Globalizations for partial (co)actions on coalgebras
aa r X i v : . [ m a t h . R A ] M a y Globalizations for partial (co)actions oncoalgebras
Felipe Castro and Glauber Quadros ∗ May 8, 2019
Abstract
In this paper, we introduce the notion of globalization for par-tial module coalgebra and for partial comodule coalgebra. We showthat every partial module coalgebra is globalizable exhibiting a stan-dard globalization. We also show the existence of globalization for apartial comodule coalgebra, provided a certain rationality condition.Moreover, we show a relationship between the globalization for the(co)module coalgebra and the usual globalization for the (co)modulealgebra.
Partial actions of groups were first considered in the context of operatorsalgebra (cf. [12]). Dokuchaev and Exel in [10] introduced partial actions ofgroups in a purely algebraic context, obtained several classical results in thesetting of partial actions of groups and covering the actions of groups. Theactions of Hopf algebras on algebras also generalize the theory of actions ofgroups (cf. [13, Example 4.1.6]). The notion of action of groups was extendedin two directions, in both contexts extensive theories were developed, wherewe highlight the Morita and Galois theories. As a natural task, Caenepeel ∗ The authors were partially supported by CNPq, Brazil. The authors would like tothank Antonio Paques, Alveri Sant’Ana and the referee, whose comments, corrections andsuggestions were very useful to improve the manuscript. We would like to thanks LourdesHaase for her corrections and suggestions about the paper writing. primary 16T15; secondary 16T99,16W22
Key words and phrases:
Hopf algebras, partial action, partial coaction, globalization,partial module coalgebra, partial comodule coalgebra. the standard globalization (see Theorem 3.25).2n the fourth section, we study partial coactions on coalgebras. We presentsome examples and important properties related to this structure. We alsoshow a correspondence among these four partial objects, asking for specialconditions, like density or finite dimension, getting a one to one correspon-dence among all of them. In a similar way as made in Section 3, we definethe induced partial coaction assuming the existence of a comultiplicative pro-jection satisfying a special condition (see Proposition 4.18). After that, wedefine globalization for partial comodule coalgebra (see Definition 4.20) andrelate it with the notion of partial module coalgebra earlier defined in Section3 (see Theorem 4.22). Supposing a rational module hypothesis, we show thatevery partial comodule coalgebra is globalizable, constructing the standard globalization for it (see Theorem 4.23).Throughout this paper k denotes a field, all objects are k -vector spaces(i.e. algebra, coalgebra, Hopf algebra, etc., mean respectively k -algebra, k -coalgebra, k -Hopf algebra, etc.), linear maps mean k -linear maps and un-adorned tensor product means ⊗ k . We use the well known Sweedler’s Nota-tion for comodules and coalgebras, in this way: given a coalgebra D , withcoproduct ∆, we will denote, for any c ∈ C ∆( c ) = c ⊗ c , where the summation is understood; and given a left C -comodule M via λ ,we will denote, for any m ∈ Mλ ( m ) = m − ⊗ m , where the summation is also understood.Moreover, we will call π a projection if it is a linear map such that π ◦ π = π . In the context of (partial) actions on algebras, it is usual to assume thatall modules are left modules. In order to respect the correspondences amongthe four partial structures (see Sections 3 and 4), we assume the followingconvention: (partial) module algebras are left (partial) module algebras; (par-tial) comodule algebras are right (partial) comodule algebras; (partial) mod-ule coalgebras are right (partial) module coalgebras; and (partial) comodulecoalgebras are left (partial) comodule coalgebras. Definition 2.1 (Module algebra) . A left H -module algebra is a pair ( A, ⊲ ) ,where A is an algebra and ⊲ : H ⊗ A → A is a linear map, such that thefollowing conditions hold, for any h, k ∈ H and a, b ∈ A : MA1) H ⊲ a = a ;(MA2) h ⊲ ( ab ) = ( h ⊲ a )( h ⊲ b ) ;(MA3) h ⊲ ( k ⊲ a ) = ( hk ) ⊲ a ; Remark 2.2.
The standard definition of module algebra (cf. [9]) containsthe additional condition, for any h ∈ H : h ⊲ A = ε ( h )1 A . This condition should be required for modules algebra over bialgebras. Since H is a Hopf algebra, so it is a consequence of the others. Definition 2.3 (Partial module algebra [7]) . A left partial H -module algebra is a pair ( A, → ) , where A is an algebra and → : H ⊗ A → A is a linear map,such that the following conditions are satisfied, for all h, k ∈ H and a, b ∈ A :(PMA1) H → a = a ;(PMA2) h → ( ab ) = ( h → a )( h → b ) ;(PMA3) h → ( k → a ) = ( h → A )( h k → a ) .The partial action → is said symmetric if the following additional conditionis satisfied:(PMA4) h → ( k → a ) = ( h k → a )( h → A ) . The conditions (
P M A
2) and (
P M A
3) can be replaced by h → ( a ( g → b )) = ( h → a )( h g → b ) , for all h, g ∈ H and a, b ∈ A . It can be useful if working with non-unitalalgebras.A natural way to get a partial module algebra is inducing from a globalaction, as follows: Proposition 2.4 ([3, Proposition 1]) . Given a (global) module algebra B anda right ideal A of B with unity A . Then A is a partial H -module algebra via h · a = 1 A ( h ⊲ a )In this context, an enveloping action (or globalization) of a partial modulealgebra A is a pair ( B, θ ), where B is a module algebra, θ : A → B is analgebra monomorphism, such that 4 i) θ ( A ) is a right ideal of B ; (ii) the partial action on A is equivalent to the induced partial actionon θ ( A ); (iii) B = H ⊲ θ ( A ).Alves and Batista showed that any partial action has an enveloping action. Theorem 2.5 ([3, Theorem 1]) . Let A be a left partial H -module algebra, ϕ : A → Hom ( H, A ) the map given by ϕ ( a )( h ) = h · a and B = H ⊲ ϕ ( A ) .Then ( B, ϕ ) is an enveloping action of A . Remark 2.6.
In the construction made above, we can highlight some pointsthat are important for this paper. The image of A should be a right ideal of an enveloping action B , butit does not need to be a left ideal, neither the partial action needs tobe symmetric (cf. [3, Proposition 4]). The construction of an enveloping action supposes that the partialmodule algebra is unital. However, this restriction may be overcomeby appropriate projections, as shown below.Let B be an H -module algebra (not necessarily unital) and A a sub-algebra of B . Given a multiplicative projection π : B −→ A such that, for all h, k ∈ H and x, y ∈ A , the condition π ( h ⊲ ( x ( k ⊲ y ))) = π ( h ⊲ ( xπ ( k ⊲ y ))) (1)holds, so we can define a structure of partial module algebra in A by h · x = π ( h ⊲ x ) . Note that the converse is also true. In fact, supposing that the pro-jection π induces a structure of partial module algebra in A , thenEquation (1) holds.Now, since we have the notion of induced partial action, we can definea globalization (or enveloping action) of A .5 efinition 2.7 (Globalization for partial module algebra) . Given aright partial H -module algebra A with partial action → , a globaliza-tion of A is a triple ( B, θ, π ) , where B is a right H -module algebra via ⊲ , θ : A → B is an algebra monomorphism and π is a multiplicativeprojection from B onto θ ( A ) , satisfying the following conditions:(GMA1) the partial action on A is equivalent to the partial actioninduced by ⊲ on θ ( A ) , that is, θ ( h → a ) = h → θ ( a ) = π ( h⊲θ ( a )) ;(GMA2) B is the H -module algebra generated by θ ( A ) , that is, B = H ⊲ θ ( A ) ,for all h ∈ H, a ∈ A and b ∈ B . It is a simple task to check that any partial module algebra has aglobalization, in this sense.Since the induced partial action defined by Alves and Batista is aparticular case of the construction above, where the projection is givenby left multiplication by the idempotent 1 A , then this construction ofglobalization generalizes the construction made in [3]. Since the notion of induction by a central idempotent is a particu-lar case of induction by projection, then it inspires us to define theinduced partial (co)module coalgebra using projections (see Defini-tions 3.18 and 4.18).
Definition 3.1 (Module coalgebra) . A right H -module coalgebra is a pair ( D, ◭ ) , where D is a coalgebra and ◭ : D ⊗ H → D is a linear map such thatfor any g, h ∈ H and d ∈ D , the following properties hold:(MC1) d ◭ H = d ;(MC2) ∆( d ◭ h ) = d ◭ h ⊗ d ◭ h ;(MC3) ( d ◭ h ) ◭ g = d ◭ hg .In this case we say that H acts on D via ◭ , or that ◭ is an action of H on D . We sometimes use the terminology global module algebra to differ theabove structure from the partial one. emark 3.2. The above definition can be seen in a categorical sense, asfollows (cf. [14, Definition 11.2.8]): a k -vector space D is said a right H -module coalgebra if it is a coalgebra object in the category of right H -modules .Note that from this categorical approach, it is required a fourth propertyfor an H -module coalgebra. More precisely, the following equality needs tohold: ε D ( d ◭ h ) = ε D ( d ) ε H ( h ) , for all h ∈ H and d ∈ D .Since H is a Hopf algebra, so it has an antipode, then this additionalcondition is a consequence from the others axioms of Definition 3.1, as statedbelow. Proposition 3.3.
Let H be a Hopf algebra and D a right H -module coalge-bra. Then ε D ( d ◭ h ) = ε D ( d ) ε H ( h ) , for any d ∈ D, h ∈ H . Remark 3.4.
From Definition 3.1 and Proposition 3.3 it follows that onecan define, without loss of generality, module coalgebra for non-counital coal-gebras, and both definitions coincide when the coalgebra is counital.Now we present some classical examples of module coalgebras (cf. [9, 13,14].
Example 3.5.
Any Hopf algebra is a right module coalgebra over itself viaright multiplication.
Example 3.6.
Let D be a right H -module coalgebra and C a coalgebra,then C ⊗ D is a right H -module coalgebra with action given by( c ⊗ d ) ◭ h = c ⊗ ( d ◭ h )for any c ⊗ d ∈ C ⊗ D and h ∈ H . Definition 3.7 (Partial module coalgebra) . [5, Definition 5.1.] A right par-tial H -module coalgebra is a pair ( C, ↼ ) , where C is a coalgebra C and ↼ : C ⊗ H → C is a linear map, such that the following conditions aresatisfied, for any g, h ∈ H and c ∈ C :(PMC1) c ↼ H = c ;(PMC2) ∆( c ↼ h ) = c ↼ h ⊗ c ↼ h ;(PMC3) ( c ↼ h ) ↼ g = ε ( c ↼ h )( c ↼ h g ) . partial module coalgebra is said symmetric if the following additional con-dition is satisfied:(PMC4) ( c ↼ h ) ↼ g = ( c ↼ h g ) ε ( c ↼ h ) . One can define left partial module coalgebra in an analogous way.The definition of right partial module coalgebra can be extended to non-counital coalgebras, as follows.
Definition 3.8. A right partial H -module coalgebra is a pair ( C, ↼ ) , where C is a (non necessarily counital) coalgebra and ↼ : C ⊗ H → C is a linearmap, such that, for any h, k ∈ H and c ∈ C , the following conditions hold:(PMC ' c ↼ H = c ;(PMC ' ( c ↼ h ) ⊗ (( c ↼ h ) ↼ k ) = ( c ↼ h ) ⊗ ( c ↼ h k ) .Moreover, it is symmetric if the following additional condition holds:(PMC’3) (( c ↼ h ) ↼ k ) ⊗ ( c ↼ h ) = ( c ↼ h k ) ⊗ ( c ↼ h ) . It is straightforward to check the following statement.
Proposition 3.9. If C is a counital coalgebra, then Definition 3.8 is equiv-alent to Definition 3.7. Batista and Vercruysse [5] related right partial module coalgebras andleft partial module algebra, using non-degenerated dual pairing between analgebra A and a coalgebra C . Here we consider a special case, where thealgebra is the dual of the coalgebra. Proposition 3.10 ([5, Theorem 5.12]) . Let C be a coalgebra, and supposethat → : H ⊗ C ∗ → C ∗ and ↼ : C ⊗ H → C are linear maps satisfying thefollowing compatibility: ( h → α )( c ) = α ( c ↼ h ) (2) for any h ∈ H and c ∈ C . Then C is a partial H -module coalgebra via ↼ ifand only if C ∗ is a partial H -module algebra via → . Remark 3.11.
The existence of a linear map ↼ : C ⊗ H −→ C implies the existence of a linear map → : H ⊗ C ∗ −→ C ∗ h ⊗ α ( h → α ) : c α ( c ↼ h ) , for all h ∈ H, α ∈ C ∗ and c ∈ C . Clearly these two maps satisfy the Equation(2). 8 emark 3.12. It is not clear to the authors if the converse of Remark 3.11is true in general. However, it holds if C ≃ C ∗∗ via ∧ : C −→ C ∗∗ c b c : α α ( c ) . In fact, given h ∈ H and c ∈ C we consider ξ h,c ∈ C ∗∗ given by ξ h,c ( α ) = ( h → α )( c ) . Hence, we can define the linear map ↼ : C ⊗ H −→ Cc ⊗ h c ↼ h = ∧ − ( ξ h,c ) , that clearly satisfies the Equation (2).The next result follows from Remarks 3.11 and 3.12. Proposition 3.13.
Let C be a coalgebra. Then the following statements hold:(i) If C is a partial H -module coalgebra, then C ∗ is a partial H -modulealgebra.(ii) If C ∗ is a partial H -module algebra and C ≃ C ∗∗ via ∧ , then C is apartial H -module coalgebra. Any global right H -module coalgebra is a partial one. Example 3.15.
Let α be a linear functional on H , then the ground field k is a right partial H -module coalgebra via x ↼ h = xα ( h ) if and only if thefollowing conditions hold, for any x ∈ k and h, k ∈ H :(i) α (1 H ) = 1 k ;(ii) α ( h ) α ( k ) = α ( h ) α ( h k ). Example 3.16.
Let G be a group and H = k G the group algebra. Consider α ∈ k G ∗ and let N = { g ∈ G | α ( g ) = 0 } . Then k is a partial k G -modulecoalgebra if and only if N is a subgroup of G . In this case, we have that α ( g ) = ( , if g lies in N , otherwise.9 xample 3.17 ([5, Theorem 5.7]) . A group G acts partially on a coalgebra C if and only if C is a symmetric partial left k G -module coalgebra. In thiscase, g · c = θ g ( P g − ( c )) and P g ( c ) = ε ( g − · c ) c = c ε ( g − · c ).Now, we construct a partial action on a coalgebra from a global one.Let D be a right H -module coalgebra and C ⊆ D a subcoalgebra. Since D is a right H -module, we could try to induce a partial action restricting theaction of D on C . But one can note that the range of this restriction doesnot need to be contained in C , thus we need to project it on C . By this way,let π : D → C be a projection from D over C (as vector spaces) and considerthe following map ı ↼ : C ⊗ H ◭ −→ D π −→ C, where ◭ denotes the right action of H on D .In the sequel, we exhibit a necessary and sufficient condition to the abovemap to be a partial action on C . In the next proposition, we denote by C ◭ H the vector space spanned by the elements c ◭ h , for c ∈ C and h ∈ H . Proposition 3.18 (Induced partial module coalgebra) . Let D be a right H -module coalgebra, C ⊆ D a subcoalgebra and π : D → C a comultiplicativeprojection satisfying π [ π ( x ) ◭ h ] = π [ ε ( π ( x )) x ◭ h ] , (3) for any x ∈ C ◭ H .Consider ı ↼ : C ⊗ H → C the linear map given by c ı ↼ h := π ( c ◭ h ) , (4) then C becomes a right partial H -module coalgebra via ı ↼ .Proof. (PMC1): Let c ∈ C , so c ı ↼ H = π ( c ◭ H ) = π ( c ) = c , where the lastequality holds because π is a projection.(PMC2): If π is a comultiplicative map (i.e., ∆ ◦ π = ( π ⊗ π ) ◦ ∆), thenfor any c ∈ C and h ∈ H , we have∆( c ı ↼ h ) = ∆( π ( c ◭ h ))= ( π ⊗ π )(∆( c ◭ h )) ( M C ) = ( π ⊗ π )( c ◭ h ⊗ c ◭ h )= π ( c ◭ h ) ⊗ π ( c ◭ h )= c ı ↼ h ⊗ c ı ↼ h . h, k ∈ H and c ∈ C , we have ε ( c ı ↼ h )( c ı ↼ h k ) = ε [ π ( c ◭ h )] π ( c ◭ h k ) ( M C ) = ε [ π ( c ◭ h )] π [( c ◭ h ) ◭ k ] ( M C ) = ε [ π (( c ◭ h ) )] π [(( c ◭ h ) ) ◭ k ]= π [ ε [ π (( c ◭ h ) )]( c ◭ h ) ◭ k ] ( ) = π [ π ( c ◭ h ) ◭ k ]= ( c ı ↼ h ) ı ↼ k. Hence, C is a partial module coalgebra. Remark 3.19.
One can note that the converse of Proposition 3.18 is alsotrue. Indeed, supposing C a subcoalgebra of a module coalgebra D and π : D → C a comultiplicative projection such that C is a partial modulecoalgebra by c ı ↼ h = π ( c ◭ h ), hence π [ π ( x ) ◭ h ] = π [ ε ( π ( x )) x ◭ h ] forany x ∈ C ◭ H .The proof of the above statement follows straight from the calculationsmade in Proposition 3.18.With the construction of induced partial action we have the necessarytools to define a globalization for a partial module coalgebra. This is ournext goal. From now on, given a left (resp., right) H -module M with the actiondenoted by ⊲ (resp., ◭ ), we consider the k -vector space H ⊲M (resp., M ◭ H )as the k -vector space generated by the elements h ⊲ m (resp., m ◭ h ) for all h ∈ H and m ∈ M . Clearly, it is an H -submodule of M . Definition 3.20 (Globalization for partial module coalgebra) . Given a rightpartial H -module coalgebra C with partial action ↼ , a globalization of C is atriple ( D, θ, π ) , where D is a right H -module coalgebra via ◭ , θ : C → D is acoalgebra monomorphism and π is a comultiplicative projection from D onto θ ( C ) , satisfying the following conditions for all h ∈ H, c ∈ C and d ∈ D :(GMC1) π [ π ( d ) ◭ h ] = π [ ε ( π ( d )) d ◭ h ] ;(GMC2) θ ( c ↼ h ) = θ ( c ) ı ↼ h ;(GMC3) D is the H -module generated by θ ( C ) , i.e., D = θ ( C ) ◭ H . emark 3.21. The first condition of Definition 3.20 says that we can inducea structure of partial module coalgebra on θ ( C ). The second one says thatthis induced partial action on θ ( C ) is equivalent to the partial action on C . The last one says that does not exists any submodule coalgebra of D containing θ ( C ). Our next aim is to establish relations between the globalization for par-tial module coalgebras, as defined in Definition 3.20, and for partial modulealgebras (cf. [3]). For this we use the fact that a partial module coalgebra C naturally induces a structure of a partial module algebra on C ∗ .Given a right partial H -module coalgebra C , it follows from Proposition3.13 that the dual C ∗ is a left partial H -module algebra. The same is truefor (global) module coalgebras (cf. [14]). Remark 3.22.
Let C be a right partial H -module coalgebra, D a right H -module coalgebra, θ : C → D a coalgebra monomorphism and π : D → θ ( C )a comultiplicative projection. Since θ is injective, hence it has an inverse θ − ,defined in θ ( C ) = π ( D ).Consider the linear map ϕ : C ∗ → D ∗ , given by the transpose of θ − ◦ π ,i.e., for any α ∈ C ∗ , we have ϕ ( α ) := ( θ − ◦ π ) ∗ ( α ) = α ◦ θ − ◦ π . This isclearly a multiplicative monomorphism.So, one can also define the following linear maps: ı ↼ : π ( D ) ⊗ H −→ π ( D ) π ( d ) ⊗ h π ( π ( d ) ◭ h )and → ı : H ⊗ ϕ ( C ∗ ) −→ D ∗ h ⊗ ϕ ( α ) ϕ ( ε C ) ∗ ( h ⊲ ϕ ( α )) . Note that, there is a correspondence between these maps, given by( h → ı ϕ ( α ))( π ( d )) = ϕ ( α )( π ( d ) ı ↼ h ) , (5)for any h ∈ H , d ∈ D and α ∈ C ∗ . In fact,( h → ı ϕ ( α ))( π ( d )) = ( ϕ ( ε ) ∗ ( h ⊲ ϕ ( α )))( π ( d ))= ϕ ( ε )( π ( d ) ) (( h ⊲ ϕ ( α )))( π ( d ) )= ϕ ( ε )( π ( d ) ) ϕ ( α )( π ( d ) ◭ h )12 ϕ ( ε )( π ( d )) ϕ ( α )( π ( d ) ◭ h )= ε ( θ − ( π ( π ( d )))) α ( θ − ( π ( π ( d ) ◭ h )))= ε θ ( C ) ( π ( d )) α ( θ − ( π ( π ( π ( d ) ◭ h ))))= ε θ ( C ) ( π ( d ) ) ϕ ( α )( π ( π ( d ) ◭ h ))= ϕ ( α )( π ( π ( d ) ◭ h ))= ϕ ( α )( π ( d ) ı ↼ h ) , for any h ∈ H , d ∈ D and α ∈ C ∗ .Notice that the maps ı ↼ and → ı are the induced partial actions on π ( D )and ϕ ( C ∗ ), respectively.In fact, the next statement shows that the map ı ↼ is the induced rightpartial action in π ( D ) if and only if the map → ı is the induced left partialaction in ϕ ( C ∗ ) and, moreover, it relates the globalization of the partialmodule coalgebra to the globalization of the dual partial module algebra.Therefore, for simplicity we will write ↼ instead of ı ↼ and → instead of → ı (even for induced partial actions). Theorem 3.23.
Let C be a partial module coalgebra. With the above nota-tions, we have that ( θ ( C ) ◭ H, θ, π ) is a globalization for C if and only if ( H ⊲ ϕ ( C ∗ ) , ϕ ) is a globalization for C ∗ .Proof. If ( θ ( C ) ◭ H, θ, π ) is a globalization for C , it follows that( ϕ ( ε C ) ∗ ( h ⊲ ϕ ( α )))( d ) = ( ϕ ( ε C )( d ))(( h ⊲ ϕ ( α ))( d ))= ε C ( θ − ( π ( d ))) ϕ ( α )( d ◭ h )= ε θ ( C ) ( π ( d )) α ( θ − ( π ( d ◭ h )))= α ( θ − ( π ( ε θ ( C ) ( π ( d )) d ◭ h ))) ( GM C ) = α ( θ − ( π ( π ( d ) ◭ h ))) ( ) = α ( θ − ( π ( d ) ↼ h )) ( GM C ) = α (( θ − ( π ( d ))) ↼ h ) ( . ) = ( h → α )( θ − ( π ( d )))= ϕ ( h → α )( d ) , for every h ∈ H, α ∈ C ∗ and d ∈ D . Thus, ϕ ( C ∗ ) is a right ideal of H ⊲ ϕ ( C ∗ )and, moreover, h → ϕ ( α ) = ϕ ( h → α ). Therefore, ( H ⊲ ϕ ( C ∗ ) , ϕ ) is aglobalization for C ∗ , as desired.Conversely, if ( H ⊲ ϕ ( C ∗ ) , ϕ ) is a globalization for C ∗ , then13GMC1): Given α ∈ C ∗ , h ∈ H and d ∈ D we have α ( θ − ( π ( π ( d ) ◭ h ))) = α ( θ − ( π ( π ( d ) ↼ h )))= ϕ ( α )( π ( d ) ↼ h ) ( ) = ( h → ϕ ( α ))( π ( d )) ( GM A ) = ϕ ( h → α )( π ( d ))= ( h → α )( θ − ( π ( π ( d ))))= ( h → α )( θ − ( π ( d )))= ϕ ( h → α )( d )= ( ϕ ( ε C ) ∗ ( h ⊲ ϕ ( α )))( d )= ϕ ( ε C )( d )( h ⊲ ϕ ( α ))( d )= ϕ ( ε C )( d ) ϕ ( α )( d ◭ h )= ϕ ( ε C )( d ) α ( θ − ( π ( d ◭ h )))= ε C ( θ − ( π ( d ))) α ( θ − ( π ( d ◭ h )))= ε ( π ( d )) α ( θ − ( π ( d ◭ h )))= α ( θ − ( π ( ε ( π ( d )) d ◭ h ))) . Since α is an arbitrary linear functional on C and θ ◦ θ − = I C , then π ( π ( d ) ◭ h ) = π ( ε ( π ( d )) d ◭ h ) . (GMC2): Given α ∈ C ∗ , h ∈ H and c ∈ C , then( α ◦ θ − )( θ ( c ) ↼ h ) = ( α ◦ θ − )( π ( θ ( c ) ◭ h ))= ( α ◦ θ − ◦ π )( θ ( c ) ◭ h )= ϕ ( α )( θ ( c ) ◭ h )= [ h ⊲ ϕ ( α )]( θ ( c ))= [ h ⊲ ϕ ( α )]( θ ( ε C ( c ) c ))= ε C ( c ) [ h ⊲ ϕ ( α )]( θ ( c ))= ( ε C ◦ θ − ◦ θ )( c ) [ h ⊲ ϕ ( α )]( θ ( c ))= ( ε C ◦ θ − ◦ π ◦ θ )( c ) [ h ⊲ ϕ ( α )]( θ ( c ))= ϕ ( ε C )( θ ( c )) [ h ⊲ ϕ ( α )]( θ ( c ))= ϕ ( ε C )( θ ( c ) ) [ h ⊲ ϕ ( α )]( θ ( c ) )= [ ϕ ( ε C ) ∗ ( h ⊲ ϕ ( α ))] ( θ ( c ))= ( h → ϕ ( α )) ( θ ( c )) ( GM A ) = ϕ ( h → α ) ( θ ( c ))14 ( h → α ) ( θ − ( π ( θ ( c ))))= ( h → α ) ( θ − ( θ ( c )))= ( h → α )( c ) ( . ) = α ( c ↼ h )= ( α ◦ θ − ◦ θ )( c ↼ h )= ( α ◦ θ − )( θ ( c ↼ h )) . Since α ∈ C ∗ is arbitrary, we obtain that θ ( c ) ↼ h = θ ( c ↼ h ). Therefore,( θ ( C ) ◭ H, θ, π ) is a globalization for C . Now our next aim is to show that every partial module coalgebra has aglobalization, constructing the standard globalization.
Remark 3.24.
Let C be a right partial H -module coalgebra. Consider thecoalgebra C ⊗ H with the natural structure of the tensor coalgebra, thecoalgebra monomorphism from C into C ⊗ H , given by the natural embedding ϕ : C −→ C ⊗ Hc c ⊗ H , and the projection from C ⊗ H onto ϕ ( C ), given by π : C ⊗ H −→ θ ( C ) c ⊗ h ( c ↼ h ) ⊗ H . We claim that π is comultiplicative. Indeed, for c ∈ C and h ∈ H we have∆( π ( c ⊗ h )) = ∆(( c ↼ h ) ⊗ H )= ( c ↼ h ) ⊗ H ⊗ ( c ↼ h ) ⊗ H ( P M C ) = c ↼ h ⊗ H ⊗ c ↼ h ⊗ H = π ( c ⊗ h ) ⊗ π ( c ⊗ h )= ( π ⊗ π )∆( c ⊗ h ) . With the above noticed we are able to construct a globalization for apartial module coalgebra.
Theorem 3.25.
Every right partial H -module coalgebra has a globalization. roof. Let C be a left partial H -module coalgebra, so from Examples 3.5 and3.6, we know that C ⊗ H is an H -module coalgebra, with action given by rightmultiplication in H . By the above noticed, we have the maps ϕ : C → C ⊗ H and π : C ⊗ H → ϕ ( C ), as required in Definition 3.20. Then we only need toshow that the conditions (GMC1) and (GMC2) hold.(GMC1): For every h, k ∈ H and c ∈ C , we have π [ ε ( π (( c ⊗ h ) ))( c ⊗ h ) ◭ k ] ( P M C ) = π [ ε ( π ( c ⊗ h ))( c ⊗ h ) ◭ k ]= ε ( c ↼ h ) π [( c ⊗ h ) ◭ k ]= ε ( c ↼ h ) π [ c ⊗ h k ]= ε ( c ↼ h )( c ↼ h k ) ⊗ H ( P M C ) = ( c ↼ h ) ↼ k ⊗ H = π [( c ↼ h ) ⊗ k ]= π [(( c ↼ h ) ⊗ H ) ◭ k ]= π [ π ( c ⊗ h ) ◭ k ] . (GMC2): Let h ∈ H and c ∈ C , then ϕ ( c ) ↼ h = π [ ϕ ( c ) ◭ h ]= π [( c ⊗ H ) ◭ h ]= π [ c ⊗ h ]= c ↼ h ⊗ H = ϕ ( c ↼ h ) . Moreover, by the definitions of π, ϕ and ◭ it follows that ϕ ( C ) ◭ H = C ⊗ H . Therefore C ⊗ H is a globalization for C .The globalization above constructed is called the standard globalization and it is close related with the standard globalization for partial modulealgebras, as follows. Theorem 3.26.
Let C be a right partial H -module coalgebra, then the stan-dard globalization for C generates the standard globalization for C ∗ as leftpartial H -module algebra.Proof. From Theorem 3.25, we have that ( C ⊗ H, ϕ, π ) is the standard glo-balization for C . Consider the multiplicative map φ : C ∗ −→ ( C ⊗ H ) ∗ α α ◦ ϕ − ◦ π. H ⊲ φ ( C ∗ ) , φ ) is a globalizationfor C ∗ , where the action on ( C ⊗ H ) ∗ is given by ⊲ : H ⊗ ( C ⊗ H ) ∗ −→ ( C ⊗ H ) ∗ h ⊗ ξ ( h ⊲ ξ )( c ⊗ k ) = ξ ( c ⊗ k h ) , for every ξ ∈ ( C ⊗ H ) ∗ , c ∈ C and h, k ∈ H .Now, consider the following algebra isomorphism given by the adjointisomorphism Ψ : ( C ⊗ H ) ∗ −→ Hom(
H, C ∗ ) ξ [Ψ( ξ )( h )]( c ) = ξ ( c ⊗ h ) , which is an H -module morphism. In fact, let h, k ∈ H , c ∈ C and ξ ∈ ( C ⊗ H ) ∗ , so { [Ψ( h ⊲ ξ )]( k ) } ( c ) = [( h ⊲ ξ )]( c ⊗ k )= ξ ( c ⊗ k h )= { [Ψ( ξ )]( k h ) } ( c )= { [ h ⊲ Ψ( ξ )]( k ) } ( c )and, therefore, Ψ is an H -module map. Moreover, composing Ψ with φ weobtain { [Ψ ◦ φ ( α )]( h ) } ( c ) = φ ( α )( c ⊗ h )= α ( ϕ − ( π ( c ⊗ h )))= α ( ϕ − ( c ↼ h ⊗ H ))= α ( c ↼ h )= ( h → α )( c )= [Φ( α )( h )]( c ) , where Φ : C ∗ → Hom(
H, C ∗ ), given by Φ( α )( h ) = h → α , for all h ∈ H and α ∈ C ∗ is the multiplicative map that appears in the construction of thestandard globalization, replacing A by C ∗ (cf. [3, Theorem 1]). Given two vector spaces V and W , we write τ V,W to denote the standardisomorphism between V ⊗ W and W ⊗ V .17 efinition 4.1 (Comodule coalgebra) . A left H -comodule coalgebra is apair ( D, λ ) , where D is a coalgebra and λ : D → H ⊗ D is a linear map, suchthat, for all d ∈ D , the following conditions hold:(CC1) ( ε H ⊗ I ) λ ( d ) = d ;(CC2) ( I ⊗ ∆ D ) λ ( d ) = ( m H ⊗ I ⊗ I )( I ⊗ τ D,H ⊗ I )( λ ⊗ λ )∆ D ( d ) ;(CC3) ( I ⊗ λ ) λ ( d ) = (∆ H ⊗ I ) λ ( d ) .In this case, we say that H coacts on D via λ , or that λ is a coaction of H on D . We will also call it a global comodule coalgebra to differ explicitly fromthe partial one. We can also see the above definition in a categorical approach, in thefollowing sense (cf. [14, Definition 11.3.7]): a k -vector space D is a left H -comodule coalgebra if it is a coalgebra object in the category of left H -comodules. From this categorical point of view, one additional condition is requiredin Definition 4.1, that is, ( I ⊗ ε D ) λ ( d ) = ε D ( d )1 H , (6)for all d ∈ D .Since H is a Hopf algebra (so it has an antipode), thus this additionalcondition may be obtained from the another ones, as stated below. Proposition 4.2.
Let D be a left H -comodule coalgebra in the sense of Def-inition 4.1. Then ( I ⊗ ε D ) λ ( d ) = ε D ( d )1 H , for any d ∈ D . Remark 4.3.
From the above proposition one can extend the notion ofcomodule coalgebra for non-counital coalgebras.Now we exhibit some classical examples of comodule coalgebras (cf. [14,Section 11.3]).
Example 4.4.
A Hopf algebra H is an H -comodule coalgebra with coaction λ : H → H ⊗ H given by λ ( h ) = h S ( h ) ⊗ h , for any h ∈ H . Example 4.5.
Any coalgebra D is an H -comodule coalgebra with coaction λ : D → H ⊗ D defined by λ ( d ) = 1 H ⊗ d , for every d ∈ D .18 xample 4.6. If the Hopf algebra H is finite dimensional, then H ∗ is an H -comodule coalgebra with structure given by λ : H ∗ → H ⊗ H ∗ with λ ( f ) = n P i =1 h i ⊗ f ∗ h ∗ i , where { h i } ni =1 and { h ∗ i } ni =1 are dual basis for H and H ∗ ,respectively. Example 4.7.
Let C be a left H -comodule coalgebra with coaction λ and D a coalgebra, then C ⊗ D is a left H -comodule coalgebra via λ ⊗ I D . (Partial comodule coalgebra) . [5, Definition 6.1] A left par-tial H -comodule coalgebra is a pair ( C, λ ′ ) , where C is a coalgebra and λ ′ : C → H ⊗ C is a linear map, such that, for any c ∈ C , the followingconditions hold:(PCC1) ( ε H ⊗ I ) λ ′ ( c ) = c ;(PCC2) ( I ⊗ ∆ C ) λ ′ ( c ) = ( m H ⊗ I ⊗ I )( I ⊗ τ C,H ⊗ I )( λ ′ ⊗ λ ′ )∆ C ( c ) ;(PCC3) ( I ⊗ λ ′ ) λ ′ ( c ) = ( m H ⊗ I ⊗ I ) {∇ ⊗ [(∆ H ⊗ I ) λ ′ ] } ∆ C ( c ) ,where ∇ : C → H is defined by ∇ ( c ) = ( I ⊗ ε C ) λ ′ ( c ) .The partial comodule coalgebra is said symmetric if the following addi-tional condition holds, for any c ∈ C :(PCC4) ( I ⊗ λ ′ ) λ ′ ( c ) = ( m H ⊗ I ⊗ I )( I ⊗ τ H ⊗ C,H ) { [(∆ H ⊗ I ) λ ′ ] ⊗ ∇} ∆ C ( c ) . Remark 4.9.
For a partial comodule coalgebra C via λ ′ we use the Sweedler’snotation λ ′ ( c ) = c − ⊗ c , where the summation is understood. The bar overthe upper index is useful to distinguish partial from global comodule coalge-bras when working with both structures in a single computation. Proposition 4.10. [5, Lemma 6.3 and Corollary 6.4]
Let C be left partial H -comodule coalgebra, then, for all c ∈ C , the following equalities hold: c − ⊗ c = ∇ ( c ) c − ⊗ c = c − ∇ ( c ) ⊗ c (7) and ∇ ( c ) ∇ ( c ) = ∇ ( c ) . (8) Example 4.11.
Every left H -comodule coalgebra is a left partial H -comodulecoalgebra.The next result gives us a simple method to construct new examplesof partial comodule coalgebras. The proof is straightforward and it will beomitted. 19 roposition 4.12. Let λ ′ : k → H ⊗ k be a linear map and h ∈ H suchthat λ ′ (1 k ) = h ⊗ k . Then the ground field k is a left partial H -comodulecoalgebra if and only if the following conditions hold:1. ε H ( h ) = 1 k ;2. h ⊗ h = ( h ⊗ H )∆( h ) . As an application of the above result, we present the next example.
Example 4.13.
Let G be a group, λ ′ : k → k G ⊗ k a linear map and x = P g ∈ G α g g in k G such that λ ′ (1) = x ⊗
1. Consider N = { g ∈ G | α g = 0 } andsuppose that the characteristic of k does not divides | N | . Then k is a leftpartial k G -comodule coalgebra via λ ′ if and only if N is a finite subgroup of G . In this case, we have that α g = 1 | N | , for all g ∈ N Proposition 4.14. [5] Let C be a left partial H -comodule coalgebra via λ ′ ,then it is a (global) H -comodule coalgebra if and only if c − ε C ( c ) = ε C ( c )1 H , for all c ∈ C . Given a Hopf algebra H , we say that the finite dual H separate points ifit is dense on H ∗ in the finite topology, i.e., if h ∈ H is such that f ( h ) = 0,for all f ∈ H , then h = 0.For a coalgebra C and a linear map λ ′ : C → H ⊗ C (denoting by λ ′ ( c ) = c − ⊗ c ) we have two induced linear maps λ ' ↼ : C ⊗ H ∗ → C and λ ' → : H ∗ ⊗ C ∗ → C ∗ , given respectively by c λ ' ↼ f = f ( c − ) c (9)( f λ ' → α )( c ) = f ( c − ) α ( c ) , (10)for all c ∈ C, α ∈ C ∗ and f ∈ H ∗ . Since, in general, H ∗ is not a Hopf algebra,thus we can restrict λ ' ↼ and λ ' → to the subspaces C ⊗ H and H ⊗ C ∗ ,respectively. Therefore, under the assumption that C is a left partial H -comodule coalgebra, we can show the following. Theorem 4.15.
With the above notations, if C is a left partial H -comodulecoalgebra via λ ′ , then the following statements hold: C is a right partial H -module coalgebra via λ ' ↼ ;(2) C ∗ is a left partial H -module algebra via λ ' → .Proof. (1) : Taking the dual pairing between H and H given by evaluation,then by Theorem 6.7 of [5] we have the desired.(2) : In this case, taking the dual pairing between C and C ∗ given byevaluation, then the desired follows from Theorem 6.8 of [5].It is not clear if the converse of the above theorem is true in general,but whenever H separate points (so the dual pairing between H and H isnon-degenerate) it holds, as stated in the next theorem. Theorem 4.16.
With the above notations, if H separate points, then thefollowing conditions are equivalent:(1) C is a right partial H -module coalgebra via λ ' ↼ ;(2) C is a left partial H -module algebra via λ ' → ;(3) C is a left partial H -comodule coalgebra via λ ′ . The above theorem can be translated in the following commutative dia-gram: (
C, λ ′ , H ) / / o o H sep points ❴❴❴❴❴❴❴❴❴❴❴❴ (cid:15) (cid:15) O O H sep points ✤✤✤✤✤✤ ( C ∗ , λ ' → , H )( C, λ ' ↼ , H ) u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ (11)Theorems 4.15 and 4.16 show relations between partial comodule coalge-bra, partial module coalgebra and partial module algebra, whenever we startfrom a partial coaction λ ′ . In general, we can not start from an action andinduce a coaction. To do this we require a more strong hypothesis on H ,more precisely, we assume that H is finite dimensional.In fact, if H is finite dimensional, then H = H ∗ (and so H separatepoints). Moreover, given a linear map ↼ : C ⊗ H ∗ → C and a dual basis { h i , h ∗ i } ni =1 for H and H ∗ , then one can induce a linear map λ ′ ↼ : C → H ⊗ C ,by λ ′ ↼ ( c ) = n X i =1 h i ⊗ c ↼ h ∗ i f ( c − ) c = c ↼ f, for all c ∈ C and f ∈ H ∗ .Thus, for a finite dimensional Hopf algebra H , we can induce a coactionof H on a coalgebra C from a given action of H ∗ on C .In [2], there is a similar construction for (right) partial H -comodule al-gebras and (left) partial H ∗ -module algebras. Hence we have that these fourpartial structures are close related, in the following sense: Theorem 4.17.
Let C be a coalgebra and suppose that the Hopf algebra H is finite dimensional. Then the following statements are equivalent:(1) C is a left partial H -comodule coalgebra;(2) C ∗ is a right partial H -comodule algebra;(3) C is a right partial H ∗ -module coalgebra;(4) C ∗ is a left partial H ∗ -module algebra.The relations between the correspondent actions and coactions are given inthe following way, for any c ∈ C, α ∈ C ∗ and f ∈ H ∗ : α ( c − ) c = α ( c ) α +1 (12)( f → α )( c ) = α ( c ↼ f ) (13) c ↼ f = f ( c − ) c (14) f → α = α f ( α +1 ) , (15) where λ ′ : c c − ⊗ c and ρ ′ : α α ⊗ α +1 are the partial coactions on C and C ∗ , respectively. Using Theorem 4.17, we can extend the Diagram (11) to the followingcommutative diagram, under the hypothesis that the Hopf algebra is finitedimensional: (
C, λ ′ , H ) o o / / O O (cid:15) (cid:15) ( C ∗ , λ ' → , H )( C, λ ' ↼ , H ) ( C ∗ , ρ ′ , H ) (cid:15) (cid:15) O O / / o o (16)22 .2 Globalization for partial comodule coalgebras In this section, our goal is to introduce the concept of globalization forpartial comodules coalgebras.Let D be a left H -comodule coalgebra via λ : d d − ⊗ d ∈ H ⊗ D and C a subcoalgebra of D . In order to induce a coaction on C we can restrictthe coaction λ to C , but in general λ ( C ) H ⊗ C . However, if there is alinear map π : D → C we consider the composite map λ ′ : C −→ H ⊗ Cc c − ⊗ π ( c ) . (17)The following result gives us conditions on the map π for the above mapbecomes a partial coaction on C . Proposition 4.18 (Induced partial comodule coalgebra) . Let ( D, λ ) be aleft H -comodule coalgebra, C a subcoalgebra of D and π : D → C a comulti-plicative projection such that ( I ⊗ I ⊗ π )( I ⊗ λπ ) λ ( c ) = ( I ⊗ I ⊗ π )( I ⊗ λ ⊗ επ )( I ⊗ τ ∆) λ ( c ) , (18) for any c ∈ C . Then C is a left partial H -comodule coalgebra, with structuregiven by Equation (17) .Proof. First of all, since π is a projection from D onto C , then λ ′ satisfiesthe condition (PCC1). In fact, given c ∈ C , we have( ε ⊗ I ) λ ′ ( c ) = ε ( c − ) π ( c )= π ( ε ( c − ) c ) ( CC ) = π ( c )= c, where the last equality holds since π is a projection (and so π ( c ) = c , for all c ∈ C ).Now, since π is a comultiplicative map (i.e. ∆ ◦ π = ( π ⊗ π ) ◦ ∆) then itfollows that λ ′ satisfies the condition (PCC2). In fact, let c ∈ C , so( I ⊗ ∆ C ) λ ′ ( c ) = c − ⊗ ∆( π ( c ))= c − ⊗ ( π ⊗ π )(∆( c ))= c − ⊗ π ( c ) ⊗ π ( c ) ( CC ) = c − c − ⊗ π ( c ) ⊗ π ( c )23 ( m H ⊗ I ⊗ I )( I ⊗ τ C,H ⊗ I )( λ ′ ⊗ λ ′ )∆ C ( c ) . Finally, since π satisfies Equation (18) we have that, for all c ∈ C ( I ⊗ λ ′ ) λ ′ ( c ) = c − ⊗ c − ⊗ c = c − ⊗ π ( c ) − ⊗ π ( π ( c ) ) ( ) = c − ⊗ c − ⊗ ε ( π ( c )) π ( c ) ( CC ) = c − c − ⊗ c − ⊗ ε ( π ( c )) π ( c ) ( CC ) = c − ε ( π ( c )) c − ⊗ c − ⊗ π ( c )= c − ε ( c ) c − ⊗ c − ⊗ c = ( m H ⊗ I ⊗ I ) {∇ ⊗ [(∆ H ⊗ I ) λ ′ ] } ∆ C ( c ) . Therefore, C is a left partial H -comodule coalgebra. Remark 4.19.
One can note that the converse of Proposition 4.18 is alsotrue. Indeed, supposing C a subcoalgebra of a comodule coalgebra D and π : D → C a comultiplicative projection such that C is a partial comodulecoalgebra by λ ′ ( c ) = c − ⊗ π ( c ), hence Equation (18) holds.The proof of the above statement follows straight from the calculationsmade in Proposition 4.18.We are now able to define a globalization for a partial comodule coalgebra,as follows. Definition 4.20 (Globalization for partial comodule coalgebra) . Let C be aleft partial H -comodule coalgebra. A globalization for C is a triple ( D, θ, π ) ,where D is an H -comodule coalgebra, θ is a coalgebra monomorphism from C into D and π is a comultiplicative projection from D onto θ ( C ) , such thatthe following conditions hold:(GCC1) x − ⊗ π ( x ) − ⊗ π ( π ( x ) ) == x − ⊗ x − ⊗ ε ( π ( x )) π ( x ) , for all x ∈ θ ( C ) ;(GCC2) θ is an equivalence of partial H -comodule coalgebra;(GCC3) D is the H -comodule coalgebra generated by θ ( C ) . Remark 4.21.
The first item in Definition 4.20 tells us that it is possible todefine the induced partial comodule coalgebra on θ ( C ).24he second one tells us that this induced partial coaction coincides withthe original, and this fact is translated in the commutative diagram bellow: C λ ′ / / θ (cid:15) (cid:15) H ⊗ C I ⊗ θ (cid:15) (cid:15) θ ( C ) λ π / / H ⊗ θ ( C ) (cid:8) (19)Moreover, the second condition can be seen as θ ( c ) − ⊗ π ( θ ( c ) ) = c − ⊗ θ ( c ) , (20)for all c ∈ C .Finally, the last condition of Definition 4.20 tells us that there is no propersubcomodule coalgebra of D containing θ ( C ). Given a left partial H -comodule coalgebra C we can induce a structureof right partial H -module coalgebra on C (see Theorem 4.15). It is also truethat a (global) H -comodule coalgebra induces a (global) H -module coalge-bra. Therefore, given a globalization ( D, θ, π ) for C , one can ask: Is theresome relation between D and C when viewed as H -module coalgebras (globaland partial, respectively)? Here we study a little bit more these structures inorder to answer this question. The notations previously used are kept.Let C be a left partial H -comodule coalgebra and suppose that ( D, θ, π )is a globalization for C . From Theorem 4.15, we have that C is a right partial H -module coalgebra with partial action given by c ↼ f = f ( c − ) c , for all c ∈ C and f ∈ H . Clearly, the same is true for D , i.e., we have astructure of H -module coalgebra on D given by d ◭ f = f ( d − ) d , for all d ∈ D and f ∈ H . Theorem 4.22.
Let C be a left partial H -comodule coalgebra and supposethat ( D, θ, π ) is a globalization for C . If H separate points, then ( D, θ, π ) isalso a globalization for C , as right partial H -module coalgebra. roof. Since θ is a coalgebra monomorphism from C into D and π is a comul-tiplicative projection from D onto θ ( C ), in order to induce a structure ofpartial H -module coalgebra on θ ( C ) we just need to check that the Equa-tion (3) holds (see Proposition 3.18). For this, let x = θ ( c ) ◭ g ∈ θ ( C ) ◭ H and f ∈ H , so π ( π ( x ) ◭ f ) = π ( π ( θ ( c ) g ( θ ( c ) − )) ◭ f )= g ( θ ( c ) − ) π ( π ( θ ( c ) ) f ( π ( θ ( c ) ) − ))= g ( θ ( c ) − ) f ( π ( θ ( c ) ) − ) π ( π ( θ ( c ) ) ) ( ) = g ( θ ( c ) − ) f ( θ ( c ) − ) π ( θ ( c ) ) επ ( θ ( c ) )= f (( θ ( c ) ◭ g ) − ) π (( θ ( c ) ◭ g ) ) επ (( θ ( c ) ◭ g ) )= f ( x − ) π ( x ) επ ( x )= π ( x ◭ f ) επ ( x )= π ( ε ( π ( x )) x ◭ f ) . Thus, θ ( C ) has a structure of partial H -module coalgebra induced from thestructure of module coalgebra of D .Now we show that θ is a morphism of partial actions. In fact, let c ∈ C ,so θ ( c ) ↼ f = π ( θ ( c ) ◭ f )= f ( θ ( c ) − ) π ( θ ( c ) ) ( ) = f ( c − ) θ ( c )= θ ( f ( c − ) c )= θ ( c ↼ f ) . Therefore, we just need to show that the (GMC3) in Definition 3.20 holds.Let M be any H -submodule coalgebra of D containing θ ( C ). We needto show that M = D , and for this it is enough to show that M is an H -subcomodule coalgebra of D .Take f ∈ H , m ∈ M , and consider { h i } a basis of H . Let { h ∗ i } be theset contained in H ∗ whose elements are all the dual maps of the h i ’s. Thenwrite λ ( m ) ∈ H ⊗ D in terms of the basis of H , i.e., λ ( m ) = n X i =0 h i ⊗ m i , where the m i ’s are non-zero elements, at least, in D .26ince D is an H -comodule, so it is an H ∗ -module via the same action of H . Moreover, the action of H on D is a restriction of the action of H ∗ .Since H separate points, it follows by Jacobson Density Theorem that if m ∈ M then there exists n h m ) i ∈ H o such that m ◭ h m ) i = m ◭ h ∗ i , foreach i . Thus we have that m ◭ h m ) j = m ◭ h ∗ j = n X i =0 h ∗ j ( h i ) m i = m j and so each m i lies in M . Then M is an H -subcomodule of D , so M is an H -subcomodule coalgebra of D containing θ ( C ), which implies M = D , andthe proof is complete. Now we construct a globalization for a left partial comodule coalgebra C in a special situation. First of all, remember that if M is a right H -moduleand H separate points, then we have a linear map ϕ : M −→ Hom( H , M ) m ϕ ( m )( f ) = m · f and an injective linear map γ : H ⊗ M −→ Hom( H , M ) h ⊗ m γ ( h ⊗ m )( f ) = f ( h ) m. In the above situation, we say that M is a rational H -module if ϕ ( M ) ⊆ γ ( H ⊗ M ) (cf. [9, Definition 2.2.2]).This definition can be seen in the following commutative diagram:Hom( H , M ) M ϕ ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) λ / / ❴❴❴❴❴❴❴❴❴❴❴❴❴ H ⊗ M γ a a ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ (21)Notice that, given a rational H -module M we have on it a structure of H -comodule via λ : M → H ⊗ M satisfying, for any m ∈ Mλ ( m ) = X h i ⊗ m i ⇐⇒ m · f = X f ( h i ) m i , for all f ∈ H . (22)27rom Theorem 4.22, it follows that exists a natural way to look for aglobalization for a partial coaction of H on C , i.e., to get a globalizationfor C we should see it as a partial module coalgebra and then consider itsstandard globalization as module, under the hypothesis that H separatepoints.Thus, given a left partial H -comodule coalgebra ( C, λ ′ ) we have fromTheorem 3.25 that ( C ⊗ H , θ, π ) is a globalization for C as right partial H -module coalgebra.We desire C ⊗ H to be a globalization for C as partial comodule coalge-bra, but, in general, it is not even an H -comodule coalgebra.In order to overcome this problem we will suppose that C ⊗ H is arational H -module and, therefore, we have that C ⊗ H is an H -comodulewith coaction satisfying Equation (22), i.e., the following holds λ ( c ⊗ f ) = X h i ⊗ c i ⊗ f i ⇐⇒ c ⊗ ( f ∗ g ) = X g ( h i ) c i ⊗ f i , (23)for any c ⊗ f ∈ C ⊗ H and g ∈ H Therefore, by Theorem 4.16, C ⊗ H is an H -comodule coalgebra. Nowwe are in position to show that C ⊗ H is a globalization for C as partialcomodule colagebra, as follows. Theorem 4.23.
Let C be a left partial H -comodule coalgebra. With the abovenotations, if C ⊗ H is a rational H -module and H separate points, then ( C ⊗ H , θ, π ) is a globalization for C .Proof. By the above discussed, C ⊗ H is an H -comodule coalgebra, θ : C → C ⊗ H is a coalgebra map and π : C ⊗ H ։ θ ( C ) is a comultiplicativeprojection.Now, we can show directly that the conditions (GCC1)-(GCC3) hold. Let x ∈ C ⊗ H , and f, g ∈ H , so( f ⊗ g ⊗ I )[ x − ⊗ π ( x ) − ⊗ π ( π ( x ) )] == f ( x − ) g ( π ( x ) − ) π ( π ( x ) )= f ( x − ) π ( g ( π ( x ) − ) π ( x ) )= f ( x − ) π [ π ( x ) ◭ g ]= π [ π ( x f ( x − )) ◭ g ]= π [ π ( x ◭ f ) ◭ g ] ( GM C ) = π [ επ (( x ◭ f ) )( x ◭ f ) ◭ g ]= π [ επ ( x ◭ f )( x ◭ f ) ◭ g ]28 π [ επ ( x f ( x − ))( x f ( x − ) ◭ g )]= f ( x − ) f ( x − ) επ ( x ) π ( x ◭ g )= f ( x − x − ) επ ( x ) π ( x ◭ g ) ( P CC ) = f ( x − ) επ ( x ) π ( x ◭ g )= f ( x − ) επ ( x ) π ( x g ( x − ))= f ( x − ) g ( x − ) επ ( x ) π ( x )= ( f ⊗ g ⊗ I )( x − ⊗ x − ⊗ επ ( x ) π ( x )) . Since H separate points, the condition (GCC1) is satisfied.To prove the condition (GCC2) take c ∈ C and note that( g ⊗ I )[ θ ( c ) − ⊗ π ( θ ( c ) )] = g ( θ ( c ) − ) π ( θ ( c ) )= π ( g ( θ ( c ) − ) θ ( c ) )= π ( θ ( c ) ◭ g )= θ ( c ↼ g )= g ( c − ) θ ( c )= ( g ⊗ I )[ c − ⊗ θ ( c )] . Since H separate points, thus θ is an equivalence of partial coactions.Finally, to show that C ⊗ H is generated by θ ( C ), consider a subco-module coalgebra M of C ⊗ H containing θ ( C ). By Theorem 4.15, M is an H -submodule coalgebra of C ⊗ H containing θ ( C ). Thus, it follows fromcondition (GMC3) that M = C ⊗ H .Therefore C ⊗ H is a globalization for C as a partial H -comodule coal-gebra.The globalization above constructed is called the standard globalization for a partial comodule coalgebra. Remark 4.24.
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