗ α ) , ( H, α )] to which the Hom-entwiningstructure [(
H, α ) , ( H, α )] ψ is associated, where we have the entwining map ψ : H ⊗ H → H ⊗ H as ψ ( g ⊗ h ) = α ( h (0) ) α − ( g ) · h (1) = α ( α ( h )) ⊗ α − ( g ) · ( α − ( ϕ ( S ( h ))) ⊗ h )= α ( h ) ⊗ ( α − ( ϕ ( S ( h ))) α − ( g )) φ ( α ( h ))= α ( h ) ⊗ ϕ ( S ( h ))( α − ( g ) φ ( h )) . For m ∈ M and h ∈ H , we have the condition (4.37) ρ M ( mh ) = m (0) h (0) ⊗ m (1) · h (1) = m (0) α ( h ) ⊗ m (1) · ( α − ( ϕ ( S ( h ))) ⊗ h )= m (0) α ( h ) ⊗ α − ( ϕ ( S ( h )) m (1) ) φ ( α ( h ))= m (0) α ( h ) ⊗ ϕ ( S ( h ))( α − ( m (1) ) φ ( h )) . By the above proposition, the (
H, α )-Hom-coring structure of ( H ⊗ H, α ⊗ α ) is immediate. Herewe only write down the right Hom-module condition( h ⊗ h ′ ) g = hg (0) ⊗ h ′ · g (1) = hα ( g ) ⊗ h ′ · ( α − ( ϕ ( S ( g ))) ⊗ g )= hα ( g ) ⊗ ϕ ( S ( g ))( α − ( h ′ ) φ ( g )) , completing the proof. (cid:3) Remark . (1) By putting φ = id H = ϕ in the compatibility condition (4.37) we get theusual condition for (right-right) Hom-Yetter-Drinfeld modules, which is(4.38) ρ M ( mh ) = m (0) α ( h ) ⊗ S ( h )( α − ( m (1) ) h ) . (2) If the antipode S of ( H, α ) is a bijection , then by taking φ = id H and ϕ = S − , we havethe compatibility condition for (right-right) anti-Hom-Yetter-Drinfeld modules as follows(4.39) ρ M ( mh ) = m (0) α ( h ) ⊗ S − ( h )( α − ( m (1) ) h ) . We get an equivalent condition for the generalized Hom-Yetter-Drinfeld modules by the follow-ing
Proposition 4.11.
The compatibility condition (4.37) for ( φ, ϕ ) -Hom-Yetter-Drinfeld modules isequivalent to the equation (4.40) m (0) α − ( h ) ⊗ m (1) φ ( α − ( h )) = ( mh ) (0) ⊗ α − ( ϕ ( h )( mh ) (1) ) . Proof.
Assume that (4.40) holds, then m (0) α ( h ) ⊗ ϕ ( S ( h ))( α − ( m (1) ) φ ( h ))= m (0) α − ( α ( h )) ⊗ ϕ ( S ( h ))( α − ( m (1) ) α − ( φ ( α ( h ))))= m (0) α − ( α ( h ) ) ⊗ ϕ ( S ( h )) α − ( m (1) α − ( φ ( α ( h ) ))) ( . ) = ( mα ( h ) ) (0) ⊗ ϕ ( S ( h ))( α − ( ϕ ( α ( h ) )) α − (( mα ( h ) ) (1) ))= ( mα ( h )) (0) ⊗ ϕ ( S ( h ))( ϕ ( h ) α − (( mα ( h )) (1) )) ( . ) = ( mα ( h )) (0) ⊗ ϕ ( S ( α ( h )))( ϕ ( h ) α − (( mα ( h )) (1) ))= ( mα ( h )) (0) ⊗ ϕ ( S ( h ) h ) α − (( mα ( h )) (1) )= ( mα ( h )) (0) ⊗ ϕ ( ε ( h )1 H ) α − (( mα ( h )) (1) ))= ε ( h )( mα ( h )) (0) ⊗ ( mα ( h )) (1) = ε ( h ) ρ M ( mα ( h )) = ρ M ( mh ) , which gives us (4.37). One can easily show that by applying the Hom-coassociativity condition(3.10) twice we have(4.41) α − ( h ) ⊗ h ⊗ α ( h ) ⊗ α ( h ) = h ⊗ h ⊗ h ⊗ h , which is used in the below computation. Thus, if we suppose that (4.37) holds, then( mh ) (0) ⊗ α − ( ϕ ( h )( mh ) (1) ) ( . ) = m (0) α ( h ) ⊗ α − ( ϕ ( h )( ϕ ( S ( h ))( α − ( m (1) ) φ ( h ))))= m (0) α ( h ) ⊗ α − (( α − ( ϕ ( h )) ϕ ( S ( h )))( m (1) α ( φ ( h )))) ( . ) = m (0) h ⊗ α − (( ϕ ( h ) ϕ ( S ( h )))( m (1) φ ( h )))= m (0) h ⊗ ( ε ( h )1 H ) α − ( m (1) φ ( h ))= m (0) h ⊗ ε ( h ) m (1) φ ( h ) ( . ) = m (0) h ε ( h ) ⊗ m (1) φ ( α − ( h ))= m (0) α − ( h ) ⊗ m (1) φ ( α − ( h )) , finishing the proof. (cid:3) Remark . The above result implies that the equations (4.38) and (4.39) are equivalent to m (0) α − ( h ) ⊗ m (1) α − ( h ) = ( mh ) (0) ⊗ α − ( h ( mh ) (1) )and m (0) α − ( h ) ⊗ m (1) α − ( h ) = ( mh ) (0) ⊗ α − ( S − ( h )( mh ) (1) ) , respectively. Example 4.13. The flip and Hom-Long dimodule
Let (
H, α ) be a monoidal Hom-bialgebra.Then:(1) [(
H, α ) , ( H, α )] ψ , where ψ : H ⊗ H → H ⊗ H, g ⊗ h h ⊗ g , is an Hom-entwining structure.(2) ( M, µ ) is an [(
H, α ) , ( H, α )] ψ -entwined Hom-module if and only if it is a right ( H, α )-Hom-module with ρ M : M ⊗ H → M, m ⊗ h mh and a right ( H, α )-Hom-comodule with ρ M : M → M ⊗ H, m m (0) ⊗ m (1) such that(4.42) ρ M ( mh ) = m (0) α − ( h ) ⊗ α ( m (1) )for all m ∈ M and h ∈ H . Such Hom-modules ( M, µ ) are called (right-right) ( H, α ) -Hom-Long dimodules (see [11]). OM-ENTWINING STRUCTURES AND HOM-HOPF-TYPE MODULES 21 (3) ( C , χ ) = ( H ⊗ H, α ⊗ α ) is an ( H, α )-Hom-coring with comultiplication ∆ C ( h ⊗ h ′ ) =( α − ( h ) ⊗ h ′ ) ⊗ H (1 H ⊗ h ′ ) and counit ε C ( h ⊗ h ′ ) = α ( h ) ε H ( h ′ ), and ( H, α )-Hom-bimodulestructure g ( h ⊗ h ′ ) = α − ( g ) h ⊗ α ( h ′ ) , ( h ⊗ h ′ ) g = hα − ( g ) ⊗ α ( h ′ )for all h, h ′ , g ∈ H . Proof. ( H, α ) itself is a right (
H, α )-Hom-comodule algebra with Hom-coaction ρ H = ∆ H : H → H ⊗ H, h h (0) ⊗ h (1) = h ⊗ h . In addition, ( H, α ) becomes a right (
H, α )-Hom-modulecoalgebra with the trivial Hom-action ρ H : H ⊗ H → H, g ⊗ h g · h = α ( g ) ε ( h ). Hencewe have [( H, α ) , ( H, α ) , ( H, α )] as Hom-Doi-Koppinen datum with the associated Hom-entwiningstructure [(
H, α ) , ( H, α )] ψ , where ψ ( h ′ ⊗ h ) = α ( h (0) ) ⊗ α − ( h ′ ) · h (1) = α ( h ) ⊗ α − ( h ′ ) · h = α ( h ) ⊗ α ( α − ( h ′ )) ε ( h ) = h ⊗ h ′ . (cid:3) Definition 4.14.
Let (
B, β ) be a monoidal Hom-bialgebra. A left ( B, β )- Hom-comodule coalgebra ( C, γ ) is a monoidal Hom-coalgebra and a left (
B, β )-Hom-comodule with a Hom-coaction ρ : C → B ⊗ C, c c ( − ⊗ c (0) such that, for any c ∈ C (4.43) c ( − ⊗ c (0)1 ⊗ c (0)2 = c − c − ⊗ c ⊗ c , c ( − ε C ( c (0) ) = 1 B ε C ( c ) ,ρ ◦ γ = ( β ⊗ γ ) ◦ ρ. We lastly introduce the below construction regarding the Hom-version of the so-called alter-native Doi-Koppinen datum given in [34]. For that we recall the definition of a Hom-modulealgebra from [10]: Let (
B, β ) be a monoidal Hom-bialgebra. A right ( B, β )- Hom-module al-gebra ( A, α ) is a monoidal Hom-algebra and a right (
B, β )-Hom-module with a Hom-action ρ A : A ⊗ B → A, a ⊗ b a · b such that, for any a, a ′ ∈ A and b ∈ B (4.44) b · ( aa ′ ) = ( b · a )( b · a ′ ) , b · A = ε ( b )1 A ,ρ A ◦ ( α ⊗ β ) = α ◦ ρ A . Proposition 4.15.
Let ( B, β ) be a monoidal Hom-bialgebra. Let ( A, α ) be a left ( B, β ) -Hom-module algebra with Hom-action A ρ : B ⊗ A → A, b ⊗ a b · a and ( C, γ ) be a left ( B, β ) -Hom-comodule coalgebra with Hom-coaction C ρ : C → B ⊗ C, c c ( − ⊗ c (0) . Define the map (4.45) ψ : C ⊗ A → A ⊗ C, c ⊗ a c ( − · α − ( a ) ⊗ γ ( c (0) ) Then the following statements hold. (1) [(
A, α ) , ( C, γ )] ψ is an Hom-entwining structure. (2) ( M, µ ) is an [( A, α ) , ( C, γ )] ψ -entwined Hom-module iff it is a right ( A, α ) -Hom-modulewith ρ M : M ⊗ A → M, m ⊗ a ma and a right ( C, γ ) -Hom-comodule with ρ M : M → M ⊗ C, m m [0] ⊗ m [1] such that (4.46) ρ M ( ma ) = ( ma ) [0] ⊗ ( ma ) [1] = m [0] ( m [1]( − · α − ( a )) ⊗ γ ( m [1](0) ) for any m ∈ M and a ∈ A . (3) ( C , χ ) = ( A ⊗ C, α ⊗ γ ) is an ( A, α ) -Hom-coring with comultiplication and counit givenby (3.22) and (3.23), respectively, and the ( A, α ) -Hom-bimodule structure a ( a ′ ⊗ c ) = α − ( a ) a ′ ⊗ γ ( c ) , ( a ′ ⊗ c ) a = a ′ ( c ( − · α − ( a )) ⊗ γ ( c (0) ) for a, a ′ ∈ A and c ∈ C .A triple [( A, α ) , ( B, β ) , ( C, γ )] satisfying the above assumptions of the proposition is called analternative Hom-Doi-Koppinen datum.Proof. The first two conditions for Hom-entwining structures will be checked and the rest of theproof can be completed by performing similar computations as in Proposition (4.3). For a, a ′ ∈ A and c ∈ C , ( aa ′ ) κ ⊗ γ ( c ) κ = γ ( c ) ( − · α − ( aa ′ ) ⊗ γ ( γ ( c ) (0) )= β ( c ( − ) · ( α − ( a ) α − ( a ′ )) ⊗ γ ( c (0) )= ( β ( c ( − ) · α − ( a ))( β ( c ( − ) · α − ( a ′ )) ⊗ γ ( c (0) )= ( β ( c ( − ) · α − ( a ))( β ( c ( − ) · α − ( a ′ )) ⊗ γ ( c (0) )= ( β ( β − ( c ( − )) · α − ( a ))( β ( c (0)( − ) · α − ( a ′ )) ⊗ γ ( γ ( c (0)(0) ))= ( c ( − · α − ( a ))( γ ( c (0) ) ( − · α − ( a ′ )) ⊗ γ ( γ ( c (0) ) (0) )= ( c ( − · α − ( a )) a ′ λ ⊗ γ ( γ ( c (0) ) λ )= a κ a ′ λ ⊗ γ ( c κλ ) ,α − ( a κ ) ⊗ c κ ⊗ c κ = α − ( c ( − · α − ( a )) ⊗ γ ( c (0) ) ⊗ γ ( c (0) ) = β − ( c ( − ) · α − ( a ) ⊗ γ ( c (0)1 ) ⊗ γ ( c (0)2 )= β − ( c − c − ) · α − ( a ) ⊗ γ ( c ) ⊗ γ ( c )= ( β − ( c − ) β − ( c − )) · α − ( a ) ⊗ γ ( c ) ⊗ γ ( c )= c − · ( β − ( c − ) · α − ( a )) ⊗ γ ( c ) ⊗ γ ( c )= c − · α − ( c − · α − ( a )) ⊗ γ ( c ) ⊗ γ ( c )= ( c − · α − ( α − ( a ))) κ ⊗ c κ ⊗ γ ( c )= α − ( a ) λκ ⊗ c κ ⊗ c λ . (cid:3) Acknowledgments
The author would like to thank Professor Christian Lomp for his valuable suggestions. Thisresearch was funded by the European Regional Development Fund through the programme COM-PETE and by the Portuguese Government through the FCT- Funda¸c˜ao para a Ciˆencia e a Tec-nologia under the project PEst-C/MAT/UI0144/2013. The author was supported by the grantSFRH/BD/51171/2010.
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Department of Mathematics, FCUP, University of Porto, Rua Campo Alegre 687, 4169-007 Porto,Portugal
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