Partial Coactions of Weak Hopf Algebras on Coalgebras
aa r X i v : . [ m a t h . R A ] S e p PARTIAL COACTIONS OF WEAK HOPF ALGEBRAS ON COALGEBRAS
GRAZIELA FONSECA, ENEILSON FONTES, AND GRASIELA MARTINI
Abstract.
It will be seen that if H is a weak Hopf algebra in the definition of coaction of weak bialgebrason coalgebras [15], then a definition property is suppressed giving rise to the (global) coactions of weak Hopfalgebras on coalgebras. The next step will be introduce the more general notion of partial coactions of weakHopf algebras on coalgebras as well as a family of examples via a fixed element on the weak Hopf algebra,illustrating both definitions: global and partial. Moreover, it will also be presented how to obtain a partialcomodule coalgebra from a global one via projections, giving another way to find examples of partial coactionsof weak Hopf algebras on coalgebras. In addition, the weak smash coproduct [15] will be studied and it will beseen under what conditions it is possible to generate a weak Hopf algebra structure from the coproduct and thecounit defined on it. Finally, a dual relationship between the structures of partial action and partial coactionof a weak Hopf algebra on a coalgebra will be established. Key words:
Weak Hopf algebra, globalization, dualization, partial comodule coalgebra, weak smash coproduct.
Mathematics Subject Classification: primary 16T99; secondary 20L05 Introduction
Partial action theory appeared firstly in [11] in the context of operator algebra. Later, in [10], M. Dokuchaevand R. Exel brought partial actions to a purely algebraic context contributing to the development of classicalresults, such as Galois theory, in the case of partial actions of groups on rings.Following this line of research, S. Caenepeel and K. Janssen introduced the notions of partial actions andcoactions of Hopf algebras on algebras in [6]. The main idea of studying partial actions for the context of Hopfalgebras is to generalize the results obtained for partial group actions to this broader context. The notions ofpartial actions and coactions of Hopf algebras on coalgebras appeared for the first time in [9], dualizing thestructures introduced in [6].As a natural task, in [8], was introduced the notion of partial actions of weak Hopf algebras on algebras. Inthis work, the authors extended many results of the classic theory for this setting.We introduced in [7] the theory of partial actions of weak Hopf algebras on coalgebras, inspired by thenotion of partial action of a Hopf algebra on a coalgebra, presented in [9]. Basically, it was constructed in [7] acorrespondence between a partial action of a groupoid G on a coalgebra C and a partial action of the groupoidalgebra k G on the coalgebra C .In the present work, we give successions to the theory of partial actions. The notion of partial and globalcoactions of weak Hopf algebras on coalgebras is introduced as well as some important properties and examples.In the sequel, we will study the weak smash coproduct presented in [15] in order to see under what conditionsthis structure is a weak Hopf algebra. We divide this paper as follows:The second section is devoted to the study of weak Hopf algebras, their properties and some examplesthat will be commonly used throughout the text. A weak bialgebra is a vector space that has a structure ofalgebra and coalgebra simultaneously, with a compatibility property between these structures. The axioms ofweak bialgebra appear for the first time in [4]. If a weak bialgebra is provided with an anti-homomorphism ofalgebras and coalgebras, them we say that is a weak Hopf algebra. The main difference between a weak Hopfalgebra and a Hopf algebra is that in the case of a Hopf algebra the counit is an algebra homomorphism. The first author was partially supported by CNPq, Brazil.
The concept of coaction of a weak bialgebra on a coalgebra was introduced in [15]. In section 3, the coactionof a weak Hopf algebra on a coalgebra is presented. Generalizing this concept, the definition of partial coactionof a weak Hopf algebra on a coalgebra is exhibited with its properties and a family of examples. It is alsoascertained what conditions are necessary and sufficient for a partial comodule coalgebra to be generated froma global comodule coalgebra via a projection.Section 4 is intended to investigate the weak smash coproduct presented in [15]. The idea is to constructa weak Hopf algebra from the existing coalgebra structure in the weak smash coproduct. Historically, theconstruction of Hopf algebras and weak Hopf algebras from global and partial (co)actions has been studied byseveral authors. This can be seen in texts such as [1], [13] and [14]. This shows a great concern in presentingnew examples of such structures. Our contribution is to make the weak smash coproduct into a weak Hopfalgebra under certain conditions.From now, some notations will be fixed. It will be denoted by k a generic field, unless some additionalspecification is made about such structure. Moreover, every tensorial product will be considered over the field k , then, it will be used the notation ⊗ instead of ⊗ k . A will always denote an algebra, C a coalgebra and H aweak Hopf algebra. Throughout the text other properties may be required over the structures A , C e H , butthey will be duly mentioned. Besides that, every map will be considered k -linear and the vector spaces will beconsidered over the field k . Finally, the isomorphism V ⊗ k ≃ V ≃ k ⊗ V will be used automatically for everyvector space V . 2. Preliminaries
In this section, we present few results of weak Hopf algebras. For more details we refer [2], [3] and [4].A weak bialgebra ( H, m, u, ∆ , ε ) (or simply H ) is a vector space such that ( H, m, u ) is an algebra, ( H, ∆ , ε )is a coalgebra, and, in addition, the following conditions are satisfied for all h, k ∈ H :(i) ∆( hk ) = ∆( h )∆( k );(ii) ε ( hkℓ ) = ε ( hk ) ε ( k ℓ ) = ε ( hk ) ε ( k ℓ );(iii) (1 H ⊗ ∆(1 H ))(∆(1 H ) ⊗ H ) = (∆(1 H ) ⊗ H )(1 H ⊗ ∆(1 H )) = ∆ (1 H ).Since ∆ is multiplicative, we conclude that ∆( h ) = ∆( h H ) = ∆(1 H h ), then h ⊗ h = h ⊗ h = 1 h ⊗ h . (1)It is possible to use ε to define the following linear maps ε t : H → H and ε s : H → Hh ε (1 h )1 h ε ( h ) . Then, it can be defined the vector spaces H t = ε t ( H ) and H s = ε s ( H ). Thus, for any weak bialgebra H , everyelement h ∈ H can be written as h = ( ε ⊗ I )∆( h ) = ( ε ⊗ I )∆(1 H h ) = ε t ( h ) h , (2) h = ( I ⊗ ε )∆( h ) = ( I ⊗ ε )∆( h H ) = h ε s ( h ) . (3) Proposition 2.1.
Let H be a weak bialgebra. Then, the following properties hold for all h, k ∈ Hε t ( ε t ( h )) = ε t ( h )(4) ε s ( ε s ( h )) = ε s ( h )(5) ε ( hε t ( k )) = ε ( hk )(6) ε ( ε s ( h ) k ) = ε ( hk )(7) ∆(1 H ) ∈ H s ⊗ H t (8) ε t ( hε t ( k )) = ε t ( hk )(9) ε s ( ε s ( h ) k ) = ε s ( hk )(10) ARTIAL COACTIONS OF WEAK HOPF ALGEBRAS ON COALGEBRAS 3 ∆( h ) = 1 h ⊗ for all h ∈ H t (11) ∆( h ) = 1 ⊗ h for all h ∈ H s (12) h ⊗ ε t ( h ) = 1 h ⊗ (13) ε s ( h ) ⊗ h = 1 ⊗ h (14) hε t ( k ) = ε ( h k ) h (15) ε s ( h ) k = k ε ( hk ) . (16)Therefore, H t and H s are subalgebras of H such that contain 1 H and hk = kh for all h ∈ H t and k ∈ H s . (17)Finally, it is still possible to show that ε t ( ε t ( h ) k ) = ε t ( h ) ε t ( k )(18) ε s ( hε s ( k )) = ε s ( h ) ε s ( k ) , (19)for all h, k ∈ H .Let H be a weak bialgebra. We say that H is a weak Hopf algebra if there is a linear map S : H −→ H ,called antipode , which satisfies:(i) h S ( h ) = ε t ( h );(ii) S ( h ) h = ε s ( h );(iii) S ( h ) h S ( h ) = S ( h ) , for all h ∈ H . The antipode of a weak Hopf algebra is anti-multiplicative , that is, S ( hk ) = S ( k ) S ( h ), and anti-comultiplicative , which means S ( h ) ⊗ S ( h ) = S ( h ) ⊗ S ( h ). Proposition 2.2.
Let H be a weak Hopf algebra. Then, the following identities hold for all h ∈ Hε t ( h ) = ε ( S ( h )1 )1 (20) ε s ( h ) = 1 ε (1 S ( h ))(21) ε t ◦ S = ε t ◦ ε s = S ◦ ε s (22) ε s ◦ S = ε s ◦ ε t = S ◦ ε t (23) h ⊗ S ( h ) h = h ⊗ S (1 )(24) h S ( h ) ⊗ h = S (1 ) ⊗ h. (25)Hence, if H is a weak Hopf algebra, S (1 H ) = 1 H , ε ◦ S = ε , S ( H t ) = H s , S ( H s ) = H t and S ( H ) is also aweak Hopf algebra, with the same counit and antipode. It is easy to see that every Hopf algebra is a weak Hopfalgebra. Conversely we have the following result. Proposition 2.3.
A weak Hopf algebra is a Hopf algebra if one of the following equivalent conditions is satisfied: (i) ∆(1 H ) = 1 H ⊗ H ; (ii) ε ( hk ) = ε ( h ) ε ( k );(iii) h S ( h ) = ε ( h )1 H ;(iv) S ( h ) h = ε ( h )1 H ;(v) H t = H s = k H ; for all h, k ∈ H . In order to construct an example of weak Hopf algebra, we present the following definition.
Definition 2.4 (Groupoid) . Consider G a non-empty set with a binary operation partially defined which isdenoted by concatenation. This operation is called product. Given g, h ∈ G , we write ∃ gh whenever the product gh is set (similarly we use ∄ gh whenever the product is not defined). Thus, G is called groupoid if: FONSECA, FONTES, AND MARTINI (i) For all g, h, l ∈ G , ∃ ( gh ) l if and only if ∃ g ( hl ), and, in this case, ( gh ) l = g ( hl );(ii) For all g, h, l ∈ G , ∃ ( gh ) l if and only if ∃ gh and ∃ hl ;(iii) For each g ∈ G there are unique elements d ( g ) , r ( g ) ∈ G such that ∃ gd ( g ), ∃ r ( g ) g and gd ( g ) = g = r ( g ) g ;(iv) For each g ∈ G there exists an element such that d ( g ) = g − g and r ( g ) = gg − .Moreover, the element g − is the only one that satisfies such property and, in addition, ( g − ) − = g , for all g ∈ G . An element e is said identity in G if for some g ∈ G , e = d ( g ) = r ( g − ). Therefore, e = e , which impliesthat d ( e ) = e = r ( e ) and e = e − . We denote G the set of all identities elements of G . Besides that, one candefine the set G = { ( g, h ) ∈ G × G | ∃ gh } of all pairs of elements composable in G . Proposition 2.5.
Let G be a groupoid. Then, for all g, h ∈ G : (i) ∃ gh if and only if d ( g ) = r ( h ) and, in this case, d ( gh ) = d ( h ) and r ( gh ) = r ( g ) ; (ii) ∃ gh if and only if ∃ h − g − and, in this case, ( gh ) − = h − g − . Example 2.6 (Groupoid Algebra) . Let G be a groupoid such that the cardinality of G is finite and k G thevector space with basis indexed by the elements of G given by { δ g } g ∈G . Then, k G is a weak Hopf algebra withthe following structures m ( δ g ⊗ δ h ) = (cid:26) δ gh , if ∃ gh , , otherwise u (1 k ) = 1 G = X e ∈G δ e ∆( δ g ) = δ g ⊗ δ g ε ( δ g ) = 1 k S ( δ g ) = δ g − . Remark that when it is assumed that the dimension of a weak Hopf algebra H is finite, it is obtainedthat the dual structure H ∗ = Hom ( H, k ) is a weak Hopf algebra with the convolution product ( f ∗ g )( h ) = m ( f ⊗ g )( h ) = f ( h ) g ( h ) for all h ∈ H, the unit u H ∗ (1 k ) = 1 H ∗ = ε H , the coprodut defined by the relation∆ H ∗ ( f ) = f ⊗ f ⇔ f ( hk ) = f ( h ) f ( k ) for all h, k ∈ H , and the counit ε H ∗ ( f ) = f (1 H ) . Besides that, wehave ( ε t ) H ∗ ( f ) = f ◦ ε t and ( ε s ) H ∗ ( f ) = f ◦ ε s . Example 2.7 (Dual Groupoid Algebra) . Let G be a finite groupoid and ( k G ) ∗ the vector space with basisindexed by the elements of G given by { p g } g ∈G , where p g ( δ h ) = (cid:26) k , if g = h , , otherwise.Then, ( k G ) ∗ is a weak Hopf algebra with the following structures p g ∗ p h = (cid:26) p g , if g = h , , otherwise 1 ( k G ) ∗ = X g ∈G p g ∆ ( k G ) ∗ ( p g ) = X h ∈G , ∃ h − g p h ⊗ p h − g ε ( k G ) ∗ ( p g ) = p g (1 k G ) S ( k G ) ∗ ( p g ) = p g ◦ S. The following example of weak Hopf algebra was presented by G. B¨ohm and J. G´omes-Torrecillas in [3].
Example 2.8.
Consider G a finite abelian group with cardinality | G | , where | G | is not a multiple of thecharacteristic of k . If we consider k G the algebra with basis indexed by the elements of G and with coalgebrastructure given by ∆( g ) = 1 | G | X h ∈ G gh ⊗ h − ε ( g ) = (cid:26) | G | , if g = 1 G , , otherwise.Then, k G is a weak Hopf algebra with antipode defined by S ( g ) = g . Besides that, ε s ( g ) = ε t ( g ) = g , for all g ∈ G , what implies that H s = H t = k G . ARTIAL COACTIONS OF WEAK HOPF ALGEBRAS ON COALGEBRAS 5 Comodule Coalgebra
Consider H a weak bialgebra. In [15], Yu. Wang and L. Zhang defined C a (left) H -comodule coalgebra whenthere exits a linear map ρ : C −→ H ⊗ Cc c − ⊗ c such that for all c ∈ C (CC1) ( ε H ⊗ I C ) ρ ( c ) = c (CC2) ( I H ⊗ ∆ C ) ρ ( c ) = ( m H ⊗ I C ⊗ I C )( I H ⊗ τ C,H ⊗ I C )( ρ ⊗ ρ )∆ C ( c )(CC3) ( I H ⊗ ρ ) ρ ( c ) = (∆ H ⊗ I C ) ρ ( c )(CC4) ( I H ⊗ ε C ) ρ ( c ) = ( ε t ⊗ ε C ) ρ ( c ) . In this case, it is said that H coacts on the coalgebra C . Proposition 3.1.
Let H be a weak Hopf algebra. If there exists a linear map ρ : C −→ H ⊗ Cc c − ⊗ c that satisfies (CC1)-(CC3), then the condition (CC4) is satisfied.Proof. Suppose that there is a linear map ρ that satisfies the conditions (CC1)-(CC3), then for every c ∈ Cε t ( c − ) ε C ( c ) = c − S H ( c − ) ε C ( c ) ( CC = c − S H ( c − ) ε C ( c ) ( CC = c − c − S H ( c − ) ε C ( c ) ε C ( c ) ( CC = c − c − S H ( c − ) ε C ( c ) ε C ( c ) ( ) = c − ε H ( c − c − ) ε C ( c ) ε C ( c ) ( CC = c − ε C ( c ) ε C ( c ) ε H ( c − c − ) ( CC = c − ε C ( c ) ε C ( c ) ε H ( c − ) ( CC = c − ε C ( c ) ε C ( c )= ( I H ⊗ ε C ) ρ ( c ) . (cid:3) Example 3.2. [15] Consider H a weak Hopf algebra finite dimensional. Then, H is a H ∗ -comodule coalgebradefined by ρ : H −→ H ∗ ⊗ Hh i h i ∗ ⊗ h i where { h i } ni =1 is a basis for H and { h i ∗ } ni =1 is the dual basis for H ∗ .3.1. Partial Comodule Coalgebra.
In this section, the main purpose is to introduce the concept of apartial coaction of a weak Hopf algebra H on a coalgebra. It is also introduced some examples that supportthe theory exposed here and some properties. FONSECA, FONTES, AND MARTINI
Definition 3.3.
We say that C is a (left) partial H -comodule coalgebra (or that H coacts partially on C ) ifthere exists a linear map ρ : C −→ H ⊗ Cc c − ⊗ c such that for all c ∈ C (CCP1) ( ε H ⊗ I C ) ρ ( c ) = c (CCP2) ( I H ⊗ ∆ C ) ρ ( c ) = ( m H ⊗ I C ⊗ I C )( I H ⊗ τ C,H ⊗ I C )( ρ ⊗ ρ )∆ C ( c )(CCP3) ( I H ⊗ ρ ) ρ ( c ) = ( m H ⊗ I H ⊗ I C )[( I H ⊗ ε C )( ρ ( c )) ⊗ (∆ H ⊗ I C )( ρ ( c ))].Moreover, C is said a (left) symmetric partial H -comodule coalgebra if, in addition, satisfies( I H ⊗ ρ ) ρ ( c ) = ( m H ⊗ I H ⊗ I C )( I H ⊗ τ H ⊗ C,H )[(∆ H ⊗ I C )( ρ ( c )) ⊗ ( I H ⊗ ε C )( ρ ( c ))] . Remark 3.4.
Every H -comodule coalgebra is a partial H -comodule coalgebra. Indeed for every c ∈ C ( I H ⊗ ρ ) ρ ( c ) = c − ⊗ c − ⊗ c CC = c − ⊗ c − ⊗ c CC = c − c − ⊗ c − c − ⊗ c ε C ( c )3 . ε t ( c − ) c − ⊗ ε t ( c − ) c − ⊗ c ε C ( c ) ( ) = 1 H ε t ( c − ) c − ⊗ H c − ⊗ c ε C ( c ) ( ) = ε t ( c − )1 H c − ⊗ H c − ⊗ c ε C ( c ) ( ) = c − c − ⊗ c − ⊗ c ε C ( c )= ( m H ⊗ I H ⊗ I C )[( I H ⊗ ε C )( ρ ( c )) ⊗ (∆ H ⊗ I C )( ρ ( c ))] . Proposition 3.5.
Let C be a partial H -comodule coalgebra. Then, C is a H -comodule coalgebra if and only if c − ε C ( c ) = ε t ( c − ) ε C ( c ) for all c ∈ C .Proof. Suppose that C is a partial H -comodule coalgebra that satisfies c − ε C ( c ) = ε t ( c − ) ε C ( c ), then it isenough to show that ( I H ⊗ ρ ) ρ ( c ) = (∆ H ⊗ I C ) ρ ( c ):( I H ⊗ ρ ) ρ ( c ) ( ) = c − ε C ( c )1 H c − ⊗ H c − ⊗ c ) = 1 H ε t ( c − ) ε C ( c ) c − ⊗ H c − ⊗ c ) = ε t ( c − ) ε C ( c ) c − ⊗ ε t ( c − ) c − ⊗ c CCP = c − ε C ( c ) ⊗ c − ⊗ c = (∆ H ⊗ I C ) ρ ( c ) . (cid:3) Example 3.6.
Consider k G a groupoid algebra, where G is generated by the disjoint union of the finite groups G and G . Therefore, the group algebra k G is a partial ( k G ) ∗ -comodule coalgebra via ρ : k G → ( k G ) ∗ ⊗ k G h p e ⊗ h, where e is the identity element of G . ARTIAL COACTIONS OF WEAK HOPF ALGEBRAS ON COALGEBRAS 7
Coactions via ρ h . In this section it is explored a specific family of examples of partial comodulecoalgebra. We say that C is a (left) H -comodule coalgebra via ρ h if, for some h ∈ H fixed, the linear map ρ h : C −→ H ⊗ Cc h ⊗ c defines a structure of comodule coalgebra on C . Note that since H is a weak Hopf algebra, the above applicationdoes not always defines a structure of H -comodule coalgebra on C . To see this, it is enough to observe that ρ H ( c ) = 1 H ⊗ c turns out C on a comodule coalgebra if and only if H is a Hopf algebra. The following resulthas the intention to characterize the properties that an element h ∈ H must to satisfy in order that ρ h be acoaction of H on a coalgebra C . Proposition 3.7.
We say that C is a (left) H -comodule coalgebra via ρ h if and only if (i) ε H ( h ) = 1 k (ii) h = h (iii) ∆ H ( h ) = h ⊗ h. Proof.
The proof follows immediately from the definition comodule coalgebra via ρ h . (cid:3) Besides that, if h ∈ H satisfies the properties of Proposition 3.7, then h = ε t ( h ). Example 3.8.
Consider k G the groupoid algebra generated by a groupoid G . Thus, fixing an element e ∈ G , ρ δ e ensure a structure of k G -comodule coalgebra on any coalgebra C by Proposition 3.7.Moreover, it is easy to see that ρ δ g gives a structure of H -comodule coalgebra on a coalgebra C if and onlyif g ∈ G . Example 3.9.
Consider k G the weak Hopf algebra given in Example 2.8, h an element in G and C a coalgebra. C is a k G -comodule coalgebra via ρ h if and only if G = { G } .Thinking on the partial case, we say that C is a (left) partial H -comodule coalgebra via ρ h , for some fixed h ∈ H , if the linear map ρ h : C −→ H ⊗ Cc h ⊗ c determines a structure of partial H -comodule coalgebra on C . Proposition 3.10. C is a (left) partial H -comodule coalgebra via ρ h if and only if (i) ε H ( h ) = 1 k (ii) ( h ⊗ H )∆ H ( h ) = h ⊗ h. Observe that if h ∈ H satisfies the properties (i) and (ii), then h = h . Proof.
The proof follows immediately from the definition of partial H -comodule coalgebra via ρ h . (cid:3) Note that C is a symmetric partial H -comodule coalgebra via ρ h if and only if(i) ε H ( h ) = 1 k (ii) ( h ⊗ H )∆ H ( h ) = h ⊗ h (iii) ∆ H ( h )( h ⊗ H ) = h ⊗ h. Remark 3.11.
If in addition h ∈ H satisfies h = ε t ( h ), then C is a H -comodule coalgebra.If k G is the weak Hopf algebra given in Example 2.8 and h is an element in G , then every parcial k G -comodulecoalgebra via ρ h is actually a (global) k G -comodule coalgebra via ρ h thanks to Remark 3.11. The followingexample was inspired by Example 3 . . k is seen as a partial k G -comodule algebra. FONSECA, FONTES, AND MARTINI
Example 3.12.
Consider k G the groupoid algebra where the groupoid G is the disjoint union of the finitesgroups G and G . Under these conditions, any coalgebra C is a partial k G -comodule coalgebra via ρ h where h = X g ∈ G | G | δ g . We can also characterize the coaction of the weak Hopf algebra k G on k when G a finite groupoid. Example 3.13. k is a partial k G -comodule coalgebra via ρ h if and only if h = X g ∈ N | N | δ g , for N some groupin G . Example 3.14. k is a partial ( k G ) ∗ -comodule coalgebra via ρ f if and only if f = X g ∈ N p g , where N is a groupin G .3.3. Induced Coaction.
Let H be a weak Hopf algebra and C a coalgebra. Suppose that C is a H -comodulecoalgebra via ρ : C −→ H ⊗ Cc c − ⊗ c Our goal in this section is to construct a symmetric partial H -comodule coalgebra from a H -comodule coalgebra.For this, consider D ⊆ C a subcoalgebra of C such that there exists a projection π : C → C onto D , i.e., π ( π ( c )) = π ( c ) for all c ∈ C, and Imπ = D. Under these conditions, it can be obtained the following result.
Proposition 3.15. D is a symmetric partial H -comodule coalgebra via ρ : D −→ H ⊗ Dd ( I H ⊗ π ) ρ ( d ) = d − ⊗ π ( d ) if and only if the projection π satisfies: (i) d − ⊗ ∆ D ( π ( d )) = d − ⊗ ( π ⊗ π )(∆ D ( d )) ; (ii) d − ⊗ π ( d ) − ⊗ π ( π ( d ) ) = d − ε D ( π ( d )) d − ⊗ d − ⊗ π ( d )= d − d − ε D ( π ( d )) ⊗ d − ⊗ π ( d ) ;for all d ∈ D . In this case we say that ρ is an induced coaction.Proof. Suppose that D is a symmetric partial H -comodule coalgebra via ρ ( c ) = ( I H ⊗ π ) ρ ( d ) = d − ⊗ π ( d ) . Therefore, d − ⊗ ∆ D ( π ( d )) = ( I H ⊗ ∆ D )( ρ ( d )) ( CCP = ( m H ⊗ I D ⊗ I D )( I H ⊗ τ H,D ⊗ I D )( ρ ⊗ ρ )∆ D ( d ) ( CC = d − ⊗ ( π ⊗ π )(∆ D ( d )) . Besides that, d − ⊗ π ( d ) − ⊗ π ( π ( d ) ) = ( I H ⊗ ρ ) ρ ( d ) ( CCP = ( m H ⊗ I H ⊗ I D )[( I H ⊗ ε D )( ρ ( d )) ⊗ (∆ H ⊗ I D )( ρ ( d ))]= d − ε D ( π ( d )) d − ⊗ d − ⊗ π ( d ) . Analogously, using the symmetry condition, d − ⊗ π ( d ) − ⊗ π ( π ( d ) ) = ( I H ⊗ ρ ) ρ ( d ) ARTIAL COACTIONS OF WEAK HOPF ALGEBRAS ON COALGEBRAS 9 = d − d − ε D ( π ( d )) ⊗ d − ⊗ π ( d ) . The converse is immediate. (cid:3)
Note that the induced coaction is a H -comodule coalgebra if and only if ε t ( d − ) ε D ( π ( d )) = d − ε D ( π ( d ))for all d ∈ D . Example 3.16.
Consider G and G finite groups, G the groupoid generated by the disjoint union of thesegroups and k G its groupoid algebra. Define ρ : k G −→ ( k G ) ∗ ⊗ k G h X g ∈ G p g ⊗ hg with { g } g ∈ G basis for the Hopf algebra k G and { p g } g ∈G the dual basis for the weak Hopf algebra ( k G ) ∗ . Thus, k G is a ( k G ) ∗ -comodule coalgebra. Define π : k G → k G g (cid:26) g, if g ∈ D , , otherwise,where D = < l > k , for some l fixed in G . Then, D is a symmetric partial ( k G ) ∗ -comodule coalgebra byProposition 3.15. Moreover, the induced coaction constructed is not global. Indeed, on the one hand, h − ε D ( π ( h )) = X g ∈ G p g ε D ( π ( hg )) hg = l = p h − l . On the other hand, ε t ( k G ) ∗ ( h − ) ε D ( π ( h )) = X g ∈ G ε t k G∗ ( p g ) ε D ( π ( hg )) hg = l = ε t ( k G ) ∗ ( p h − l ) . Note that p h − l = ε t ( k G ) ∗ ( p h − l ). Therefore, D is not a global ( k G ) ∗ -comodule coalgebra.4. Weak Smash Coproduct
Yu. Wang and L. Yu. Zhang, in [15], introduced a new structure generated from a H - comodule coalgebra,the weak smash coproduct. Therefore, a natural question arises: “ Under what conditions the weak smashcoproduct becomes a weak Hopf algebra? ”This section is destined to answer this question and to make a contribution to the existing theory. The weaksmash coproduct was defined being the vector space C × H = < c ⊗ h ε H ( c − h ) > k that has a specific structureof coalgebra. Initially we start showing that this structure of coalgebra is inherited from some properties of thevector space C ⊗ H . Proposition 4.1.
The vector space C ⊗ H has a coassociative coproduct ∆( c ⊗ h ) = c ⊗ c − h ⊗ c ⊗ h . Moreover, if C is an algebra, then C ⊗ H is an algebra with product given by ( c ⊗ h )( d ⊗ k ) = ( cd ⊗ hk ) , and unit ( C ⊗ H ) = 1 C ⊗ H . Proof.
Indeed for all c ∈ C and h ∈ H ( I ⊗ ∆)∆( c ⊗ h ) = c ⊗ c − h ⊗ c ⊗ c − h ⊗ c ⊗ h CC = c ⊗ c − c − h ⊗ c ⊗ c − h ⊗ c ⊗ h CC = c ⊗ c − c − h ⊗ c ⊗ c − h ⊗ c ⊗ h = (∆ ⊗ I )∆( c ⊗ h ) . The properties of associativity and unit follow naturally. (cid:3)
Proposition 4.2.
Let H a weak Hopf algebra and C a H -comodule coalgebra. Then, the vector space C × H = < c ⊗ h ε H ( c − h ) > k is a coalgebra with counit ε ( c × h ) = ε C ( c ) ε H ( c − h ) . Proof.
Indeed for all c ∈ C and h ∈ H ∆( c × h ) = ∆( c ⊗ h ε H ( c − h ))= c ⊗ c − h ε H ( c − h ) ⊗ c ⊗ h CC = c ⊗ c − h ε H ( c − c − h ) ⊗ c ⊗ h CC = c ⊗ c − h ε H ( c − c − h ) ⊗ c ⊗ h = c ⊗ ( c − ) h ε H ( c − ( c − ) h ) ⊗ c ⊗ h ε H ( c − h ) ( CC = c ⊗ c − h ε H ( c − c − h ) ⊗ c ⊗ h ε H ( c − h )= c × c − h ⊗ c × h . Besided that, C × H is counitary since( ε ⊗ I )∆( c × h ) = ( ε ⊗ I )(∆( c ⊗ h ε H ( c − h )))= ( ε ⊗ I )( c ⊗ c − h ε H ( c − h ) ⊗ c ⊗ h )= ε C ( c ) ε H ( c − h ) ε H ( c − h )( c ⊗ h )= ε C ( c − h ) ε H ( c − h )( c ⊗ h ) ( CC = ε H ( c − h ) ε H ( c − h )( c ⊗ h )= ε H ( c − h )( c ⊗ h )= c × h. Similarly, ( I ⊗ ε )∆( c × h ) = ( c ⊗ c − h ) ε H ( c − h ) ε C ( c ) ε H ( h )= ( c ⊗ c − h ) ε H ( c − h ) ε C ( c ) ( CC = ( c ⊗ c − h ) ε H ( c − c − h ) ε C ( c ) ( CC = ( c ⊗ c − h ) ε H ( c − c − h ) ε C ( c )= ( c ⊗ ε t ( c − ) h ) ε H ( c − ε t ( c − ) ) ε C ( c ) ( ) = ( c ⊗ H h ) ε H ( c − H ε t ( c − )) ε C ( c ) ( ) = ( c ⊗ H h ) ε H ( c − ε t ( c − )1 H ) ε C ( c ) ( ) = ( c ⊗ H h ) ε H ( c − ε s (1 H )) ( ) = ( c ⊗ H h ) ε H ( c − ε t ( ε s (1 H ))) ARTIAL COACTIONS OF WEAK HOPF ALGEBRAS ON COALGEBRAS 11( ) = ( c ⊗ H h ) ε H ( c − ε t ( S H (1 H ))) ( ) = ( c ⊗ H h ) ε H ( c − S H (1 H )) ( ) = ( c ⊗ h ) ε H ( c − ε t ( h )) ( ) = ( c ⊗ h ) ε H ( c − h )= c × h. Therefore, C × H = < c ⊗ h ε H ( c − h ) > k is a coalgebra. (cid:3) Definition 4.3.
Let C a weak bialgebra and H a weak Hopf algebra. We say that C is a (left) H - comodulebialgebra if there exists a linear map ρ : C → H ⊗ C such that ρ defines simultaneously a structure of (left) H -comodule coalgebra and (left) H -comodule algebra in C .From now, suppose that C is a H -comodule bialgebra (then C is a weak bialgebra) where H is a commutativeweak Hopf algebra. It follows from the commutativity of H that ε t and ε s are multiplicatives. Moreover, S.Caenepeel and E. De Groot showed in [5] that ε t ◦ ε t = ε t (26) ε s ◦ ε s = ε s (27)where ε t ( h ) = 1 H ε H (1 H h ) and ε s ( h ) = 1 H ε H ( h H ) for all h ∈ H . Since H is commutative ε t ( h ) = 1 H ε H (1 H h )= 1 H ε H ( h H )= ε s ( h ) . And, analogously ε s ( h ) = ε t ( h ) . Therefore, ε t ◦ ε s = ε t (28) ε s ◦ ε t = ε s . (29)These properties for ε t and ε s will be commonly used throughout this section. Lemma 4.4.
The product in C ⊗ H induces a product in C × H given by ( c × h )( b × k ) = ( cb × hk ) , for all c ∈ C and h ∈ H .Proof. Let c, b ∈ C and h, k ∈ H . Then,( c × h )( b × k ) = ( c ⊗ h ε H ( c − h ))( b ⊗ k ε H ( b − k )) ( ) = c b ⊗ h k ε H ( c − ε t ( h )) ε H ( b − ε t ( k )) ( ) = c b ⊗ H h H ′ kε H ( c − S H (1 H )) ε H ( b − S H (1 H ′ )) ( ) = c b ⊗ H h H ′ kε H ( c − ε t ( S H (1 H ))) ε H ( b − ε t ( S H (1 H ′ ))) ( ) = c b ⊗ H h H ′ kε H ( c − ε t ( ε s (1 H ))) ε H ( b − ε t ( ε s (1 H ′ ))) ( ) = c b ⊗ H h H ′ kε H ( c − ε t (1 H )) ε H ( b − ε t (1 H ′ )) ( ) = c b ⊗ H h H ′ kε H ( c − H ) ε H ( b − H ′ ) ( ) = ( cb ) ⊗ H ε H ( ε s (1 H )( cb ) − ) hk ( ) = ( cb ) ⊗ H ε H (( cb ) − ε t ( ε s (1 H ))) hk ) = ( cb ) ⊗ H ε H (( cb ) − ε t ( S H (1 H ))) hk ( ) = ( cb ) ⊗ H ε H (( cb ) − S H (1 H )) hk ( ) = ( cb ) ⊗ ( hk ) ε H (( cb ) − ε t (( hk ) ))= ( cb × hk ) . (cid:3) Note that the associativity of the algebras C and H guarantee the associativity of C × H . We also have that1 C × H = 1 C × H .For the next result consider C × H with the product given in Lemma 4.4 and the coproduct given by∆( c × h ) = c × c − h ⊗ c × h for all c ∈ C and h ∈ H . Lemma 4.5.
Under these conditions, the coproduct defined on C × H is multiplicative.Proof. Given b, c ∈ C and h, k ∈ H ∆(( c × h )( b × k )) = ∆( cb × hk )= ( cb ) × ( cb ) − ( hk ) ⊗ ( cb ) × ( hk ) = c b × ( c b ) − h k ⊗ ( c b ) × h k = c b × c − b − h k ⊗ c b × h k = ( c × c − h )( b × b − k ) ⊗ ( c × h )( b × k )= [( c × c − h ) ⊗ ( c × h )][( b × b − k ) ⊗ ( b × k )]= ∆( c × h )∆( b × k ) . (cid:3) Once proved the multiplicativity of the coproduct of C × H and knowing that the coassociativity follows bythe fact that the coproduct of the weak smash coproduct is inherited from C ⊗ H , we are able to show theproperties of weak bialgebra for ε C × H as we can see in the following lemma. Lemma 4.6.
Let C × H be the weak smash coproduct. Then, the counity ε C × H satisfies ε (( a × k )( c × h )( b × l )) = ε (( a × k )( c × h ) ) ε (( c × h ) ( b × l ))= ε (( a × k )( c × h ) ) ε (( c × h ) ( b × l )) for all a, b, c ∈ C and h, k, l ∈ H .Proof. Let a, b, c ∈ C and h, k, l ∈ H , ε (( a × k )( c × h )( b × l )) = ε ( acb × khl )= ε C (( acb ) ) ε H (( acb ) − khl )= ε C ( a c b ) ε H ( a − c − b − khl )On the one hand, ε (( a × k )( c × h ) ) ε (( c × h ) ( b × l ))= ε C (( ac ) ) ε H (( ac ) − kc − h ) ε C (( c b ) ) ε H (( c b ) − h l )= ε C ( a c ) ε H ( a − c − kc − h ) ε C ( c b ) ε H ( c − b − h l ) ( CC = ε C ( a c ) ε H ( a − c − kc − h ) ε C ( c b ) ε H ( c − b − h l ) ( CC = ε C ( a c ) ε H ( a − c − khb − l ) ε C ( c b ) ARTIAL COACTIONS OF WEAK HOPF ALGEBRAS ON COALGEBRAS 13 = ε C ( a c b ) ε H ( a − c − b − khl ) . On the other hand, ε (( a × k )( c × h ) ) ε (( c × h ) ( b × l ))= ε ( ac × kh ) ε ( c b × c − h l )= ε C (( ac ) ) ε H (( ac ) − kh ) ε C (( c b ) ) ε H (( c b ) − c − h l )= ε C ( a c ) ε H ( a − c − kh ) ε C ( c b ) ε H ( c − b − c − h l ) ( CC = ε C ( a c ) ε H ( a − c − kh ) ε C ( c b ) ε H ( c − b − c − h l ) ( CC = ε C ( a c ) ε H ( a − khlc − b − ) ε C ( c b )= ε C ( a c b ) ε H ( a − c − b − khl ) . (cid:3) Lemma 4.7.
Let C × H be the weak smash coproduct. Then, (1 C × H ⊗ ∆(1 C × H ))(∆(1 C × H ) ⊗ C × H ) = (∆(1 C × H ) ⊗ C × H )(1 C × H ⊗ ∆(1 C × H ))= ( I ⊗ ∆)∆(1 C × H )4 .
5= (∆ ⊗ I )∆(1 C × H ) . Proof.
Indeed,(∆(1 C × H ) ⊗ C × H )(1 C × H ⊗ ∆(1 C × H ))= 1 C ⊗ C − H ε H (1 C − C − H ) ⊗ C C ′ ⊗ H C ′ − ε H (1 C − C ′ − H C ′ − ) ⊗ C ′ ⊗ H ε H (1 C ′ − H )= 1 C ⊗ C − H ε H (1 C − C − ) ε H (1 C − H ) ⊗ C C ′ ⊗ H C ′ − ε H (1 C − C ′ − H C ′ − ) ⊗ C ′ ⊗ H ε H (1 C ′ − H ) ( CC = 1 C ⊗ C − H ε H (1 C − C − ) ⊗ C C ′ ⊗ H C ′ − ε H (1 C − C ′ − C ′ − ) ⊗ C ′ ⊗ H ε H (1 C ′ − H ) ( ) = 1 C ⊗ C − H ε H (1 C − C − ) ⊗ C C ′ ⊗ H C ′ − ε H (1 C ′ − C ′ − ε t (1 C − )) ⊗ C ′ ⊗ H ) = 1 C ⊗ H ′ C − H ε H (1 C − C − ) ⊗ C C ′ ⊗ H C ′ − ε H (1 C ′ − C ′ − H ′ ) ⊗ C ′ ⊗ H = 1 C ⊗ ε s (1 H ′ C ′ − C ′ − ) ε H (1 H ′ C ′ − C ′ − )1 C − H ε H (1 C − C − ) ⊗ C C ′ ⊗ C ′ − H ⊗ C ′ ⊗ H = 1 C ⊗ C − ε s (1 H ′ C ′ − C ′ − )1 H ε H (1 C − C − ) ⊗ C C ′ ⊗ C ′ − H ε H (1 C ′ − C ′ − H ′ ) ⊗ C ′ ⊗ H ) = 1 C ⊗ H ′ C − ε s (1 C ′ − C ′ − )1 H ε H (1 C − C − ) ⊗ C C ′ ⊗ C ′ − H ε H (1 C ′ − C ′ − H ′ ) ⊗ C ′ ⊗ H ) = 1 C ⊗ C − ε s (1 C ′ − C ′ − )1 H ε H (1 C − C − ) ⊗ C C ′ ⊗ C ′ − H ε H (1 C ′ − C ′ − ε t (1 C − )) ⊗ C ′ ⊗ H ) = 1 C ⊗ C − ε s (1 C ′ − C ′ − )1 H ε H (1 C − C − ) ⊗ C C ′ ⊗ C ′ − H ε H (1 C ′ − C ′ − C − ) ⊗ C ′ ⊗ H CC = 1 C ⊗ C − ε s (1 C ′ − C ′ − )1 H ε H (1 C − C − ) ⊗ C C ′ ⊗ C ′ − H ε H (1 C − C ′ − C ′ − ) ⊗ C ′ ⊗ H CC = 1 C ⊗ C − ε s (1 C − )1 H ε H (1 C − C − ) ⊗ C C ⊗ C − H ε H (1 C − C − C − ) ⊗ C ⊗ H ∗ ) = 1 C ⊗ C − C − H ε H (1 C − C − ) ⊗ C C ⊗ C − H ε H (1 C − C − C − ) ⊗ C ⊗ H = 1 C ⊗ C − C − H ε H (1 C − C − ) ⊗ C C ⊗ C − H ε H (1 C − H ) ε H (1 C − C − C − ) ⊗ C ⊗ H CC = 1 C ⊗ C − C − H ε H (1 C − C − ) ⊗ C C ⊗ C − H ε H (1 C − C − C − ) ⊗ C ⊗ H ε H (1 C − H ) ( ) = 1 C ⊗ C − H ′ C − H ε H (1 C − C − H ′ ) ⊗ C C ⊗ C − H ε H (1 C − C − C − ) ⊗ C ⊗ H ε H (1 C − H ) ( ) = 1 C ⊗ C − C − H ε H ( ε s (1 C − )1 C − C − ) ⊗ C C ⊗ C − H ε H (1 C − C − C − ) ⊗ C ⊗ H ε H (1 C − H ) ( ) = 1 C ⊗ C − C − H ε H (1 C − C − C − ) ⊗ C C ⊗ C − H ε H (1 C − C − C − ) ⊗ C ⊗ H ε H (1 C − H )= 1 C ⊗ C − C − H ε H (1 C − H ) ε H (1 C − C − C − ) ⊗ C C ⊗ C − H ε H (1 C − H ) ε H (1 C − C − C − ) ⊗ C ⊗ H ε H (1 C − H )= 1 C × C − C − H ⊗ C C × C − H ⊗ C × H CC = 1 C × C − C ′ − C ′ − H ⊗ C C ′ × C ′ − H ⊗ C ′ × H = 1 C × C − C − H ⊗ C × C − H ⊗ C × H CC = 1 C × C − H ⊗ C × C − H ⊗ C × H = ( I ⊗ ∆)∆(1 C × H ) . In ( ∗ ) it was used the property ρ (1) ∈ H s ⊗ C of Proposition 4.11 of [5]. Similarly it is possible to show(1 C × H ⊗ ∆(1 C × H ))(∆(1 C × H ) ⊗ C × H ) = ( I ⊗ ∆)∆(1 C × H ) . (cid:3) Therefore, we obtain the following result.
Proposition 4.8.
Let C be a weak bialgebra and H a commutative weak Hopf algebra such that C is a H -comodule bialgebra. Then, C × H is a weak bialgebra. The next step is to know if we can give a weak Hopf algebra structure to C × H . However, it is necessary toimpose a natural condition on C , as it can be seen in Theorem 4.9. Teorema 4.9.
Let C and H be two weak Hopf algebras such that C is a H -comodule bialgebra and H iscommutative. Then, C × H is a weak Hopf algebra. ARTIAL COACTIONS OF WEAK HOPF ALGEBRAS ON COALGEBRAS 15
Proof.
By Proposition 4.8 we know that C × H is a weak bialgebra, so, it is enough to define a map S C × H in C × H and show that S C × H satisfies the properties of antipode of a weak Hopf algebra. Define S C × H by S C × H ( c × h ) = S C ( c ) × S H ( c − h ) , for all c ∈ C and h ∈ H .(I) ( c × h ) S (( c × h ) ) = ε t ( c × h ), indeed for all c ∈ C and h ∈ H ( c × h ) S (( c × h ) ) ( CC = c S C ( c ) × c − h S H ( c − h ) ( ) = c S C ( c ) ⊗ H ε H ( c − S H ( c ) − H ε t ( c − h )) ( ) = c S C ( c ) ⊗ ε t ( c − S H ( c ) − c − h ) ( ) = c S C ( c ) ⊗ ε t ( ε t ( c − ) c − ) ε t ( c − S H ( c ) − h ) ( ) = c S C ( c ) ⊗ ε t ( c − ) ε t ( c − ) ε t ( c − S H ( c ) − h ) ( CC = c S C ( c ) ⊗ ε t ( c − S H ( c ) − c − c − h ) ( CC = c S C ( c ) ⊗ ε t ( c − S H ( c ) − c − h ) ( ) = c S C ( c ) ⊗ H ε H (1 H c − S H ( c ) − ε t ( c − h )) ( ) = c S C ( c ) ⊗ ε t ( c − h ) ε H ( c − S H ( c ) − ε t ( c − h ) ) ( CC = c S C ( c ) × ε t ( c − c − h ) ( CC = 1 C c S C ( c ) × ε t (1 C − c − h ) ( ) = ε C ( ε s (1 C ) c )(1 C × ε t (1 C − c − h )) ( ) = ε C (1 C c )(1 C × ε t (1 C − c − h )) ( CC = ε C (1 C c )(1 C × ε t (1 C − C − c − h ))= ε t ( c × h ) . (II) S (( c × h ) )( c × h ) = ε s ( c × h ), indeed for all c ∈ C and h ∈ HS (( c × h ) )( c × h ) CC = S C ( c ) c × S H ( c − h ) h = 1 C ε C ( c C ) ⊗ S H ( c − ) ε s ( h ) ε H (1 C − S H ( c − ) ε s ( h ) ) ( ) = 1 C ε C ( c C ) ⊗ S H ( c − ) ε s ( h )1 H ε H (1 C − S H ( c − ) H )= 1 C ε C ( c C ) ⊗ ε t ( S H ( c − ) ) S H ( c − ) ε s ( h ) ε t (1 C − ) ( ) = 1 C ε C ( c C ) ⊗ S H ( c − ) ε s ( h ) ε t (1 C − ) ( ∗ ) = 1 C ε C ( c C ) ⊗ S H ( c − ε t (1 C − )) ε s ( h ) ε t (1 C − ) ε t = ε s = 1 C ε C ( c C ) ⊗ S H ( c − ε s (1 C − )) ε s ( h ) ε t (1 C − ) ( ) = 1 C ε C ( c C ) ⊗ ε t ( ε s (1 C − )) S H ( c − ) ε s ( h ) ε t (1 C − ) ( ) = 1 C ε C ( c C ) ⊗ ε t (1 C − ) S H ( c − ) ε s ( h ) ε t (1 C − ) ( CC = 1 C ε C ( c C ) ⊗ S H ( c − ) ε s ( h ) ε t (1 C − ) ) = 1 C ε C ( c C ) ⊗ S H ( c − ) ε s ( h ) ε t ( ε s (1 C − )) ( ) = 1 C ε C ( c C ) ⊗ S H ( c − ) ε s ( h ) S H ( ε s (1 C − )) ( ∗∗ ) = 1 C ε C ( c C ) ⊗ S H ( c − C − ) ε s ( h ) ( CC = 1 C ε C ( c C ) ⊗ S H ( c − C − C − ) ε s ( h ) ( ) = 1 C ε C ( c C ) ⊗ S H (1 C − ) ε s ( ε t (1 C − )) ε s ( ε t ( c − )) ε s ( h ) ( ) = 1 C ε C ( c C ) ⊗ S H (1 C − ) ε s (1 C − ) ε s ( c − ) ε s ( h ) ( CC = 1 C ε C ( c C ) ⊗ S H (1 C − ) S H (1 C − )1 C − ε s ( c − h ) ( CC = 1 C ε C ( c C ) ⊗ S H (1 C − )1 C − ε s ( c − h ) ( ∗∗ ) = 1 C ε C ( c C ) ⊗ S H ( ε s (1 C − ))1 C − ε s ( c − h ) ( ) = 1 C ε C ( c C ) ⊗ ε t ( ε s (1 C − ))1 C − ε s ( c − h ) ( ) = 1 C ε C ( c C ) ⊗ ε t (1 C − )1 C − ε s ( c − h ) ( CC = 1 C ε C ( c C ) ⊗ ε t (1 C − C − )1 C − ε s ( c − h ) ( CC = 1 C ε C ( c C ) ⊗ ε t (1 C − C − )1 C − ε s ( c − h ) ( ) = 1 C ⊗ C − ε s ( c − h ) ε t (1 C − ) ε t ( ε t (1 C − )) ε C ( c C ) ( ) = 1 C ⊗ C − ε s ( c − h ) ε t (1 C − C − ) ε C ( c C )= 1 C ⊗ C − ε s ( c − h )1 H ε H (1 C − C − H ) ε C ( c C ) ( ) = 1 C ⊗ C − ε s ( c − h ) ε H (1 C − C − ε s ( c − h ) ) ε C ( c C )= 1 C × C − ε s ( c − h ) ε C ( c C ) ( ) = 1 C × C − ε s (1 C − ) ε s ( c − h ) ε C ( c C ) ( CC = (1 C × C − H ) ε C ( c C ) ε H ( c − C − h H )= ε s ( c × h ) . In ( ∗ ) it was used Proposition 4.27 of [5]. In ( ∗∗ ) it was used the property ρ (1) ∈ H s ⊗ C of Proposition 4.11of [5].(III) S (( c × h ) )( c × h ) S (( c × h ) ) = S ( c × h ), indeed for all c ∈ C and h ∈ HS (( c × h ) )( c × h ) S (( c × h ) ) ( ) = S C ( c ) ε t ( c ) × S H ( c − c − h ) ε t ( ε s ( c − h )) ( ) = S C ( c ) ε t ( c ) × S H ( c − c − h ) S H ( ε s ( c − h )) ( CC = S C ( c ) ε t ( c ) × S H ( c − c − h ε s ( c − ) ε s ( h )) ( ) = S C ( c ) ε t ( c ) × S H ( c − c − h )= S ( c × h ) . Therefore C × H is a weak Hopf algebra. (cid:3) ARTIAL COACTIONS OF WEAK HOPF ALGEBRAS ON COALGEBRAS 17 Dualization
In order to show that a partial coaction on a coalgebra can generate a partial action on a coalgebra, and viceversa, we need to assume the additional hypothesis that the weak Hopf algebra H is finite dimensional. Thus,we know that the dual H ∗ of H is also a weak Hopf algebra. Teorema 5.1.
Let C be a coalgebra and H be a weak Hopf algebra finite dimensional. Then, the followingaffirmations are equivalent: (i) C is a left partial H -comodule coalgebra; (ii) C is a right partial H ∗ -module coalgebra.Moreover, to say that C is a left symmetric partial H -comodule coalgebra is equivalent to say that C is a rightsymmetric partial H ∗ -module coalgebra.Proof. Suppose that C is a left symmetric partial H -comodule coalgebra via ρ : C → H ⊗ Cc c − ⊗ c . Then, C is a right symmetric partial H ∗ -module coalgebra via ↼ : C ⊗ H ∗ → Cc ⊗ f c ↼ f = f ( c − ) c . Conversely, suppose that C is a right symmetric partial H ∗ -module coalgebra via ↼ : C ⊗ H ∗ → Cc ⊗ f c ↼ f. Then, C is a left symmetric partial H -comodule coalgebra via ρ : C → H ⊗ Cc n X i =1 ( h i ⊗ c ↼ h i ∗ )where, { h i } ni =1 is a basis of H and { h i ∗ } ni =1 is the dual basis of H ∗ . (cid:3) Let λ be an element in H ∗ . We say that C is a (right) partial H -module coalgebra via λ if c ↼ h = cλ ( h )defines a partial action of H on the coalgebra C for all h ∈ H and c ∈ C . Moreover, in [7] it was proved that C is a partial H -module coalgebra via λ if and only if for all h, k ∈ H (i) λ (1 H ) = 1 k (ii) λ ( h ) λ ( k ) = λ ( h ) λ ( h k ) . Another result of dualization obtained is the one that says that the element λ defined from the partial actionvia λ is equal to the element defined from the partial coaction ρ λ presented in Proposition 3.10 as can be seenas follows. Corollary 5.2. C is a right partial H -module coalgebra via c ↼ h = cλ ( h ) if and only if C is a left partial H ∗ -comodule coalgebra via ρ λ ( c ) = λ ⊗ c .Proof. It is enough to show that(i) ε H ∗ ( λ ) = 1 k , since ε H ∗ ( λ ) = λ (1 H ) . = 1 k (ii) ( λ ⊗ H ∗ )∆( λ ) = λ ⊗ λ , since for all h, k ∈ H ( λ ⊗ H ∗ )∆( λ )( h ⊗ k ) = ( λλ )( h )( λ )( k )= λ ( h ) λ ( h ) λ ( k )= λ ( h ) λ ( h k )= λ ( h ) λ ( k )= ( λ ⊗ λ )( h ⊗ k ) . The converse is immediate. (cid:3)
References [1] N. Andruskiewitsch and S. Natale, Double categories and quantum groupoids, math.QA/0308228 (2003).[2] G. B¨ohm, Doi-Hopf modules over weak Hopf algebras. Communications in Algebra (28), 4687 - 4698 (2000).[3] G. B¨ohm, J. G´omes-Torrecillas, On the Double Crossed Product of Weak Hopf Algebras. Contemporary Mathematics 585,153-174 (1999).[4] G. B¨ohm, F. Nill, K. Szlach´anyi, Weak Hopf Algebras I: Integral Theory and C ∗ -Structure. Journal of Algebra 221 (2), 385-438(1999).[5] S. Caenepeel, E. De Groot, Modules Over Weak Entwining Structures in ”New trends Hopf Algebra Theory”, ContemporaryMathematics 267, 31-54 (2000).[6] S. Caenepeel, K. Janssen, Partial (Co)Actions of hopf algebras and Partial Hopf-Galois Theory, Communications in Algebra(36), 2923-2946 (2008).[7] E. Campos, G. Fonseca, G. Martini, Partial Actions of Weak Hopf Algebras on Coalgebras, http://arxiv.org/abs/1810.02872.[8] F. Castro, A. Paques, G. Quadros, A. Sant’Ana, Partial actions of weak Hopf algebras: smash products, globalization andMorita theory, Journal of Pure and Applied Algebra 29, 5511 - 5538 (2015).[9] F. Castro, G. Quadros, Globalizations for partial (co)actions on coalgebras, http://arxiv.org/abs/1510.01388v1.[10] M. Dokuchaev, R. Exel, Associativity of Crossed Products by Partial Actions, Enveloping Actions and Partial Representations,Trans. Amer. Math. Soc. 357 (5), 1931-1952 (2005).[11] R. Exel, Circle Actions on C ∗ − Algebras, Partial Automorphisms and Generalized PimsnerVoiculescu Exect Sequences, J.Funct. Anal.122 (3), 361-401 (1994).[12] G. Quadros, Partial (co)actions of weak Hopf algebras: globalizations, Galois theory and Morita theory. Tese de doutorado,UFRGS (2016).[13] S. Majid,
Foundations of quantum group theory , Cambridge University Press (1995).[14] M. Takeuchi,
Matched pairs of groups and bismash products of Hopf algebras , Communications in Algebra (9), 841-882 (1981).[15] Yu. Wang, L. Yu. Zhang, The Structure Theorem for Weak Module Coalgebras, Mathematical Notes 88 (1), 3 - 17 (2010).(Fonseca)
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