Rota-Bater paried comodule and Rota-Bater paired Hopf module
aa r X i v : . [ m a t h . QA ] S e p Rota-Bater paried comodule and Rota-Bater paired Hopfmodule ∗ Huihui Zheng, Yuxin Zhang, Liangyun Zhang ∗∗ College of Science, Nanjing Agricultural University, Nanjing 210095, China
Abstract : In this paper, we introduce the conception of Rota-Baxter paired comodules, which is dual to Rota-Baxter paired modules in [14]. We mainly discuss some properties of Rota-Baxter paired comodules, especiallywe give the characterization of generic Rota-Baxter paired comodules, which has important application forthe construction of Rota-Baxter comodules. Moreover, we construct Rota-Baxter paired comodules on Hopfalgebras, weak Hopf algebras, weak Hopf modules, dimodules, relative Hopf modules and Rota-Baxter pairedcomodules. And then we finally introduce the conception of Rota-Baxter paired Hopf modules by combiningRota-Baxter paired module with Rota-Baxter paired comodule, and give the structure theorem of generic Rota-Baxter paired Hopf modules.
Keywords : Rota-Baxter coalgebra, Rota-Baxter paired comodule, bialgebra, Hopf algebra, Rota-Baxter pairedHopf module. : 16T05; 16T15; 17B38. § A Rota-Baxter algebra (first known as a Baxter algebra) is an algebra A with a linear operator P on A that satisfies the Rota-Baxter identity P ( x ) P ( y ) = P ( P ( x ) y ) + P ( xP ( y )) + λP ( xy ) , for all x, y ∈ A, where λ ∈ k (the field), called the weight [2, 11].Rota-Baxter algebra originated from the 1960 paper [2] of Baxter based on his probabilitystudy to understand Spitzer’s identity in fluctuation theory. It wasn’t long before the concept at-tracted the attention of many mathematicians, especially Rota, whose fundamental papers around1970 brought the subject into the areas of algebra and combinatorics. In [1], a connection withmathematical physics was also established that related a Rota-Baxter algebra of weight 0 to theassociative analog of classical Yang-Baxter equation.To study the representations of Rota-Baxter algebras, the authors in [7] introduced the con-ception of Rota-Baxter modules related to the ring of Rota-Baxter operators. By the definition,a Rota-Baxter module over a Rota-Baxter algebra ( A, P ) is a pair (
M, T ) where M is a (left) A -module and T : M → M a k -linear operator such that P ( a ) · T ( m ) = T ( P ( a ) · m ) + T ( a · T ( m )) + λT ( a · m ) , for all a ∈ A, m ∈ M. ∗ This work is supported by Natural Science Foundation (11571173). ∗∗ Corresponding author: [email protected] 1ater, Rota-Baxter paired modules were introduced in [14], without requiring (
A, P ) to be aRota-Baxter algebra, which is a natural generalization of Rota-Baxter modules. Many propertiesof Rota-Baxter modules, even of Rota-Baxter algebras, are naturally generalized to Rota-Baxterpaired modules. Rota-Baxter paired module has broader connections and applications, especiallyto Hopf algebras. We have constructed a large number of Rota-Baxter paired modules from Hopfalgebra related structures in [14].Representation theory of coalgebras and comodules is a very extensive subject. On the basisof the comodule theory, we can naturally consider Rota-Baxter operators on comodules. In thispaper, we naturally introduce the conception of Rota-Baxter paired comodules, which is dualto Rota-Baxter paired modules, and give some properties of Rota-Baxter paired comodules. Inaddition, we also give its construction from Hopf algebra related coalgebras and comodules.A Hopf module on a bialgebra H is also an H -module and an H -comodule, whose action andcoaction satisfy a compatibility condition.As is well known, the structure theorem on Hopf modules is concerned by many experts andscholars. Especially, this structure theorem can describe the integrals of Hopf algebras. CombiningRota-Baxter paired modules and Rota-Baxter paired comodules, we can naturally introduce theconception of Rota-Baxter paired Hopf modules, and study Rota-Baxter operator on it, and proveits structure theorem.This article is organized as follows. In Section 2, we recall the definition of Rota-Baxter coal-gebras, and then give the notion of Rota-Baxter paired comodules, which is dual to Rota-Baxterpaired modules in [14]. Moreover, we provide a large number of examples of Rota-Baxter pairedcomodules. In Section 3, we discuss some properties of Rota-Baxter paired comodules, especiallywe give the characterization of generic Rota-Baxter paired comodules (see Theorem 3.1), whichhas an important application for the construction of Rota-Baxter comodules. In Section 4, we con-struct Rota-Baxter paired comodules on Hopf algebras, weak Hopf algebras, weak Hopf modules,dimodules, relative Hopf modules and Rota-Baxter paired comodules, respectively. Especially,we find some Rota-Baxter coalgebras and Rota-Baxter paired comodules by applying (co)integralin bialgebra, antipode and idempotent element in (weak) Hopf algebras and R -matrix in quasi-triangular Hopf algebra. In Section 5, we construct pre-Lie comodules from Rota-Baxter pairedcomodules. In Section 6, we introduce the conception of Rota-Baxter paired Hopf modules bycombining Rota-Baxter paired module with Rota-Baxter paired comodule, and give the structuretheorem of generic Rota-Baxter Hopf modules.Throughout this paper, let k be a fixed field. Unless otherwise specified, linearity, modules and ⊗ are all meant over k . And we freely use the Hopf algebras terminology introduced in [12]. For acoalgebra C , we write its comultiplication ∆( c ) with c ⊗ c , for any c ∈ C ; for a left C -comodule M , we denote its coaction by ρ ( m ) = m ( − ⊗ m (0) , for any m ∈ M ; for a right C -comodule M ,we denote its coaction by ρ ( m ) = m (0) ⊗ m (1) , for any m ∈ M , in which we omit the summationsymbols for convenience. § In this section, we firstly recall the definition of Rota-Baxter coalgebras, and then give the notion2f Rota-Baxter paired comodules, which is dual to the definition of Rota-Baxter paired modulesin [14]. Moreover, we provide a large number of examples of Rota-Baxter paired comodules.
Let ( C, ∆ , ε ) be a coalgebra. We call ( C, P ) a
Rota-Baxter coalgebra ofweight λ [8], if the linear map P : C → C satisfies the following( P ⊗ P )∆ = ( P ⊗ id )∆ P + ( id ⊗ P )∆ P + λ ∆ P, where λ ∈ k and id denotes the identical map.We refer the reader to [8] for further discussions on Rota-Baxter coalgebra and only give thefollowing simple examples of Rota-Baxter coalgebras which will be revisited later. Example 2.2 (a) Let C be an augmented coalgebra, that is, there exists a coalgebra homo-morphism f : k → C . Then, it is easy to see that f (1 k ) is a group-like element in C . So, ( C, P ) isa Rota-Baxter coalgebra of weight − P is given by P : C → C, c ε ( c ) f (1 k ) . Furthermore, if C is a bialgebra with the unit µ , then, C is an augmented coalgebra since µ : k → C is a coalgebra map. So, ( C, P ) is a Rota-Baxter coalgebra of weight − P : C → C, c ε ( c )1 C . (b) Let C be a coalgebra and λ ∈ C ∗ (the linear dual space of C ). Define P : C → C, c λ ( c ) c . Then, (
C, P ) is a Rota-Baxter coalgebra of weight − λ = λ , that is, λ ( c ) λ ( c ) = λ ( c ) for all c ∈ C . Let ( C, ∆ , ε ) be a coalgebra, and M a left C -comodule with coaction ρ . A pair( P, T ) of linear maps P : C → C and T : M → M is called a Rota-Baxter paired operator ofweight λ on M if ( P ⊗ T ) ρ = ( P ⊗ id ) ρT + ( id ⊗ T ) ρT + λρT. We also call the triple (
M, P, T ) a
Rota-Baxter paired (left) C -comodule of weight λ .Given a linear map T : M → M , if ( M, P, T ) is a Rota-Baxter paired C -comodule of weight λ forevery linear map P : C → C , then ( M, T ) is called a generic Rota-Baxter paired C -comoduleof weight λ . Example 2.4 (1) Let C be a coalgebra, regarded also as a left C -comodule via its comultipli-cation ∆. If ( C, P ) is a Rota-Baxter coalgebra of weight λ , then ( C, P, P ) is a Rota-Baxter paired C -comodule of weight λ .In particular, if C is a bialgebra with the unit µ , then, by Example 2.2, ( C, P, P ) is a Rota-Baxter paired C -comodule of weight λ , where P : C → C, c ε ( c )1 C .
32) Let (
M, P, T ) be a Rota-Baxter paired C -comodule of weight λ . Then, for any µ ∈ k ,( M, µP, µT ) is a Rota-Baxter paired C -comodule of weight λµ .(3) Let H be a bialgebra, and ( M, P, T ) a Rota-Baxter paired H -comodule of weight λ . If P isidempotent and a bialgebra homomorphism from H to H , then ( H ⊗ M, ρ, P, T ′ ) is a Rota-Baxterpaired H -comodule of weight λ , where T ′ : H ⊗ M → H ⊗ M and ρ : H ⊗ M → H ⊗ H ⊗ M aredefined by T ′ ( h ⊗ m ) = P ( h ) ⊗ T ( m ) and ρ ( h ⊗ m ) = h m ( − ⊗ h ⊗ m (0) , respectively.In fact, it is easy to prove that ( H ⊗ M, ρ ) is a left H -comodule. Moreover, for any h ∈ H, m ∈ M, we have P (( h ⊗ m ) ( − ) ⊗ T ′ (( h ⊗ m ) (0) )= P ( h m ( − ) ⊗ P ( h ) ⊗ T ( m (0) )= P ( h ) P ( m ( − ) ⊗ P ( h ) ⊗ T ( m (0) )= P ( h ) P ( T ( m ) ( − ) ⊗ P ( h ) ⊗ T ( m ) (0) + P ( h ) T ( m ) ( − ⊗ P ( h ) ⊗ T ( T ( m ) (0) )+ λP ( h ) T ( m ) ( − ⊗ P ( h ) ⊗ T ( m ) (0) ,P ( T ′ ( h ⊗ m ) ( − ) ⊗ T ′ ( h ⊗ m ) (0) + T ′ ( h ⊗ m ) ( − ⊗ T ′ ( T ′ ( h ⊗ m ) (0) )+ λT ′ ( h ⊗ m ) ( − ⊗ T ′ ( h ⊗ m ) (0) = P ( P ( h ) T ( m ) ( − ) ⊗ P ( h ) ⊗ T ( m ) (0) + P ( h ) T ( m ) ( − ⊗ P ( P ( h ) ) ⊗ T ( T ( m ) (0) )+ λP ( h ) T ( m ) ( − ⊗ P ( h ) ⊗ T ( m ) (0) = P ( P ( h ) ) P ( T ( m ) ( − ) ⊗ P ( h ) ⊗ T ( m ) (0) + P ( h ) T ( m ) ( − ⊗ P ( P ( h ) ) ⊗ T ( T ( m ) (0) )+ λP ( h ) T ( m ) ( − ⊗ P ( h ) ⊗ T ( m ) (0) = P ( h ) P ( T ( m ) ( − ) ⊗ P ( h ) ⊗ T ( m ) (0) + P ( h ) T ( m ) ( − ⊗ P ( h ) ⊗ T ( T ( m ) (0) )+ λP ( h ) T ( m ) ( − ⊗ P ( h ) ⊗ T ( m ) (0) , so, by Definition 2.3, we know that ( H ⊗ M, ρ, P, T ′ ) is a Rota-Baxter paired H -comodule of weight λ . (4) Let M be a left C -comodule with the coaction ρ , and V a vector space. Then, M ⊗ V has aleft C -comodule structure, whose comodule structure map is given by ρ ⊗ id . So, if ( M, P, T ) is aRota-Baxter paired C -comodule of weight λ , we easily see that ( M ⊗ V, P, T ⊗ id ) is a Rota-Baxterpaired C -comodule of weight λ .In particular, if ( C, P ) is a Rota-Baxter coalgebra of weight λ , then ( C ⊗ V, P, P ⊗ id ) is aRota-Baxter paired C -comodule of weight λ .Furthermore, if ( M, T ) is a generic Rota-Baxter paired C -comodule of weight λ , ( M ⊗ V, T ⊗ id )is also a generic Rota-Baxter paired C -comodule of weight λ .(5) Let M be a left C -comodule, and T an idempotent epimorphism in End( M ). Then,( M, id, T ) is a Rota-Baxter paired C -comodule of weight − m ∈ M, we have(( id ⊗ id ) ρT + ( id ⊗ T ) ρT − ρT )( m ) = (( id ⊗ id ) ρT + ( id ⊗ T ) ρT − ρT )( m )= ( id ⊗ T ) ρT ( m ) . T is an idempotent epimorphism, we have( id ⊗ T ) ρ = ( id ⊗ id ) ρT + ( id ⊗ T ) ρT − ρT. Hence, (
M, id, T ) is a Rota-Baxter paired C -comodule of weight − Rota-Baxter paired C -subcomodule N of a Rota-Baxter paired C -comodule ( M, P, T ) is a C -subcomodule of M such that T ( N ) ⊆ N . A Rota-Baxter paired comodule map f : ( M, P, T ) → ( M ′ , P ′ , T ′ ) of the same weight λ is a C -comodule map such that f T = T ′ f . Proposition 2.5
Let f : ( M, P, T ) → ( M ′ , P ′ , T ′ ) be a Rota-Baxter paired comodule map ofweight λ . Then the following conclusions hold.(a) Ker f is a Rota-Baxter paired C -subcomodule of M .(b) If K is a Rota-Baxter paired C -subcomodule of M , then f ( K ) is a Rota-Baxter paired C -subcomodule of M ′ .In particular, if T is C -colinear, then T ( M ) is a Rota-Baxter paired C -subcomodule of M .(c) If L is a Rota-Baxter paired C -subcomodule of M ′ , then f − ( L ) is a Rota-Baxter paired C -subcomodule of M . Proof. (a) Since f is a C -comodule map, Ker f is a C -subcomodule of M ′ . In addition, forany x ∈ Ker f , f T ( x ) = T ′ f ( x ) = 0, so T (Ker f ) ⊆ Ker f . Hence Ker f is a Rota-Baxter paired C -subcomodule of M .(b) It is obvious that f ( K ) is a subcomodule of M ′ , so, we have only to verify that T ′ ( f ( K )) ⊆ f ( K ). Since T ( K ) ⊆ K , and f T = T ′ f , we have T ′ f ( K ) = f T ( K ) ⊆ f ( K ).(c) We consider the composition πf of comodule maps π and f , where π : N → N/L is aprojection. By (a), we know that Ker f is a C -subcomodule of M , so, Ker( πf ) = f − ( L ) and asubcomodule of M . In addition, we have f T ( f − ( L )) = T ′ f ( f − ( L )) = T ′ ( L ) ⊆ L . So, we canget T ( f − ( L )) ⊆ f − ( L ). (cid:3) § In this section, we will discuss some properties of Rota-Baxter paired comodules.Recall that a linear operator T : M → M is called quasi-idempotent [14] of weight λ if T = − λT . We have the following characterization of generic Rota-Baxter paired comodules, which hasimportant application for the construction of Rota-Baxter comodules. Theorem 3.1
Let C be a coalgebra, and M a left C -comodule. If there exists a colinear map T : M → M , then the following are equivalent.(1) ( M, T ) is a generic Rota-Baxter paired C -comodule of weight λ .(2) There is a linear operator P : C → C such that ( M, P, T ) is a Rota-Baxter paired C -comodule of weight λ .(3) T is quasi-idempotent of weight λ . Proof.
Under the C -colinearity condition of T , for any linear operator P : A → A and m ∈ M ,we have P ( m ( − ) ⊗ T ( m (0) ) = P ( T ( m ) ( − ) ⊗ T ( m ) (0) + T ( m ) ( − ⊗ T ( T ( m ) (0) ) + λT ( m ) ( − ⊗ T ( m ) (0) ⇐⇒ P ( m ( − ) ⊗ T ( m (0) ) = P ( m ( − ) ⊗ T ( m (0) ) + m ( − ⊗ T ( m (0) ) + λm ( − ⊗ T ( m (0) )5 ⇒ m ( − ⊗ T ( m (0) ) + λm ( − ⊗ T ( m (0) ).If (1) holds, applying ε ⊗ id to both sides of the above equation, we get T = − λT . Conversely,if T = − λT , it is obvious that (1) holds.In a similar way, we can prove that (2) ⇐⇒ (3) . (cid:3) Proposition 3.2
Let M be a left C -comodule. Then, there exists a left C -comodule map T : M → M such that ( M, T ) is a generic Rota-Baxter C -comodule of weight −
1, if and only ifthere is a C -comodule direct sum decomposition M = M ⊕ M such that T : M → M ⊆ M isthe project of M onto M : T ( m + m ) = m for m ∈ M and m ∈ M . Proof.
Suppose M has a direct sum decomposition M = M ⊕ M of C -comodules, where M and M are subcomodule of M . Then the projection T of M onto M is idempotent, since, for m = m + m ∈ M with m ∈ M and m ∈ M , we have T ( m ) = T ( m + m ) = T ( m ) = m = T ( m ).Furthermore, we have( id ⊗ T ) ρ ( m ) = ( id ⊗ T ) ρ ( m + m )= m − ⊗ T ( m + 0) + m − ⊗ T (0 + m )= m − ⊗ m = ρ ( m )= ρT ( m ) , so, T is a left C -comodule map. Again by Theorem 3.1, we know ( M, T ) is a generic Rota-Baxterpaired C -comodule of weight − M, T ) is a generic Rota-Baxter paired C -comodule of weight − T a left C -comodule map, then by Theorem 3.1, we know T is idempotent.Let M = T ( M ) and M = ( id − T )( M ). Because T is a left C -comodule map, both M and M are subcomodule of M . Also, for any m ∈ M , m = T ( m ) + ( id − T )( m ), so M = M + M .Furthermore, if n ∈ M ∩ M , then n = T ( x ) = ( id − T )( y ), for some x, y ∈ M . Thus n = T ( x ) = T ( x ) = T ( id − T )( y ) = ( T − T )( y ) = 0. Therefore M = M ⊕ M .Finally, since m = T ( m ) + ( id − T )( m ) is the decomposition of m ∈ M as m = m + m with m ∈ M and m ∈ M , we see that T is the projection of M onto M . (cid:3) Proposition 3.3
Let M be a C -comodule and P : C → C , T : M → M linear maps. Then( M, P, T ) is a Rota-Baxter paired C -comodule of weight λ = 0 if and only if there is a map f : M → C ⊗ M such that ( P ⊗ T ) ρ = f T, ( P ⊗ T ) ρ = − f T , where P = − P − λid and T = − T − λid. Proof.
Let (
M, P, T ) be a Rota-Baxter paired C -comodule of weight λ . Then, we have( P ⊗ T ) ρ = ( P ⊗ id ) ρT + ( id ⊗ T ) ρT + λρT. Let f = ( P ⊗ id ) ρ + ( id ⊗ T ) ρ + λρ . Then the above equation gives( P ⊗ T ) ρ = f T, P ⊗ T ) ρ = − f T . Now we consider the converse. Suppose that there exists a map f : M → C ⊗ M such that( P ⊗ T ) ρ = f T and ( P ⊗ T ) ρ = − f T . Then we have − λf = f T + f T = ( P ⊗ T ) ρ − ( P ⊗ T ) ρ = ( P ⊗ T ) ρ − (( − λid − P ) ⊗ ( − λid − T )) ρ = ( P ⊗ T ) ρ − ( λ id ⊗ id + λid ⊗ T + λP ⊗ id + P ⊗ T ) ρ = − λ ( λid ⊗ id + id ⊗ T + P ⊗ id ) ρ. So, we get f = ( P ⊗ id ) ρ + ( id ⊗ T ) ρ + λρ. Furthermore, ( P ⊗ T ) ρ = (( P ⊗ id ) ρ + ( id ⊗ T ) ρ + λρ ) T . Hence ( M, P, T ) is a Rota-Baxterpaired C -comodule of weight λ = 0. (cid:3) By the above definition of T and P in Proposition 3.3, there are also the following relationships. Proposition 3.4
Let (
M, P, T, ρ ) be a Rota-Baxter paired C -comodule of weight λ . Then wehave P ( m ( − ) ⊗ T ( m (0) ) = T ( m ) ( − ⊗ T ( T ( m ) (0) ) + P ( T ( m ) ( − ) ⊗ T ( m ) (0) ,P ( m ( − ) ⊗ T ( m (0) ) = P ( T ( m ) ( − ) ⊗ T ( m ) (0) + T ( m ) ( − ⊗ T ( T ( m ) (0) )for any m ∈ M . Proof.
By using the compatible condition of Rota-Baxter paired comodules, we can directlyverify that the equation holds. (cid:3)
The following result shows that how close it is for an idempotent Rota-Baxter operator ofcomodule to have weight − Proposition 3.5
Let (
M, P, T ) be a Rota-Baxter paired C -comodule of weight λ .(a) If T is idempotent, then (1 + λ ) T ( m ) ( − ⊗ T ( T ( m ) (0) ) = 0, for any m ∈ M .(b) If P and T are idempotent, then (1 + λ ) P ( T ( m ) ( − ) ⊗ T ( m ) (0) = 0, for any m ∈ M .(c) If P and T are idempotent, then (1+ λ )( P ( T ( m ) ( − ) ⊗ T ( m ) (0) − λT ( m ) ( − ⊗ T ( m ) (0) ) = 0,for any m ∈ M .Thus an idempotent Rota-Baxter operator of comodule must have weight − Proof. (a) Since T = T , for any m ∈ M we obtain P ( m ( − ) ⊗ T ( m (0) ) = P ( m ( − ) ⊗ T ( T ( m (0) ))= P ( T ( m ) ( − ) ⊗ T ( T ( m ) (0) ) + T ( m ) ( − ⊗ T ( T ( m ) (0) ) + λT ( m ) ( − ⊗ T ( T ( m ) (0) )= P ( T ( m ) ( − ) ⊗ T ( T ( m ) (0) ) + T ( m ) ( − ⊗ T ( T ( m ) (0) ) + λT ( m ) ( − ⊗ T ( T ( m ) (0) ) , that is, ( P ⊗ T ) ρ = ( P ⊗ T ) ρT + ( id ⊗ T ) ρT + λ ( id ⊗ T ) ρT .Applying T = T to the above equality, we have ( P ⊗ T ) ρT = ( P ⊗ T ) ρ . Thus (1+ λ ) T ( m ) ( − ⊗ T ( T ( m ) (0) ) = 0 for any m ∈ M . 7b) Since P = P , for any m ∈ M we obtain P ( m ( − ) ⊗ T ( m (0) ) = P ( P ( m ( − )) ⊗ T ( m (0) )= P ( T ( m ) ( − ) ⊗ T ( m ) (0) + P ( T ( m ) ( − ) ⊗ T ( T ( m ) (0) ) + λP ( T ( m ) ( − ) ⊗ T ( m ) (0) = P ( T ( m ) ( − ) ⊗ T ( m ) (0) + P ( T ( m ) ( − ) ⊗ T ( T ( m ) (0) ) + λP ( T ( m ) ( − ) ⊗ T ( m ) (0) , that is, ( P ⊗ T ) ρ = ( P ⊗ id ) ρT + ( P ⊗ T ) ρT + λ ( P ⊗ id ) ρT .Applying T = T to the above equality, we have ( P ⊗ T ) ρT = ( P ⊗ T ) ρ . Thus (1 + λ ) P ( T ( m ) ( − ) ⊗ T ( m ) (0) = 0 for any m ∈ M .(c) By (a) and (b), we can prove that (c) holds. (cid:3) § In this section, we construct Rota-Baxter paired comodule by deformation, direct sum, Hopfalgebra, Rota-Baxter paired module, Hopf module, co-Hopf module, dimodule.Firstly, we construct Rota-Baxter paired comodule by deformation.
Let C be a coalgebra, and M a left C -comodule. Define two maps T : M → M and P : C → C by T ( m ) = χ ( m ( − ) m (0) and P ( c ) = χ ( c ) c , for any m ∈ M, c ∈ C , respectively.Then, ( M, P, T ) is a Rota-Baxter paired C -comodule of weight − χ ∈ C ∗ is idempotent underthe convolution product. Proof.
For any m ∈ M , we have P ( m ( − ) ⊗ T ( m (0) ) = χ ( m ( − ) m ( − ⊗ χ ( m (0)( − ) m (0)(0) = χ ( m ( − ) m ( − ⊗ χ ( m ( − ) m (0) . Moreover, for any m ∈ M , ρ ( T ( m )) = ρ ( χ ( m − ) m ) = χ ( m − ) m (0)( − ⊗ m (0)(0) = χ ( m ( − ) m ( − ⊗ m (0) , so, we obtain that P ( T ( m ) ( − ) ⊗ T ( m ) (0) + T ( m ) ( − ⊗ T ( T ( m ) (0) ) − T ( m ) ( − ⊗ T ( m ) (0) = χ ( m ( − ) P ( m ( − ) ⊗ m (0) + χ ( m ( − ) m ( − ⊗ T ( m (0) ) − χ ( m ( − ) m ( − ⊗ m (0) = χ ( m ( − ) χ ( m ( − ) m ( − ⊗ m (0) + χ ( m ( − ) m ( − ⊗ χ ( m (0)( − ) m (0)(0) − χ ( m ( − ) m ( − ⊗ m (0) = χ ( m ( − ) χ ( m ( − ) m ( − ⊗ m (0) + χ ( m ( − ) m ( − ⊗ χ ( m ( − ) m (0) − χ ( m ( − ) m ( − ⊗ m (0) = χ ( m ( − ) m ( − ⊗ m (0) + χ ( m ( − ) m ( − ⊗ χ ( m ( − ) m (0) − χ ( m ( − ) m ( − ⊗ m (0) = χ ( m ( − ) m ( − ⊗ m (0) + χ ( m ( − ) m ( − ⊗ χ ( m ( − ) m (0) − χ ( m ( − ) m ( − ⊗ m (0) = χ ( m ( − ) m ( − ⊗ χ ( m ( − ) m (0) . Hence P ( m ( − ) ⊗ T ( m (0) ) = P ( T ( m ) ( − ) ⊗ T ( m ) (0) + T ( m ) ( − ⊗ T ( T ( m ) (0) ) − T ( m ) ( − ⊗ T ( m ) (0) , ( M, P, T ) is a Rota-Baxter paired H -comodule of weight − (cid:3) Let H be a bialgebra. If there exists λ ∈ H ∗ , such that f λ = ε H ∗ ( f ) λ for any f ∈ H ∗ , thenwe call λ a left cointegral of H ∗ . Furthermore, if ε H ∗ ( λ ) = 1, we call H a cosemisimple bialgebra,and easily see λ = λ, that is, λ is idempotent. 8o, by the above proposition, we have Corollary 4.2
Let H be a cosemisimple bialgebra with cointegral λ , and M a left H -comodule.Define two maps T : M → M and P : H → H by T ( m ) = λ ( m ( − ) m (0) and P ( h ) = λ ( h ) h , forany m ∈ M, h ∈ H , respectively. Then, ( M, P, T ) is a Rota-Baxter paired H -comodule of weight − H, P ) is a Rota-Baxter coalgbra of weight − Let H be both an algebra and a coalgebra. Then H is called a weak bialgebra in [4] if it satisfies the following conditions:(1) ∆( xy ) = ∆( x )∆( y ) , for all x, y ∈ H ,(2) ε ( xyz ) = ε ( xy ) ε ( y z ) = ε ( xy ) ε ( y z ) , for any x, y, z ∈ H ,(3) ∆ (1 H ) = (∆(1 H ) ⊗ H )(1 H ⊗ ∆(1 H )) = 1 ⊗ ′ ⊗ ′ = (1 H ⊗ ∆(1 H ))(∆(1 H ) ⊗ H ) = 1 ⊗ ′ ⊗ ′ , where ∆(1 H ) = 1 ⊗ = 1 ′ ⊗ ′ and ∆ = (∆ ⊗ id H ) ◦ ∆.Moreover, if there exists a linear map S : H → H , called antipode, satisfying the followingaxioms for all h ∈ H : h S ( h ) = ε (1 h )1 , S ( h ) h = ε ( h )1 , S ( h ) h S ( h ) = S ( h ) , then the weak bialgebra H is called a weak Hopf algebra .For any weak bialgebra H , defines the maps ⊓ L , ⊓ R : H → H by the formulas ⊓ L ( h ) = ε (1 h )1 , ⊓ R ( h ) = ε ( h )1 . Denote by H L the image ⊓ L and by H R the image ⊓ R , where H L and H R are respectivelycalled the target algebra and the source algebra of the weak bialgebra H .By [4], if H is a weak Hopf algebra with antipode S , we have the following conclusions:( W ⊓ L ◦ ⊓ L = ⊓ L , ⊓ R ◦ ⊓ R = ⊓ R ;( W ⊓ L ( h ) ⊗ h = S (1 ) ⊗ h, h ⊗ ⊓ R ( h ) = h ⊗ S (1 ), for any h ∈ H ;( W ′ ) ⊓ L (1 ) ⊗ = S (1 ) ⊗ , ⊗ ⊓ R (1 ) = 1 ⊗ S (1 );( W ⊓ L ( ⊓ L ( h ) g ) = ⊓ L ( h ) ⊓ L ( g ) , ⊓ R ( h ⊓ R ( g )) = ⊓ R ( h ) ⊓ R ( g ), for any h, g ∈ H. Note that ⊓ L (1 ) ⊗ and 1 ⊗⊓ R (1 ) are separable idempotents of H L and H R by Proposition2.11 in [4], respectively. So, by ( W W h ⊓ L (1 ) ⊗ = ⊓ L (1 ) ⊗ h, h ⊗ ⊓ R (1 ) = 1 ⊗ ⊓ R (1 ) h , for any h ∈ H .Again according to [10], we have the following conclusions, that is, for any x ∈ H L , y ∈ H R :( W
5) ∆(1) = 1 ⊗ ∈ H R ⊗ H L , xy = yx ;( W
6) ∆( x ) = 1 x ⊗ , ∆( y ) = 1 ⊗ y ;( W xS (1 ) ⊗ = S (1 ) ⊗ x, y ⊗ S (1 ) = 1 ⊗ S (1 ) y .Again by ( W
5) and ( W W
8) ∆( xy ) = 1 x ⊗ y ∈ H L H R ⊗ H L H R , for any x ∈ H L , y ∈ H R .According to ( W H L H R is a subcoalgebra of H .9 roposition 4.4 Let H be a weak Hopf algebra with antipode S . Then ( H L H R , ⊓ L , ⊓ L ) isa Rota-Baxter paired H -comodule of weight −
1, whose comodule structure map is given by thecomultiplication ∆ of H .In particular, ( H L H R , ⊓ L ) is a Rota-Baxter coalgebra of weight − Proof.
By ( W H L H R is a left H -comodule via the comultiplication ∆ of H . Moreover, forany x ∈ H L , y ∈ H R , we have( ⊓ L ⊗ ⊓ L )∆( xy ) ( W = ( ⊓ L ⊗ ⊓ L )(1 x ⊗ y )= ⊓ L (1 x ) ⊗ ⊓ L (1 y ) ( W = ⊓ L ( x ) ⊗ ⊓ L (1 y ) ( W = x ⊓ L (1 ) ⊗ ⊓ L ( y ) ( W = x ⊓ L ( y ) ⊓ L (1 ) ⊗ , ( ⊓ L ⊗ id )∆ ⊓ L ( xy ) = ( ⊓ L ⊗ id )∆( x ⊓ L ( y ))= ( ⊓ L ⊗ id )( x ⊓ L ( y ) ⊗ x ⊓ L ( y ) ) ( W = ( ⊓ L ⊗ id )(1 x ′ ⊓ L ( y ) ⊗ ′ )= ( ⊓ L ⊗ id )(1 ′ x ⊓ L ( y ) ⊗ ′ )= ( ⊓ L ⊗ id )(1 x ⊓ L ( y ) ⊗ )= ⊓ L (1 x ⊓ L ( y )) ⊗ W = x ⊓ L ( y ) ⊓ L (1 ) ⊗ , ( id ⊗ ⊓ L )∆ ⊓ L ( xy ) = ( id ⊗ ⊓ L )∆( x ⊓ L ( y ))= ( id ⊗ ⊓ L )(1 x ⊓ L ( y ) ⊗ )= 1 x ⊓ L ( y ) ⊗ ⊓ L (1 ) ( W = 1 x ⊓ L ( y ) ⊗ = ∆ ⊓ L ( xy ) . Hence, we get that( ⊓ L ⊗ id )∆ ⊓ L ( xy ) + ( id ⊗ ⊓ L )∆ ⊓ L ( xy ) − ∆ ⊓ L ( xy ) = ( id ⊗ ⊓ L )∆ ⊓ L ( xy ) , that is, ( H L H R , ⊓ L , ⊓ L ) is a Rota-Baxter paired H -comodule of weight − (cid:3) In a similar way in Proposition 4.4, we have
Remark 4.5
Let H be a weak Hopf algebra with antipode S . Then the following hold.(1) ( H L H R , ⊓ R , ⊓ R ) is a Rota-Baxter paired H -comodule of weight −
1, and so ( H L H R , ⊓ R )is a Rota-Baxter coalgebra of weight − H, ⊓ L , ⊓ L ) is a Rota-Baxter paired H -comodule of weight −
1, and so ( H, ⊓ L ) is a Rota-Baxter coalgebras of weight − H, ⊓ R , ⊓ R ) is a Rota-Baxter paired H -comodule of weight −
1, and so ( H, ⊓ R ) is a Rota-Baxter coalgebra of weight − In this subsection, we always assume that H is a weak Hopf algebra with antipose S . Then, S
10s both an antimultiplication map and an anticomultiplication map, that is, for any h, g ∈ H , S ( hg ) = S ( g ) S ( h ) , S (1) = 1 , ∆ S ( h ) = S ( h ) ⊗ S ( h ) , εS ( h ) = ε ( h ) , and we have( W h ⊗ ⊓ L ( h ) = 1 h ⊗ , ⊓ R ( h ) ⊗ h = 1 ⊗ h for any h ∈ H . Definition 4.6
Suppose that H is a weak Hopf algebra with antipode S . A weak right H -Hopf module is a triple ( M, · , ρ ), where ( M, · ) is a right H -module and ( M, ρ ) a right H -comodule,such that ρ ( m · h ) = m [0] · h ⊗ m [1] h for any m ∈ M, h ∈ H .Define a map T : M → M given by T ( m ) = m [0] · S ( m ([1] ) for any m ∈ M Then, according to Proposition 3.8 in [14], T is idempotent. Moreover, we have( W ρT ( m ) = T ( m ) · ⊗ , for any m ∈ M .In fact, for any m ∈ M , we have ρT ( m ) = m [0][0] · S ( m [1] ) ⊗ m [0][1] S ( m [1] ) = m [0] · S ( m [1]2 ) ⊗ m [1]1 S ( m [1]2 ) = m [0] · S ( m [1]3 ) ⊗ m ([1]1 S ( m [1]2 )= m [0] · S ( m [1]2 ) ⊗ ⊓ L ( m [1]1 ) ( W = m [0] · S (1 m [1] ) ⊗ S (1 )= m [0] · S ( m [1] ) S (1 ) ⊗ S (1 )= T ( m ) · S (1 ) ⊗ S (1 )= T ( m ) · ⊗ . A weak Hopf algebra H is called a quantum commutative if h g ⊓ R ( h ) = hg for any h, g ∈ H . Then, by Proposition 4.1 in [3], H is quantum commutative if and only if H R ⊆ Z ( H )(the center of H ). Proposition 4.7
Let H be a quantum commutative weak Hopf algebra, and M a weak right H -Hopf module. Then ( M, ⊓ L , T ) is a Rota-Baxter paired H -comodule of weight − Proof.
According to ( W m ∈ M , we have( T ⊗ ⊓ L ) ρ ( m ) = T ( m [0] ) ⊗ ⊓ L ( m [1] )= m [0][0] · S ( m [0][1] ) ⊗ ⊓ L ( m [1]2 )= m [0] · S ( m [1]1 ) ⊗ ⊓ L ( m [1]2 )11 W = m [0] · S (1 m [1] ) ⊗ = m [0] · S ( m [1] ) S (1 ) ⊗ = T ( m ) · S (1 ) ⊗ , ( id ⊗ ⊓ L ) ρT ( m ) ( W = T ( m ) · ⊗ ⊓ L (1 ) ( W = T ( m ) · ⊗ = ρT ( m ) , ( T ⊗ id ) ρT ( m ) = T ( T ( m ) · ) ⊗ = T ( m ) [0] · S ( T ( m ) [1] ) ⊗ = T ( m ) [0] · S (1 ) S ( T ( m ) [1] ) ⊗ = T ( m ) [0] · ⊓ L (1 ) S ( T ( m ) [1] ) ⊗ W ′ ) = T ( m ) [0] · S (1 ) S ( T ( m ) [1] ) ⊗ = T ( m ) [0] · S ( T ( m ) [1] ) ⊗ = T ( m ) [0] · S (1 T ( m ) [1] ) ⊗ ( H R ⊆ Z ( H ))= T ( m ) [0] · S ( T ( m ) [1] ) S (1 ) ⊗ = T ( m ) · S (1 ) ⊗ , so, we get that ( T ⊗ ⊓ L ) ρ ( m ) = ( id ⊗ ⊓ L ) ρT ( m ) + ( T ⊗ id ) ρT ( m ) − ρT ( m ) . Hence (
M, T, ⊓ L ) is a Rota-Baxter paired H -comodule of weight − Remark 4.8
Let H be a weak Hopf algebra. Then H is a weak right H -Hopf module whoseaction and coaction are given by its multiplication and comultiplication of H . If H is quantumcommutative, then, by the above proposition and h S ( h ) = ⊓ L ( h ) for h ∈ H , we know that( H, ⊓ L , ⊓ L ) is a Rota-Baxter paired H -comodule of weight − In this subsection, we construct Rota-Baxter paired comodules on dimodules.
Definition 4.9
Assume that H is a bialgebra. A k -module M which is both a left H -moduleand a right H -comodule is called a left, right H -dimodule [6] if for any h ∈ H, m ∈ M , thefollowing equality holds: ρ ( h · m ) = h · m [0] ⊗ m [1] , where ρ is the right H -comodule structure map of M . Proposition 4.10
Let H be a bialgebra with an idempotent element e , and M a left-right H -dimodule. Define a map T : M → M by T ( m ) = e · m . Then ( M, T ) is a generic Rota-Baxterpaired H -comodule of weight − Proof.
For any m ∈ M , we have T ( m ) = T ( e · m ) = e · m = e · m = T ( m ) , T is idempotent. Since M is a left-right H -dimodule, for any m ∈ M , we have( T ⊗ id ) ρ ( m ) = T ( m [0] ) ⊗ m [1] = e · m [0] ⊗ m [1] = ρ ( e · m ) = ρT ( m ) , that is, T is a comodule map. Thus by Theorem 3.1, we know that ( M, T ) is a generic Rota-Baxterpaired H -comodule of weight − (cid:3) Remark 4.11 (1) Let G be a finite group. Then, H = ( kG ) ∗ =Hom k ( kG, k ) is a Hopf algebrawith dual basis { p g | p g ( h ) = δ gh } . According to [5], p g are orthogonal idempotents, for any g ∈ G .Thus, by the above proposition, for any left-right H -dimodule M , ( M, T g ) are a generic Rota-Baxter paired H -comodule of weight − g ∈ G , where T g ( m ) = p g · m for m ∈ M .(2) Let H be a bialgebra. If there is an element x ∈ H such that hx = ε ( h ) x for any h ∈ H ,then we call x a left integral of H .Suppose that H is a finite dimensional semisimple Hopf algebra. Then by [12, Theorem 5.1.8],there exists a non-zero left integral e such that ε ( e ) = 1. It is obvious that e = e . Hence thefollowing conclusions hold.(i) If ( H, R ) is a quasitriangular Hopf algebra. Then, by Example 3.12 in [14], ( H, ρ ) is a left,right H -dimodule, whose action is given by its multiplication and coaction ρ : H → H ⊗ H givenby ρ ( h ) = h R i ⊗ R j . So, by Proposition 4.10, ( H, T ) is a generic Rota-Baxter paired H -comoduleof weight −
1, where T : H → H is given by T ( h ) = e · h .(ii) Let M be a left, right H -dimodule. Define T ( m ) = e · m , for m ∈ M . Then, by Proposition4.10, ( M, T ) is a generic Rota-Baxter paired H -comodule of weight − T is a left H -module map: for any h ∈ H, m ∈ M , T ( h · m ) = ε ( h ) e · m = he · m = h · T ( m ) . Again T is idempotent by Proposition 4.10, thus, ( M, T ) is a generic Rota-Baxter paired H -module of weight − M, T ) is a generic Rota-Baxter paired left,right H -dimodule of weight − H, R ) is a quasitriangular Hopf algebra, then, according to the above con-clusions, we know that ( H, T ) is a generic Rota-Baxter paired left, right H -dimodule of weight − In this subsection, we construct Rota-Baxter paired comodules on relative Hopf modules.
Definition 4.12
Let H be a bialgebra, and C a right H -module coalgebra. A relative [ C, H ] -Hopf module in [13] M is a right C -comodule which is also a right H -module such that thefollowing compatible condition holds: for all m ∈ M and h ∈ H , ρ ( m · h ) = m [0] · h ⊗ m [1] · h . Proposition 4.13
Let H be a Hopf algebra with an antipode S , and M a relative [ C, H ]-Hopfmodule. If there is a right H -module coalgebra map φ : C → H , we define E C : C → C and E M : M → M given by E C ( c ) = c · Sφ ( c ) , M ( m ) = m [0] · Sφ ( m [1] ) , for any c ∈ C, m ∈ M , then, ( M, E C , E M ) is a Rota-Baxter paired right C -comodule of weight − H is a right H -module coalgebra whose action is given by its multiplication of H . Proof.
By the definition of Rota-Baxter paired comodule, we only need to prove that( E M ⊗ E C ) ρ ( m ) = ( id ⊗ E C ) ρE M + ( E M ⊗ id ) ρE M − ρE M , for any m ∈ M .In fact, firstly, for any m ∈ M, c ∈ C , we have ρE M ( m ) = m [0][0] · ( Sφ ( m [1] )) ⊗ m [0][1] · ( Sφ ( m [1] )) = m [0] · ( Sφ ( m [1]2 )) ⊗ m [1]1 · ( Sφ ( m [1]2 )) = m [0] · Sφ ( m [1]3 ) ⊗ m [1]1 · Sφ ( m [1]2 )= m [0] · Sφ ( m [1]2 ) ⊗ E C ( m [1]1 ) ,E C ( c ) = E C ( c · Sφ ( c ))= ( c · Sφ ( c )) · Sφ (( c · Sφ ( c )) )= c · ( Sφ ( c ) Sφ ( c · Sφ ( c )))= c · ( Sφ ( c ) S ( Sφ ( c )) Sφ ( c ))= c · S ( φ ( c ) Sφ ( c ) φ ( c ))= c · Sφ ( c ) = E C ( c ) . By the above equation, we get( id ⊗ E C ) ρE M ( m ) + ( E M ⊗ id ) ρE M ( m ) − ρE M ( m )= ( id ⊗ E C )( m [0] · Sφ ( m [1]2 ) ⊗ E C ( m [1]1 )) + ( E M ⊗ id )( m [0] · Sφ ( m [1]2 ) ⊗ E C ( m [1]1 )) − m [0] · Sφ ( m [1]2 ) ⊗ E C ( m [1]1 )= m [0] · Sφ ( m [1]2 ) ⊗ E C ( m [1]1 ) + E M ( m [0] · Sφ ( m [1]2 )) ⊗ E C ( m [1]1 ) − m [0] · Sφ ( m [1]2 ) ⊗ E C ( m [1]1 )= E M ( m [0] · Sφ ( m [1]2 )) ⊗ E C ( m [1]1 ) . Finally, we prove that E M ( m [0] · Sφ ( m [1]2 )) ⊗ E C ( m [1]1 ) = ( E M ⊗ E C ) ρ ( m ). E M ( m [0] · Sφ ( m [1]2 )) ⊗ E C ( m [1]1 )= ( m [0] · Sφ ( m [1]2 )) [0] · Sφ (( m [0] · Sφ ( m [1]2 )) [1] ) ⊗ E C ( m [1]1 )= ( m [0][0] · ( Sφ ( m [1]2 )) ) · Sφ (( m [0][1] · ( Sφ ( m [1]2 )) ) ⊗ E C ( m [1]1 )= m [0] · ( Sφ ( m [1]4 ) S ( Sφ ( m [1]3 )) Sφ ( m [1]1 )) ⊗ E C ( m [1]2 )= m [0] · ( S ( Sφ ( m [1]3 ) φ ( m [1]4 )) Sφ ( m [1]1 )) ⊗ E C ( m [1]2 )= m [0] · Sφ ( m [1]1 ) ⊗ E C ( m [1]2 )= m [0][0] · Sφ ( m [0][1] ) ⊗ E C ( m [1] )= E M ( m [0] ) ⊗ E C ( m [1] ) . M, E C , E M ) is a Rota-Baxter paired right C -comodule of weight − (cid:3) Remark 4.14
Let Let H be a Hopf algebra and and M a relative [ C, H ]-Hopf module, Then,it is easy to show that H ⊗ M is a relative [ C, H ]-Hopf module by ρ : H ⊗ M → H ⊗ M ⊗ C, ρ ( h ⊗ m ) = h ⊗ m [0] ⊗ m [1] · h , · : H ⊗ M ⊗ H → H ⊗ M, ( h ⊗ m ) · g = hg ⊗ m. Then, by Proposition 4.13, ( H ⊗ M, E C , E H ⊗ M ) is a right Rota-Baxter paired C -comodule ofweight − H -module coalgebra map φ : C → H , where E C ( c ) = c · Sφ ( c ), E H ⊗ M ( h ⊗ m ) = h Sφ ( m [1] · h ) ⊗ m [0] . Let (
M, P, T ) be a Rota-Baxter paired C -comodule of weight λ . Define P = − P − λid, T = − T − λid Then (
M, P , T ) is also a Rota-Baxter paired C -comodule of weight λ . Proof.
We have only to verify that( P ⊗ T ) ρ = ( P ⊗ id ) ρT + ( id ⊗ T ) ρT + λρT . Actually, for any m ∈ M , we have( P ⊗ T ) ρ ( m ) = ( P ⊗ T )( m ( − ⊗ m (0) ) = P ( m ( − ) ⊗ T (( m (0) ))= ( − P ( m ( − ) − λm ( − ) ⊗ ( − T ( m (0) ) − λm (0) )= P ( m ( − ) ⊗ T ( m (0) ) + λP ( m ( − ) ⊗ m (0) + λm ( − ⊗ T ( m (0) ) + λ m ( − ⊗ m (0) = P ( T ( m ) ( − ) ⊗ T ( m ) (0) + T ( m ) ( − ⊗ T ( T ( m ) (0) ) + λT ( m ) ( − ⊗ T ( m ) (0) + λP ( m ( − ) ⊗ m (0) + λm ( − ⊗ T ( m (0) ) + λ m ( − ⊗ m (0) , (( P ⊗ id ) ρT + ( id ⊗ T ) ρT + λρT )( m ) = ( P ⊗ id + id ⊗ T + λ ) ρ ( − T ( m ) − λm )= − ( P ⊗ id )( T ( m ) ( − ⊗ T ( m ) (0) ) − λ ( P ⊗ id )( m ( − ⊗ m (0) ) − ( id ⊗ T )( T ( m ) ( − ⊗ T ( m ) (0) ) − λ ( id ⊗ T )( m ( − ⊗ m (0) ) − λT ( m ) ( − ⊗ T ( m ) (0) − λ m ( − ⊗ m (0) = − P ( T ( m ) ( − ) ⊗ T ( m ) (0) − λP ( m ( − ) ⊗ m (0) − T ( m ) ( − ⊗ T ( T ( m ) (0) ) − λm ( − ⊗ T ( m (0) ) − λT ( m ) ( − ⊗ T ( m ) (0) − λ m ( − ⊗ m (0) = ( P ( T ( m ) ( − ) + λT ( m ) ( − ) ⊗ T ( m ) (0) − λ ( − P ( m ( − ) − λm ( − ) ⊗ m (0) − T ( m ) ( − ⊗ ( − T ( T ( m ) (0) ) − λT ( m ) (0) ) − λm ( − ⊗ ( − T ( m (0) ) − λm (0) ) − λT ( m ) ( − ⊗ T ( m ) (0) − λ m ( − ⊗ m (0) = P ( T ( m ) ( − ) ⊗ T ( m ) (0) + λT ( m ) ( − ⊗ T ( m ) (0) + λP ( m ( − ) ⊗ m (0) + T ( m ) ( − ⊗ T ( T ( m ) (0) ) + λm ( − ⊗ T ( m (0) ) + λ m ( − ⊗ m (0)
15s desired. (cid:3)
Proposition 4.16
Let (
C, P ) be a Rota-Baxter coalgebra of weight λ , and ( M, P, T ) a Rota-Baxter paired comodule of weight λ . Define another comultiplication ∆ ′ on C by∆ ′ = ( id ⊗ P )∆ + ( P ⊗ id )∆ + λ ∆ , and another operation of M by ρ ′ = ( P ⊗ id ) ρ + ( id ⊗ T ) ρ + λρ. Then the following conclusion hold.(a) ( C, ∆ ′ , P ) is also a (noncounitary) Rota-Baxter coalgebra of weight λ .(b) ρ ′ T = ( P ⊗ T ) ρ .(c) ( M, ρ ′ ) is a noncounitary ( C, ∆ ′ )-comodule.(d) ( M, P, T ) is a Rota-Baxter paired ( C, ∆ ′ )-comodule of weight λ , whose comodule structuremap is given by ρ ′ . Proof. (a) Since ( C, ∆ , P ) is a Rota-Baxter coalgebra of weight λ , we have( P ⊗ P )∆ = ( P ⊗ id )∆ P + ( id ⊗ P )∆ P + λ ∆ P. Hence we get ∆ ′ P = ( P ⊗ P )∆.Moreover, for any c ∈ C , we have( P ⊗ P )∆ ′ ( c ) = ( P ⊗ P )(( P ⊗ id )∆ + ( id ⊗ P )∆ + λ ∆)( c )= ( P ⊗ P )( P ( c ) ⊗ c + c ⊗ P ( c ) + λc ⊗ c )= P ( c ) ⊗ P ( c ) + P ( c ) ⊗ P ( c ) + λP ( c ) ⊗ P ( c )= ( P ⊗ id + id ⊗ P + λ )( P ( c ) ⊗ P ( c ))= ( P ⊗ id + id ⊗ P + λ )( P ⊗ P )∆( c )= (( P ⊗ id )∆ ′ P + ( id ⊗ P )∆ ′ P + λ ∆ ′ P )( c ) . (b) It is direct to check ρ ′ T = ( P ⊗ id ) ρT + ( id ⊗ T ) ρT + λρT = ( P ⊗ T ) ρ .(c) We only need to prove that ( id ⊗ ρ ′ ) ρ ′ = (∆ ′ ⊗ id ) ρ ′ : by ∆ ′ P = ( P ⊗ P )∆ and ( b ), for any m ∈ M , we have( id ⊗ ρ ′ ) ρ ′ ( m ) = P ( m ( − ) ⊗ ρ ′ ( m (0) ) + m ( − ⊗ ρ ′ ( T ( m (0) )) + λm ( − ⊗ ρ ′ ( m (0) )= P ( m ( − ) ⊗ P ( m (0)( − ) ⊗ m (0)(0) + P ( m ( − ) ⊗ m (0)( − ⊗ T ( m (0)(0) )+ P ( m ( − ) ⊗ λm (0)( − ⊗ m (0)(0) + m ( − ⊗ ρ ′ ( T ( m (0) )) + λm ( − ⊗ P ( m (0)( − ) ⊗ m (0)(0) + λm ( − ⊗ m (0)( − ⊗ T ( m (0)(0) ) + λm ( − ⊗ λm (0)( − ⊗ m (0)(0) = P ( m ( − ) ⊗ P ( m ( − ) ⊗ m (0) + P ( m ( − ) ⊗ m ( − ⊗ T ( m (0) )+ P ( m ( − ) ⊗ λm ( − ⊗ m (0) | {z } + m ( − ⊗ ρ ′ ( T ( m (0) )) + λm ( − ⊗ P ( m ( − ) ⊗ m (0) | {z } + λm ( − ⊗ m ( − ⊗ T ( m (0) ) + λm ( − ⊗ λm ( − ⊗ m (0) | {z } = P ( m ( − ) ⊗ P ( m ( − ) ⊗ m (0) + P ( m ( − ) ⊗ m ( − ⊗ T ( m (0) ) + ∆ ′ ( m ( − ) ⊗ λm (0) | {z } + m ( − ⊗ ρ ′ ( T ( m (0) )) + λm ( − ⊗ m ( − ⊗ T ( m (0) )16 P ( m ( − ) ⊗ P ( m ( − ) ⊗ m (0) + P ( m ( − ) ⊗ m ( − ⊗ T ( m (0) ) + ∆ ′ ( m ( − ) ⊗ λm (0) + m ( − ⊗ ( P ⊗ T ) ρ ( m (0) )) + λm ( − ⊗ m ( − ⊗ T ( m (0) )= P ( m ( − ) ⊗ P ( m ( − ) ⊗ m (0) + P ( m ( − ) ⊗ m ( − ⊗ T ( m (0) ) | {z } +∆ ′ ( m ( − ) ⊗ λm (0) + m ( − ⊗ P ( m ( − ) ⊗ T ( m (0) ) | {z } + λm ( − ⊗ m ( − ⊗ T ( m (0) ) | {z } = P ( m ( − ) ⊗ P ( m ( − ) ⊗ m (0) + ∆ ′ ( m ( − ) ⊗ T ( m (0) ) | {z } +∆ ′ ( m ( − ) ⊗ λm (0) = ∆ ′ P ( m ( − ) ⊗ m (0) + ∆ ′ ( m ( − ) ⊗ T ( m (0) ) + ∆ ′ ( m ( − ) ⊗ λm (0) = (∆ ′ ⊗ id )(( P ⊗ id ) ρ ( m ) + ( id ⊗ T ) ρ ( m ) + λρ ( m ))= (∆ ′ ⊗ id ) ρ ′ ( m ) . (d) By (b) and (c), we have only to prove that( P ⊗ T ) ρ ′ ( m ) = ( P ⊗ T )( P ( m ( − ) ⊗ m (0) + m ( − ⊗ T ( m (0) ) + λm ( − ⊗ m (0) )= P ( m ( − ) ⊗ T ( m (0) ) + P ( m ( − ) ⊗ T ( m (0) ) + λP ( m ( − ) ⊗ T ( m (0) )= ( P ⊗ id + id ⊗ T + λ )( P ⊗ T ) ρ ( m )= ( P ⊗ id + id ⊗ T + λ ) ρ ′ T ( m )= ( P ⊗ id ) ρ ′ T ( m ) + ( id ⊗ T ) ρ ′ T ( m ) + λρ ′ T ( m )for any m ∈ M , so, (d) holds. (cid:3) By the above propositions, we easily get the following corollary.
Corollary 4.17
Let (
M, P, T ) be a Rota-Baxter paired C -comodule of weight λ . Then( M, P , T ) is also a Rota-Baxter paired ( C, ∆ ′ )-comodule of weight λ .Here the coaction ρ ′ of M and the comultiplication ∆ ′ of C are defined in Proposition 4.16,and P , T are defined in Proposition 4.12. § In this section, we mainly construct pre-Lie comodules from Rota-Baxter paired comodules.
Definition 5.1 A pre-Lie coalgebra is ( C, ∆) consisting of a linear space C , a linear map∆ : C → C ⊗ C and satisfying ∆ C − Φ (12) ∆ C = 0 , where ∆ C = (∆ ⊗ id )∆ − ( id ⊗ ∆)∆ and Φ (12) ( c ⊗ c ⊗ c ) = c ⊗ c ⊗ c . Definition 5.2
Let ( C, ∆) be a pre-Lie coalgebra. A left C -pre-Lie comodule ( M, ρ ) is aspace M together with a map ρ : M → C ⊗ M , such that ρ M − ( τ ⊗ id ) ρ M = 0 , where ρ M = ( id ⊗ ρ ) ρ − (∆ ⊗ id ) ρ , and τ ( c ⊗ d ) = d ⊗ c , for any c, d ∈ C . Lemma 5.3
Let (
C, Q ) be a Rota-Baxter coalgebra of weight −
1. Define the operation e ∆ on C by e ∆( c ) = Q ( c ) ⊗ c − Q ( c ) ⊗ c − c ⊗ c . e C = ( C, e ∆) is a pre-Lie coalgebra. Proposition 5.4
Let (
C, Q ) be a Rota-Baxter coalgebra of weight −
1, and (
M, Q, T ) a Rota-Baxter paired C -comodule of weight −
1. Define a map e ρ : M → C ⊗ M by e ρ ( m ) = Q ( m ( − ) ⊗ m (0) + m ( − ⊗ T ( m (0) ) − m ( − ⊗ m (0) . Then ( M, e ρ ) is a left e C -pre-Lie comodule, where e C is defined as in Lemma 5.3. Proof.
By Lemma 5.3, we know e C = ( C, e ∆) is a pre-Lie coalgebra, so, we only need to prove e ρ M − ( τ ⊗ id ) e ρ M = 0 holds.As a matter of fact, for any m ∈ M , we have e ρ M ( m ) = ( id ⊗ e ρ ) e ρ ( m ) − ( e ∆ ⊗ id ) e ρ ( m )= ( id ⊗ e ρ )( Q ( m ( − ) ⊗ m (0) + m ( − ⊗ T ( m (0) ) − m ( − ⊗ m (0) ) − ( e ∆ ⊗ id )( Q ( m ( − ) ⊗ m (0) + m ( − ⊗ T ( m (0) ) − m ( − ⊗ m (0) )= Q ( m ( − ) ⊗ e ρ ( m (0) ) + m ( − ⊗ e ρ ( T ( m (0) )) − m ( − ⊗ e ρ ( m (0) ) − e ∆( Q ( m ( − )) ⊗ m (0) − e ∆( m ( − ) ⊗ T ( m (0) ) + e ∆( m ( − ) ⊗ m (0) = Q ( m ( − ) ⊗ Q ( m (0)( − ) ⊗ m (0)(0) + Q ( m ( − ) ⊗ m (0)( − ⊗ T ( m (0)(0) ) − Q ( m ( − ) ⊗ m (0)( − ⊗ m (0)(0) + m ( − ⊗ Q ( T ( m (0) ) ( − ) ⊗ T ( m (0) ) (0) + m ( − ⊗ T ( m (0) ) ( − ⊗ T ( T ( m (0) ) (0) ) − m ( − ⊗ T ( m (0) ) ( − ⊗ T ( m (0) ) (0) − m ( − ⊗ Q ( m (0)( − ) ⊗ m (0)(0) − m ( − ⊗ m (0)( − ⊗ T ( m (0)(0) )+ m ( − ⊗ m (0)( − ⊗ m (0)(0) − Q ( Q ( m ( − ) ) ⊗ Q ( m ( − ) ⊗ m (0) + Q ( Q ( m ( − ) ) ⊗ Q ( m ( − ) ⊗ m (0) + Q ( m ( − ) ⊗ Q ( m ( − ) ⊗ m (0) − Q ( m ( − ) ⊗ m ( − ⊗ T ( m (0) ) + Q ( m ( − ) ⊗ m ( − ⊗ T ( m (0) )+ m ( − ⊗ m ( − ⊗ T ( m (0) ) + Q ( m ( − ) ⊗ m ( − ⊗ m (0) − Q ( m ( − ) ⊗ m ( − ⊗ m (0) − m ( − ⊗ m ( − ⊗ m (0) = Q ( m ( − ) ⊗ Q ( m ( − ) ⊗ m (0) + Q ( m ( − ) ⊗ m ( − ⊗ T ( m (0) ) − Q ( m ( − ) ⊗ m ( − ⊗ m ((0) + m ( − ⊗ Q ( T ( m (0) ) ( − ) ⊗ T ( m (0) ) (0) + m ( − ⊗ T ( m (0) ) ( − ⊗ T ( T ( m (0) ) (0) ) − m ( − ⊗ T ( m (0) ) ( − ⊗ T ( m (0) ) (0) − m ( − ⊗ Q ( m (0)( − ) ⊗ m (0)(0) − m ( − ⊗ m ( − ⊗ T ( m (0) )+ m ( − ⊗ m ( − ⊗ m (0) − Q ( Q ( m ( − ) ) ⊗ Q ( m ( − ) ⊗ m (0) + Q ( Q ( m ( − ) ) ⊗ Q ( m ( − ) ⊗ m (0) + Q ( m ( − ) ⊗ Q ( m ( − ) ⊗ m (0) − Q ( m ( − ) ⊗ m ( − ⊗ T ( m (0) ) + Q ( m ( − ) ⊗ m ( − ⊗ T ( m (0) )+ m ( − ⊗ m ( − ⊗ T ( m (0) ) + Q ( m ( − ) ⊗ m ( − ⊗ m (0) − Q ( m ( − ) ⊗ m ( − ⊗ m (0) − m ( − ⊗ m ( − ⊗ m (0) = Q ( m ( − ) ⊗ Q ( m ( − ) ⊗ m (0) + m ( − ⊗ Q ( T ( m (0) ) ( − ) ⊗ T ( m (0) ) (0) + m ( − ⊗ T ( m (0) ) ( − ⊗ T ( T ( m (0) ) (0) ) − m ( − ⊗ T ( m (0) ) ( − ⊗ T ( m (0) ) (0) − m ( − ⊗ Q ( m (0)( − ) ⊗ m (0)(0) − Q ( Q ( m ( − ) ) ⊗ Q ( m ( − ) ⊗ m (0) Q ( Q ( m ( − ) ) ⊗ Q ( m ( − ) ⊗ m (0) + Q ( m ( − ) ⊗ Q ( m ( − ) ⊗ m (0) + Q ( m (0)( − ) ⊗ m ( − ⊗ T ( m (0)(0) ) − Q ( m (0)( − ) ⊗ m ( − ⊗ m (0)(0) . So, by the above equality and (
C, Q ) being a Rota-Baxter coalgebra of weight −
1, we easilyprove that e ρ M ( m ) = ( τ ⊗ id ) e ρ M ( m ) for any m ∈ M . Hence ( M, e ρ ) is a left e C -pre-Lie comodule. (cid:3) Lemma 5.5
Let ( C, ∆ , Q ) be a Rota-Baxter coalgebra of weight 0. Define the operation e ∆ on C by e ∆( c ) = Q ( c ) ⊗ c − Q ( c ) ⊗ c . Then, by [9], e C = ( C, e ∆) is a pre-Lie coalgebra.According to Lemma 5.5, we can prove the following proposition using a similar way as inProposition 5.4. Proposition 5.6
Let (
C, Q ) be a Rota-Baxter coalgebra of weight 0, and (
M, Q, T ) a Rota-Baxter paired C -comodule of weight 0. Define a map e ρ : M → C ⊗ M by e ρ ( m ) = Q ( m ( − ) ⊗ m (0) + m ( − ⊗ T ( m (0) ) , for any m ∈ M . Then ( M, e ρ ) is a left e C -pre-Lie comodule, where e C is defined as in Lemma 5.5. § In this section, we will combine Rota-Baxter paired modules and Rota-Baxter paired comodules,and introduce the conception of Rota-Baxter pared Hopf modules, and give the structure theoremof Rota-Baxter pared Hopf modules.
Definition 6.1
Let H be a bialgera, and M a left H -Hopf module. A triple ( M, P, T ) is calleda
Rota-Baxter paired left H -Hopf module of weight λ , if ( M, P, T ) is both a Rota-Baxterpaired left H -module of weight λ , and a Rota-Baxter paired left H -comodule of weight λ .A Rota-Baxter H -Hopf submodule N of a Rota-Baxter paired H -Hopf-module ( M, P, T )is an H -Hopf submodule of M such that T ( N ) ⊆ N . Then ( N, P, T ) is a Rota-Baxter paired H -Hopf module.Let ( M, P, T ) and ( M ′ , P ′ , T ′ ) be Rota-Baxter paired H -Hopf modules of the same weight λ . A Rota-Baxter H -Hopf module map f : ( M, P, T ) → ( M ′ , P ′ , T ′ ) of weight λ is a Hopf modulemap such that f ◦ T = T ′ ◦ f . Example 6.2 (1) Let H be a bialgebra. Then, H is not only an augmented coalgebra (thereis a coalgebra map µ : k → H ) and an augmented algebra (there is an algebra map ε : H → k ).Define a map P : H → H given by P ( h ) = ε ( h )1 H . Then, by Example 2.2, ( H, P ) is aRota-Baxter coalgebra of weight −
1, and a Rota-Baxter algebra of weight − H, P, P ) is not only a Rota-Baxter paired H -comodule of weight −
1, and a Rota-Baxterpaired H -module of weight − H is a right H -Hopf module via its multiplication and its comultiplication.Hence ( H, P, P ) is a Rota-Baxter paired H -Hopf module of weight − H be a bialgebra, and ( H, P, P ) a Rota-Baxter bialgebra of weight ( λ, λ ) given in [9],that is, (
H, P ) is both a Rota-Baxter algebra of weight λ , and ( H, P ) a Rota-Baxter coalgebraof weight λ . Then, ( H, P, P ) is both a Rota-Baxter paired H -module of weight λ , whose actionis given by the multiplication of H , and ( H, P, P ) a Rota-Baxter paired H -comodule of weight λ , whose coaction is given by the comultiplication of H . So, ( H, P, P ) is a Rota-Baxter paired H -Hopf module of weight λ .(3) Let H be a quantum commutative weak Hopf algebra, and M a weak H -Hopf module.Then, by Remark 3.17 in [14], ( M, ⊓ L , T ) is a Rota-Baxter paired H -module of weight −
1, where T is given in Proposition 4.7.Again by Proposition 4.7, ( M, ⊓ L , T ) is also a Rota-Baxter paired H -comodule of weight − M, ⊓ L , T ) is a Rota-Baxter paired H -Hopf module of weight − H, ⊓ L , ⊓ L ) is a Rota-Baxter paired H -Hopf module of weight − H .Combining Theorem 3.1 and Theorem 2.4 in [14], we get the following result. Proposition 6.3
Let H be a bialgebra, and M a left H -Hopf module. Suppose that there isa Hopf module map T from M to M . Then the following are equivalent:(1) ( M, T ) is a generic Rota-Baxter paired H -Hopf module of weight λ ;(2) There is a linear operator P : H → H such that ( M, P, T ) is a Rota-Baxter paired H -Hopfmodule of weight λ ;(3) T is quasi-idempotent of weight λ . Example 6.4
Let H be a bialgebra, and C a coalgebra. Then H ⊗ C is left H -Hopf moduleby h · ( g ⊗ c ) = hg ⊗ c and ρ ( h ⊗ c ) = h ⊗ h ⊗ c , for any h, g ∈ H, c ∈ C . Define a map T : H ⊗ C → H ⊗ C by T ( h ⊗ c ) = h ⊗ ε ( c ) e , where e ∈ C satisfying ε ( e ) = 1. It isn’t difficultto prove that T = T and T a Hopf module map, so by Proposition 6.3, ( H ⊗ C, T ) is a genericRota-Baxter paired H -Hopf module of weight − Theorem 6.5
Let H be a Hopf algebra, and ( M, T ) a generic Rota-Baxter pared H -Hopfmodule of weight λ in Proposition 6.3. Then, there is an isomorphism:( M, T ) ∼ = ( H ⊗ M coH , T ′ )as generic Rota-Baxter left H -Hopf modules of weight λ , where T ′ is defined by T ′ ( h ⊗ m ) = h ⊗ T ( m ) , h ∈ H, m ∈ M coH , and M coH = { m ∈ M | ρ ( m ) = 1 ⊗ m } , and H ⊗ M coH is left H -Hopf module by h · ( g ⊗ m ) = hg ⊗ m and ρ ( h ⊗ m ) = h ⊗ h ⊗ m , for any h, g ∈ H, m ∈ M coH . Proof.
Since T is a left H -comodule map, we easily see that T ( m ) ∈ M coH , for m ∈ M coH .Hence T ′ is well defined.According to [12], it is obvious that H ⊗ M coH is a left H -Hopf module.In what follows, by Proposition 6.3, we prove that ( H ⊗ M coH , T ′ ) is a generic Rota-Baxterleft H -Hopf module of weight λ . 20s a matter of fact, for any h ∈ H, m ∈ M coH , we have T ′ ( h ⊗ m ) = T ′ ( h ⊗ T ( m )) = h ⊗ T ( m ) = − λh ⊗ T ( m ) = − λT ′ ( h ⊗ m ) , so T ′ is quasi-idempotent of weight λ . Again, for any h, g ∈ H, m ∈ M coH , we get T ′ ( h · ( g ⊗ m )) = T ′ ( hg ⊗ m ) = hg ⊗ T ( m )= h · T ′ ( g ⊗ m ) ,T ′ ( h ⊗ m ) ( − ⊗ T ′ ( h ⊗ m ) (0) = ( h ⊗ T ( m )) ( − ⊗ ( h ⊗ T ( m )) (0) = h ⊗ ( h ⊗ T ( m )) = h ⊗ T ′ ( h ⊗ m )= ( h ⊗ m ) ( − ⊗ T ′ (( h ⊗ m ) (0) ) , so, T ′ is a left H -Hopf module map. Hence ( H ⊗ M coH , T ′ ) is a generic Rota-Baxter left H -Hopfmodule of weight λ by Proposition 6.3.According to Theorem 4.1.1 in [12], we have an H -Hopf module isomorphisms as follows: α : H ⊗ M coH → M, h ⊗ m h · m, with the inverse β : M → H ⊗ M coH , m m ( − ⊗ E M ( m (0) ) , where E M ( m ) is given by S ( m ( − ) · m (0) , for m ∈ M .Moreover, for any m ∈ M , we obtain T ′ ◦ β ( m ) = T ′ ( m ( − ⊗ E M ( m (0) ))= m ( − ⊗ T ( E M ( m (0) ))= m ( − ⊗ T ( S ( m (0)( − ) · m (0)(0) )= m ( − ⊗ S ( m (0)( − ) · T ( m (0)(0) )= m ( − ⊗ S ( T ( m (0) ) ( − ) · T ( m (0) ) (0) = m ( − ⊗ E M ( T ( m (0) ))= T ( m ) ( − ⊗ E M ( T ( m ) (0) ))= β ◦ T ( m ) , so T ′ ◦ β = β ◦ T . In a similar way, we can prove that α ◦ T ′ = T ◦ α . Hence ( M, T ) ∼ = ( H ⊗ M coH , T ′ )as generic Rota-Baxter left H -Hopf module of weight λ . (cid:3) References [1] M. Aguiar. Pre-Poisson Algebras. Lett. Math. Phys., 54(2000), 263-277.[2] G. Baxter. An analytic problem whose solution follows from a simple algebraic identity, Pac.J. Math., 10(1960), 731-742.[3] D. Bagio, D. Flores, A. Santana. Inner actions of weak Hopf algebras, J. Algebra Appl.,12(2015), 393-403. 214] G. B¨ a hm, F. Nill, K. Szlach´ a nyi. Weak Hopf algebras (I): integral theory and C ∗∗