Classification of operator extensions, monad liftings and distributive laws for differential algebras and Rota-Baxter algebras
aa r X i v : . [ m a t h . A C ] F e b CLASSIFICATION OF OPERATOR EXTENSIONS, MONAD LIFTINGS ANDDISTRIBUTIVE LAWS FOR DIFFERENTIAL ALGEBRAS AND ROTA-BAXTERALGEBRAS
SHILONG ZHANG, LI GUO, AND WILLIAM KEIGHERA bstract . Generalizing the algebraic formulation of the First Fundamental Theorem of Calculus(FFTC), a class of constraints involving a pair of operators was considered in [27]. For a givenconstraint, the existences of extensions of di ff erential and Rota-Baxter operators, of liftings ofmonads and comonads, and of mixed distributive laws are shown to be equivalent. In this paper,we give a classification of the constraints satisfying these equivalent conditions. C ontents
1. Introduction 12. Background and the statement of the main theorem 32.1. Free Rota-Baxter algebras and cofree di ff erential algebras 42.2. Covers and extensions of operators 62.3. Main results 93. Proof of the main theorem 103.1. Proof of Theorem 3.1.(i): Case 1 113.2. Proof of Theorem 3.1.(i): Case 2 133.3. Proof of Theorem 3.1.(i): Case 3 163.4. Proof of Theorem 3.1.(ii) 22References 241. I ntroduction The algebraic study of analysis has a long history. In the 1930s, the notion of a di ff erentialring or algebra was introduced by Ritt [20] to give an algebraic study of di ff erential analysisand di ff erential equations. Here a di ff erential algebra is an (associative) algebra R with a linearoperator d satisfying the Leibniz rule(1) d ( xy ) = d ( x ) y + xd ( y ) for all x , y ∈ R . Through the later work of Kolchin and many other mathematicians, di ff erential algebra has beendeveloped into a vast area including di ff erential Galois groups, di ff erential algebraic groups anddi ff erential algebraic geometry, with broad applications in number theory, logic and mechanicalproof of mathematical theorems [17, 23, 25].The algebraic abstraction of the integral analysis came much later, as a byproduct of the workof G. Baxter in probability in 1960 [2]. A Baxter algebra, later called Rota-Baxter algebra , isan algebra R with a linear operator P such that(2) P ( x ) P ( y ) = P ( P ( x ) y ) + P ( xP ( y )) + λ P ( xy ) for all x , y ∈ R . Date : February 12, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Rota-Baxter algebra, di ff erential algebra, cover of operators, extension of operators,monad, distributive law. Here λ is a given scalar in the base ring, called the weight of the Rota-Baxter operator. Afterthe pioneering work of Cartier and Rota [3, 22] in combinatorics, the recent developments ofRota-Baxter algebras have ranged from multiple zeta values in number theory to renormalizationof perturbation quantum field theory [1, 5, 8, 10, 14, 21, 22].As a di ff erential analog of a Rota-Baxter operator of weight λ , a di ff erential operator ofweight λ [12] is defined to satisfy the equation(3) d ( xy ) = d ( x ) y + xd ( y ) + λ d ( x ) d ( y ) for all x , y ∈ R and d ( R ) = . With the algebraizations of both di ff erential and integral analyses in place, it is natural toformulate an algebraic abstraction of the two analyses through the well-known First FundamentalTheorem of Calculus (FFTC), leading to the notion of a di ff erential Rota-Baxter algebra withweight. To be precise, a di ff erential Rota-Baxter algebra of weight λ is a triple ( R , d , P ) consistingof (i) an algebra R ,(ii) a di ff erential operator d of weight λ on R , and(iii) a Rota-Baxter operator P of weight λ on R ,such that(4) dP = id R , reflecting the FFTC. See [7, 13, 21] for a variation, called an integro-di ff erential algebra.This natural algebraic abstraction of the FFTC has quite remarkable categorical implicationsin terms of liftings of monads and mixed distributive laws, as shown in [26] which has attractedinterests from combinatorics, di ff erential algebra, probability and computer science [4, 6, 15, 24].We fix an algebra R and let R N denote the Hurwitz series algebra over R [16, 12]. Then R N ,with a natural di ff erential operator ∂ R , is the cofree di ff erential algebra on R . Further, we have acomonad, denoted by C , giving di ff erential algebras [26]. Also let X ( R ) be the mixable shu ffl eproduct algebra [10]. Then ( X ( R ) , P R ), where P R is a naturally defined Rota-Baxter operator, isthe free Rota-Baxter algebra on R which results in a monad, denoted by T , giving Rota-Baxteralgebras [26]. In [12], a di ff erential operator on R is uniquely extended to X ( R ), enriching X ( R )to be the free di ff erential Rota-Baxter algebra and giving a lifting of the monad T . Further, by thelifting, we obtain a mixed distributive law of the monad T over the comonad C [26]. Similarly,given a Rota-Baxter operator on R , we construct a cover of the operator on R N , and enrich R N tobe the cofree di ff erential Rota-Baxter algebra which gives a lifting of the comonad C [26]. Thesame mixed distributive law also follows.These results show that the coupling of a di ff erential operator and a Rota-Baxter operator viaFFTC leads to the existences of extensions of di ff erential operators, of covers of Rota-Baxteroperators, of liftings of monads, and of mixed distributive laws. Then there are the followingnatural problems. Problem 1.
How are the categorical properties of extensions of di ff erential operators, coversof Rota-Baxter operators, liftings of monads, and mixed distributive laws interrelated to oneanother? Problem 2.
Are these equivalent categorical properties unique to this coupling via FFTC? Inother words, are there other examples where these equivalent properties hold? The term “coextension” in [27] is replaced by “cover” in this paper.
LASSIFICATION OF EXTENSIONS, LIFTINGS AND DISTRIBUTIVE LAWS 3
To investigate Problem 1, a follow-up study of [26] was carried out in [27] where the identityin the FFTC is viewed as an example of a polynomial identity in two noncommutative variablessymbolizing the di ff erential operator and Rota-Baxter operator. Thus we work in the noncommu-tative polynomial algebra k h x , y i in two variables x , y and regard each polynomial ω = ω ( x , y ) in k h x , y i as a constraint between two linear operators q and Q , both defined on an algebra R , givenby a formal identity ω ( q , Q ) =
0. When q and Q are the di ff erential operator and Rota-Baxteroperator respectively, ω ( x , y ) = xy − Ω ⊂ k h x , y i of constraints inwhich to investigate Problem 1. The class of constraints is(5) Ω : = xy + k [ x ] + y k [ x ] = { xy − ( φ ( x ) + y ψ ( x )) | φ, ψ ∈ k [ x ] } . It was a pleasant surprise to find out in [27] that, for these constrains, the following categor-ical properties are in fact equivalent: extensions of di ff erential operators, covers of Rota-Baxteroperators, liftings of monads, and existence of mixed distributive laws. See Theorem 2.15 for aprecise statement. The theorem provides an answer to Problem 1.The purpose of this paper is to address Problem 2, namely the dependence of these categoricalproperties on the constraints. According to [26], the polynomial xy − Ω satisfying these equivalent properties. In Section 3,Theorem 2.16 is rephrased as Theorem 3.1 and proved in several steps. It would be interesting todetermine whether this equivalence of categorical properties holds for more general polynomialsin k h x , y i and to achieve a classification of such polynomials. Conjecture 3.2 gives a formulationin this direction.Throughout the paper, we fix a commutative ring k with identity and an element λ ∈ k . Unlessotherwise noted, we work in the categories of commutative k -algebras with identity, with orwithout linear operators. All operators and tensor products are also taken over k . Thus referencesto k will be suppressed unless doing so can cause confusion. We let N denote the additive monoidof natural numbers { , , , . . . } and N + = { n ∈ N | n > } the positive integers. Let δ i , j , i , j ∈ N denote the Kronecker delta. For categorical notations, we follow [19].2. B ackground and the statement of the main theorem In this section we provide background to state the main theorem and prove preliminary resultsrequired in the proof of the main theorem. In Section 2.1, we review free Rota-Baxter algebrasand cofree di ff erential algebras, and the corresponding adjoint functor pairs. We also prove aproperty in a special case which will be applied repeatedly in the proof of the main theorem.In Section 2.2, we start with the category of operated algebras and consider its enrichments byadding di ff erential operators or Rota-Baxter operators. We give covers of operators or extensionsof operators in a operated algebra to certain objects in these enriched categories. Building onthese preparations, we state in Section 2.3 the theorem on the equivalence of extensions or coversof operators, liftings of monads and existence of distributive laws, and the main theorem which SHILONG ZHANG, LI GUO, AND WILLIAM KEIGHER gives a classification of the constraints for which each and hence all the equivalent conditionshold.Additional details can be found in [8, 12, 27]. The proof of the main theorem will be given inthe next section.2.1.
Free Rota-Baxter algebras and cofree di ff erential algebras. Recall from [8, 10] the con-struction of the free commutative Rota-Baxter algebra X ( A ) of weight λ on a commutative alge-bra A with identity A . As a module, we have X ( A ) = M i ∈ N + A ⊗ i = A ⊕ ( A ⊗ A ) ⊕ ( A ⊗ A ⊗ A ) ⊕ · · · . Define an operator P A on X ( A ) by assigning P A ( x ⊗ x ⊗ · · · ⊗ x n ) : = A ⊗ x ⊗ x ⊗ · · · ⊗ x n for all x ⊗ x ⊗ · · · ⊗ x n ∈ A ⊗ ( n + and extending by additivity.Then by [10, Theorem 4.1], the module X ( A ), with the mixable shu ffl e product, the operator P A and the natural embedding j A : A → X ( A ), is a free Rota-Baxter algebra of weight λ on A .More precisely, for any Rota-Baxter algebra ( R , P ) of weight λ and any algebra homomorphism ϕ : A → R , there exists a unique Rota-Baxter algebra homomorphism ˜ ϕ : ( X ( A ) , P A ) → ( R , P )such that ϕ = ˜ ϕ j A .For later use, we give a class of Rota-Baxter algebras with non-zero Rota-Baxter operators. Example 2.1.
Take A = k in X ( A ). Then [10, Proposition 6.1] states that X ( k ) is an algebrawith basis z i : = ⊗ ( i + ∈ k ⊗ ( i + for each i ∈ N . The multiplication on X ( k ) is given by(6) z m z n = m X j = m + n − jn ! nj ! λ j z m + n − j for all m , n ∈ N . In particular, when λ =
0, one sees(7) z m z n = m + nn ! z m + n , giving the divided power algebra.The identity element of X ( k ) is z and the operator P k : X ( k ) → X ( k ) is given by P k ( z i ) = z i + for each i ∈ N . For a given m ∈ N + , I m : = ⊕ i ≥ m k z i is a Rota-Baxter ideal of ( X ( k ) , P k ), that is, X ( k ) I m ⊆ I m , P k ( I m ) ⊆ I m , giving rise to the quotient Rota-Baxter algebra ( X ( k ) / I m , P k ). Then for m ≥ P k ( z m − ) = z m − , , P k ( z m − ) = z m = ∈ X ( k ) / I m . We let
ALG denote the category of commutative algebras, and let
RBA λ , or simply RBA ,denote the category of commutative Rota-Baxter algebras of weight λ . Then we have a functor F : ALG → RBA given on objects A by F ( A ) = ( X ( A ) , P A ) and on morphisms ϕ : A → B by F ( ϕ ) k X i = a i ⊗ a i ⊗ · · · ⊗ a in i = k X i = ϕ ( a i ) ⊗ ϕ ( a i ) ⊗ · · · ⊗ ϕ ( a in i ) LASSIFICATION OF EXTENSIONS, LIFTINGS AND DISTRIBUTIVE LAWS 5 for any k P i = a i ⊗ a i ⊗ · · · ⊗ a in i ∈ X ( A ).Then the universal property of the free Rota-Baxter algebras X ( A ) has the following formula-tion. Proposition 2.2. ([26, Corollary 2.4])
The functor F : ALG → RBA defined above is the leftadjoint of the forgetful functor U : RBA → ALG . Furthermore, the adjunction in Proposition 2.2 provides a monad T = T RBA = h T , η, µ i on ALG giving Rota-Baxter algebras [26, § A , T ( A ) = X ( A ), η A isthe natural embedding from A to X ( A ), and µ A : X ( X ( A )) → X ( A ) is extended additively from µ A (( a ⊗ · · · ⊗ a n ) ⊗ · · · ⊗ ( a k ⊗ · · · ⊗ a kn k )) = ( a ⊗ · · · ⊗ a n ) P A ( · · · P A ( a k ⊗ · · · ⊗ a kn k ) · · · ) , where ( a ⊗ · · · ⊗ a n ) ⊗ · · · ⊗ ( a k ⊗ · · · ⊗ a kn k ) ∈ X ( X ( A )) with a i ⊗ · · · ⊗ a in i ∈ A ⊗ ( n i + for n , . . . , n k ≥ ≤ i ≤ k .Next we review some background on di ff erential algebras with weights, defined in (3), andrefer the reader to [12] for details.For any algebra A , let A N denote the k -module of functions f : N → A . We also view f ∈ A N as a sequence ( f n ) = ( f , f , · · · ) with f n : = f ( n ) ∈ A . Following [12, § λ -Hurwitz product on A N is given by(8) ( f g ) n = n X k = n − k X j = nk ! n − kj ! λ k f n − j g k + j for all f , g ∈ R N . In the special case when λ =
0, we have(9) ( f g ) n = n X j = nj ! f n − j g j . With this product, A N is called the algebra of λ -Hurwitz series over A . Further define ∂ A : A N → A N , ∂ A ( f ) n = f n + for all f ∈ A N , n ∈ N . Then ∂ A is a di ff erential operator of weight λ on A N , making ( A N , ∂ A ) into a di ff erential algebra ofweight λ . We also obtain a recursive formula for ( f g ) n :(10) ( f g ) n + = ( ∂ A ( f g )) n = ( ∂ A ( f ) g ) n + ( f ∂ A ( g )) n + ( λ∂ A ( f ) ∂ A ( g )) n for all f , g ∈ A N , n ∈ N . Let
DIF denote the category of di ff erential algebras of weight λ , and G : ALG → DIF be afunctor given on objects A by G ( A ) : = ( A N , ∂ A ) and on morphisms ϕ : A → B by G ( ϕ ) : = ϕ N : ( A N , ∂ A ) → ( B N , ∂ B ) , ( ϕ N ( f )) n = ϕ ( f n ) , f ∈ A N , n ∈ N . Proposition 2.3. ( See [12, Proposition 2.8])
The functor G : ALG → DIF is the right adjoint ofthe forgetful functor V : DIF → ALG . In other words, the di ff erential algebra ( A N , ∂ A ) , togetherwith the algebra homomorphism (11) ε A : A N → A , ε A ( f ) : = f for all f ∈ A N , is a cofree di ff erential algebra of weight λ on the algebra A. SHILONG ZHANG, LI GUO, AND WILLIAM KEIGHER
The adjunction in Proposition 2.3 gives rise to a comonad C = h C , ε, δ i on ALG giving dif-ferential algebra [26, § A ∈ ALG , C ( A ) = A N , ε A : A N → A is a surjection with ε A ( f ) = f , and δ A : A N → ( A N ) N is defined by( δ A ( f ) m ) n = f m + n , f ∈ A N , m , n ∈ N . Fix a comonad ˜ C = h ˜ C , ˜ ε, ˜ δ i on RBA . If the following identities hold U ˜ C = CU , U ˜ ε = ε U and U ˜ δ = δ U , then ˜ C is said to lift the comonad C . Dually, a monad ˜ T = h ˜ T , ˜ η, ˜ µ i on DIF lifting the monad T from Proposition 2.2 satisfies V ˜ T = T V , V ˜ η = η V and V ˜ µ = µ V . The monad T , comonad C and their lifting forms will be used in the formulation of Theorem 2.15. Example 2.4.
On the Rota-Baxter algebra ( X ( k ) , P k ) in Example 2.1, define d : X ( k ) → X ( k ) , d ( z ) = , d ( z n ) = z n − for all n ∈ N + . Then ( X ( k ) , d ) is a di ff erential algebra of weight λ . By [11, Corollary 3.7], the completion of( X ( k ) , d ) is isomorphic to the algebra k N of Hurwitz series over k .2.2. Covers and extensions of operators.
In [27], we also introduced a class of polynomialsthat serve as relations between a pair of operators as follows. In the noncommutative polynomialalgebra k h x , y i in two variables x and y , consider the subset(12) Ω : = xy + k [ x ] + y k [ x ] = { xy − ( φ ( x ) + y ψ ( x )) | φ, ψ ∈ k [ x ] } . Let q and Q be two operators on an algebra R . Each ω : = ω ( x , y ) ∈ Ω is regarded as a relation ω ( q , Q ) = q and Q . As a special case, ω = xy − ω ( d , P ) = dP − id R = d and P in a di ff erential Rota-Baxter algebra ( R , d , P ) [12]reflecting the First Fundamental Theorem of Calculus. Definition 2.5. An operated algebra [9, 18] is an algebra R with a linear operator Q on R , thusdenoted as a pair ( R , Q ). Let OA denote the category of operated algebras.As its enrichment, we have Definition 2.6. ([27, Definition 2.6]) For a given ω ∈ Ω and λ ∈ k , we say that the triple ( R , d , Q )is a type ω operated di ff erential algebra of weight λ if(i) ( R , d ) is a di ff erential algebra of weight λ ,(ii) ( R , Q ) is an operated algebra, and(iii) ω ( d , Q ) =
0, that is,(13) dQ = φ ( d ) + Q ψ ( d ) . There is a one-to-one correspondence between operators P on R N and sequences ( P n ) of linearmaps where, for each n ∈ N , P n : R N → R is given by P n ( f ) : = P ( f ) n for all f ∈ R N . For any operators Q , J on R N , and each f ∈ R N , n ∈ N , we obtain(14) ( ∂ R Q ) n ( f ) = ( ∂ R ( Q ( f ))) n = Q n + ( f ) , ( QJ ) n ( f ) = ( Q ( J ( f ))) n = Q n ( J ( f )) . We now recall the notion of a cover (called coextension in [27]) of an operator on an algebrato the cofree di ff erential algebra generated by this algebra. LASSIFICATION OF EXTENSIONS, LIFTINGS AND DISTRIBUTIVE LAWS 7
Definition 2.7.
For a given operator Q : R → R , we call an operator b Q : R N → R N a cover of Q on R N if for all f ∈ R N , we have b Q ( f ) = Q ( f ). That is, the following diagram commutes R N ε R (cid:15) (cid:15) b Q / / R N ε R (cid:15) (cid:15) R Q / / R where ε R is given in (11).The operator b Q is called a cover of Q because ε R is surjective. For each ω ∈ Ω , we establishedthe existence and uniqueness of a cover as the following proposition shows. Proposition 2.8. ([27, Proposition 2.6])
Let Q be an operator on an algebra R. For a given ω = xy − ( φ ( x ) + y ψ ( x )) ∈ Ω with φ, ψ ∈ k [ x ] , Q has a unique cover b Q ω : ( R N , ∂ R ) → ( R N , ∂ R ) such that ω ( ∂ R , b Q ω ) = , that is :(15) ∂ R b Q ω = φ ( ∂ R ) + b Q ω ψ ( ∂ R ) . Thus the triple ( R N , ∂ R , b Q ω ) is a type ω operated di ff erential algebra. For a given ω ∈ Ω , let ODA ω denote the category of type ω operated di ff erential algebras ofweight λ in Definition 2.6. Thanks to Proposition 2.8, we obtain a functor(16) G ω : OA → ODA ω . Applying (14), (15) is equivalent to(17) b Q ω n + = φ ( ∂ R ) n + b Q ω n ψ ( ∂ R ) for all n ∈ N . The following equivalent characterizations of a Rota-Baxter algebra in terms of covers will beuseful in the proof of our main result Theorem 2.16.
Proposition 2.9.
Let ( R , Q ) be an operated algebra and b Q be any cover of Q to R N . (i) ( R , Q ) is a Rota-Baxter algebra of weight λ if and only if (18) b Q ( f ) b Q ( g ) = b Q ( b Q ( f ) g ) + b Q ( f b Q ( g )) + λ b Q ( f g ) for all f , g ∈ R N . (ii) ( R N , b Q ) is a Rota-Baxter algebra of weight λ if and only if for all f , g ∈ R N , n ∈ N , (19) n X k = n − k X j = nk ! n − kj ! λ k b Q n − j ( f ) b Q k + j ( g ) = b Q n ( b Q ( f ) g ) + b Q n ( f b Q ( g )) + λ b Q n ( f g ) . The proof of Proposition 2.9 is straightforward.The following special case of Proposition 2.9.(ii) will be used repeatedly in the proof of Theo-rem 3.1.(i). When λ =
0, (19) becomes n X j = nj ! b Q n − j ( f ) b Q j ( g ) = b Q n ( b Q ( f ) g ) + b Q n ( f b Q ( g )) for all f , g ∈ R N , n ∈ N . As a consequence, we have
SHILONG ZHANG, LI GUO, AND WILLIAM KEIGHER
Corollary 2.10.
Let ( R , P ) be a Rota-Baxter algebra of weight . If there are f , g ∈ R N such that (20) (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) , , then the cover b P ω of P to ( R N , ∂ R ) is not a Rota-Baxter operator of weight . Next we recall the definition of extensions of operators in [27].
Definition 2.11.
For a given operator q : R → R satisfying q ( R ) =
0, we call an operatorˆ q : X ( R ) → X ( R ) an extension of q to X ( R ) if ˆ q | R = q . That is, the following diagramcommutes R j R (cid:15) (cid:15) q / / R j R (cid:15) (cid:15) X ( R ) ˆ q / / X ( R )where j R : R → X ( R ) is a natural embedding.Since X ( R ) = R by the definition of the mixable shu ffl e product on X ( R ), we have ˆ q ( X ( R ) ) = q ( R ) = q of q to X ( R ). Definition 2.12. ([27, Definition 2.9]) For a given ω ∈ Ω and λ ∈ k , we say that the triple ( R , q , P )is a type ω operated Rota-Baxter algebra of weight λ if(i) ( R , q ) is an operated algebra with the property q ( R ) = R , P ) is a Rota-Baxter algebra of weight λ , and(iii) ω ( q , P ) =
0, that is, qP = φ ( q ) + P ψ ( q ) . Proposition 2.13. ([27, Proposition 2.11])
Let ( R , q ) be an operated algebra where the operatorq satisfies q ( R ) = . For a given ω = xy − ( φ ( x ) + y ψ ( x )) ∈ Ω with φ, ψ ∈ k [ x ] , q has a uniqueextension ˆ q ω : ( X ( R ) , P R ) → ( X ( R ) , P R ) with the following property: for u = u ⊗ u ′ ∈ R ⊗ ( n + with u ′ ∈ R ⊗ n , (21) ˆ q ω ( u ) = q ( u ) ⊗ u ′ + ( u + λ q ( u ))( φ ( ˆ q ω ) + P R ψ ( ˆ q ω ))( u ′ ) and (22) ˆ q ω ( ⊕ ni = R ⊗ i ) ⊆ ⊕ ni = R ⊗ i for each n ∈ N + . The triple ( X ( R ) , ˆ q ω , P R ) is a type ω operated Rota-Baxter algebra. We let OA denote the category of operated algebras ( R , q ) with the property q ( R ) =
0. Thus
DIF is a subcategory of OA . Let ORB ω denote the category of type ω operated Rota-Baxteralgebras of weight λ . Proposition 2.13 gives a functor(23) F ω : OA → ORB ω . As a generalization of a di ff erential Rota-Baxter algebra, we introduced in [27] the concept oftype ω di ff erential Rota-Baxter algebras. Definition 2.14.
For a given ω ∈ Ω and λ ∈ k , we say that the triple ( R , d , P ) is a type ω di ff erential Rota-Baxter algebra of weight λ if(i) ( R , d ) is a di ff erential algebra of weight λ ,(ii) ( R , P ) is a Rota-Baxter algebra of weight λ , and LASSIFICATION OF EXTENSIONS, LIFTINGS AND DISTRIBUTIVE LAWS 9 (iii) ω ( d , P ) =
0, that is,(24) dP = φ ( d ) + P ψ ( d ) . See [27, Example 3.5] for examples of type ω di ff erential Rota-Baxter algebras from analysis.The category of type ω di ff erential Rota-Baxter algebras of weight λ will be denoted by DRB ω .Note that DRB ω is a subcategory of ORB ω (resp., ODA ω ).In the subsequent subsection, we will provide conditions ensuring that the restriction of thefunctor G ω : OA → ODA ω to the subcategory RBA of OA gives a functor RBA → DRB ω .Likewise for F ω , as indicated in the following diagram. ODA ω DRB ω ? _ o o (cid:31) (cid:127) / / ORB ω OA G ω C C ✝✝✝✝✝✝✝✝✝✝✝✝✝✝ RBA ? _ o o G ω A A ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ DIF (cid:31) (cid:127) / / F ω ] ] ❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀ OA F ω \ \ ✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾ Main results.
In this section, assume that k is a domain of characteristic 0. As in (12), let Ω : = xy + k [ x ] + y k [ x ] . The following theorem shows that the existences of covers of Rota-Baxteroperators, and of extensions of di ff erential operators, of liftings of monads and comonads, and ofmixed distributive laws are equivalent. Theorem 2.15. ([27, Theorem 3.15])
Let ω ∈ Ω be given. The following statements are equiva-lent: (i) For every Rota-Baxter operator P on every algebra R, the unique cover b P ω of P to R N given in Proposition 2.8 is a Rota-Baxter operator. (ii) For every di ff erential operator d on every algebra R, the unique extension ˆ d ω of d to X ( R ) given in Proposition 2.13 is a di ff erential operator. (iii) The functor G ω : OA → ODA ω in (16) restricts to a functor G ω : RBA → DRB ω . (iv) The functor F ω : OA → ORB ω in (23) restricts to a functor F ω : DIF → DRB ω . (v) There exist a comonad ˜ C = h ˜ C , ˜ ε, ˜ δ i on RBA lifting the comonad C from Proposition 2.3,where ˜ C ( R , P ) : = ( R N , e P ) , and a category isomorphism e H : DRB ω → RBA ˜ C over RBA given by e H ( R , d , P ) : = h ( R , P ) , θ ( R , d , P ) : ( R , P ) → ( R N , e P ) i . Here θ ( R , d , P ) ( u ) n : = d n ( u ) for all u ∈ R , n ∈ N . (vi) There exist a monad ˜ T = h ˜ T , ˜ η, ˜ µ i on DIF lifting the monad T from Proposition 2.2,where ˜ T ( R , d ) : = ( X ( R ) , ˜ d ) , and a category isomorphism e K : DRB ω → DIF ˜ T over DIF given by e K ( R , d , P ) : = h ( R , d ) , ϑ ( R , d , P ) : ( X ( R ) , ˜ d ) → ( R , d ) i . Here for every v ⊗ v ⊗ · · · ⊗ v m ∈ X ( R ) , ϑ ( R , d , P ) ( v ⊗ v ⊗ · · · ⊗ v m ) : = v P ( v P ( · · · P ( v m ) · · · )) . (vii) There is a mixed distributive law β : T C → CT such that ( ALG C ) ˜ T β is isomorphic to thecategory DRB ω , where ˜ T β is a lifting monad of T given by the mixed distributive law β . It is important to classify in concrete terms the elements ω ∈ Ω that satisfy the equivalentconditions in the theorem. We obtain two results in this direction, first in the case when theweight is zero and then in the case when the weight is arbitrary. As we will see, the conditionsimposed on ω are very strict.Consider the following subsets of Ω : Ω : = { xy − a | a ∈ k } ∪ { xy − ( b y + yx ) | b ∈ k } , Ω k : = { xy , xy − , xy − yx } . Then we give the main theorem in this paper:
Theorem 2.16.
Let k be a domain of characteristic zero, and ω ∈ Ω be given. (i) The equivalent conditions in Theorem 2.15 hold in the case of λ = if and only if ω is in Ω . (ii) The equivalent conditions in Theorem 2.15 hold for all weights λ ∈ k if and only if ω isin Ω k .
3. P roof of the main theorem
Recall that all the seven statements in Theorem 2.15 are equivalent for a given ω ∈ Ω , and thefirst statement amounts to saying that the cover of a Rota-Baxter operator is again a Rota-Baxteroperator. So we just need to prove the following theorem and then Theorem 2.16 follows. Theorem 3.1.
Let ω = xy − ( φ ( x ) + y ψ ( x )) ∈ Ω be given. (i) The following statements are equivalent. (a)
For every Rota-Baxter algebra ( R , P ) of weight , the cover b P ω of P on the di ff er-ential algebra ( R N , ∂ R ) of weight given in Proposition 2.8 is again a Rota-Baxteroperator of weight ; (b) ω is in Ω . (ii) The following statements are equivalent. (a)
For every Rota-Baxter algebra ( R , P ) of arbitrary weight λ , the cover b P ω of P on thedi ff erential algebra ( R N , ∂ R ) of weight λ given in Proposition 2.8 is again a Rota-Baxter operator of weight λ ; (b) ω is in Ω k . Proof. (Summary) The proof is divided into four parts. Let ω = xy − ( φ ( x ) + y ψ ( x )) with φ, ψ ∈ k [ x ]. The first three parts cover the proof of Item (i), partitioned into the following three cases of ω . Case 1. ψ = . This is proved in Section 3.1;
Case 2. ψ , and φ = . This is proved in Section 3.2;
Case 3. ψ , and φ , . This is proved in Section 3.3.The fourth part, given in Section 3.4, proves Item (ii) of the theorem. (cid:3)
Based on this result and computations in some other cases, we propose the following
Conjecture 3.2.
Let ω ∈ k h x , y i of the form ω = xy − ∞ P i = y i φ i ( x ) be given. (i) The following statements are equivalent. (a)
For every Rota-Baxter algebra ( R , P ) of weight , there is a cover b P ω of P on thedi ff erential algebra ( R N , ∂ R ) of weight that satisfies ω ( ∂ R , b P ω ) = and is a Rota-Baxter operator of weight ; LASSIFICATION OF EXTENSIONS, LIFTINGS AND DISTRIBUTIVE LAWS 11 (b) ω is in Ω . (ii) The following statements are equivalent. (a)
For every Rota-Baxter algebra ( R , P ) of arbitrary weight λ , there is a cover b P ω of Pon the di ff erential algebra ( R N , ∂ R ) of weight λ that satisfies ω ( ∂ R , b P ω ) = and is aRota-Baxter operator of weight λ ; (b) ω is in Ω k . Theorem 3.1 provides two classes of type ω di ff erential Rota-Baxter algebras by the followingcorollary. Corollary 3.3.
Fix an ω ∈ Ω ( resp., ω ∈ Ω k ) and assume that ( R , P ) is a Rota-Baxter algebra ofweight (resp., weight λ ) and ( R , d ) is a di ff erential algebra of weight (resp., weight λ ), where λ ∈ k is arbitrary. Then we obtain two type ω di ff erential Rota-Baxter algebra ( R N , ∂ R , b P ω ) and ( X ( R ) , ˆ d ω , P R ) of weight (resp., weight λ ).Proof. Applying Theorem 3.1, b P ω is a Rota-Baxter operator of weight 0 (resp., weight λ ) on R N . Also by (15), ∂ R and b P ω satisfy the required relation. Thus ( R N , ∂ R , b P ω ) is a type ω dif-ferential Rota-Baxter algebra of weight 0 (resp., weight λ ). Similarly, by Proposition 2.13 andTheorem 2.15, ( X ( R ) , ˆ d ω , P R ) is a type ω di ff erential Rota-Baxter algebra. (cid:3) For a given ω = xy − ( φ ( x ) + y ψ ( x )) ∈ Ω with φ ( x ) : = r P i = a i x i , ψ ( x ) : = s P j = b j x j , and each f ∈ R N , n ∈ N + , we obtain b P ω n ( f ) = ( φ ( ∂ R ) n − + b P ω n − ψ ( ∂ R ))( f ) (by (17)) = (cid:16)(cid:16) r X i = a i ∂ iR (cid:17) n − + s X j = b j b P ω n − ∂ jR (cid:17) ( f ) = r X i = a i f n − + i + s X j = b j b P ω n − ( ∂ jR f ) . (25)Recall from Example 2.1 that, for m ∈ N + , P k is a Rota-Baxter operator on the quotient algebra X ( k ) / I m . These Rota-Baxter algebras ( X ( k ) / I m , P k ) will be used extensively with Corollary 2.10to give counterexamples in the later proofs.3.1. Proof of Theorem 3.1.(i): Case 1.
In this case ω : = xy − φ ( x ) ∈ Ω , where φ ∈ k [ x ]. Thus toprove Case 1 of Theorem 3.1.(i), we only need to prove the following proposition which providesan additional equivalent condition. Proposition 3.4.
Let ω : = xy − φ ( x ) with φ ( x ) : = r P i = a i x i . The following statements are equivalent. (i) For every Rota-Baxter algebra ( R , P ) of weight , the cover b P ω of P on the di ff erentialalgebra ( R N , ∂ R ) of weight is again a Rota-Baxter operator of weight ; (ii) φ = a , that is, ω = xy − a ; (iii) For every Rota-Baxter algebra ( R , P ) of weight , we have (26) b P ω ( f ) = ( P ( f ) , a f , a f , · · · ) for all f ∈ R N . Proof.
By (25), we have(27) b P ω n ( f ) = r X i = a i f n − + i for all f ∈ R N , n ∈ N + . In particular, when n = b P ω ( f ) = r X i = a i f i for all f ∈ R N . (ii) = ⇒ (iii). When ω = xy − a , by (27), we obtain b P ω n ( f ) = a f n − for all f ∈ R N , n ∈ N + .Together with b P ω ( f ) = P ( f ) from the definition of a cover, (26) follows.(iii) = ⇒ (ii). Suppose r : = deg φ ≥
1, so a r ,
0. Take the Rota-Baxter algebra ( R , P ) tobe ( X ( k ) / I , P k ) in Example 2.1, and let f : = ( f ℓ ) ∈ ( X ( k ) / I ) N with f ℓ : = δ ℓ, r z . Then (28)gives b P ω ( f ) = r P i = a i δ i , r z = a r z , b P ω ( f ) = a f = a δ , r z =
0. This is acontradiction. Therefore, φ = a .(i) = ⇒ (ii). We just need to show that if r : = deg φ ≥
1, then there is a Rota-Baxter algebra( R , P ) such that the cover b P ω of P is not a Rota-Baxter operator on R N . When r ≥
1, we see a r ,
0. Let M n denote the maximum of the subscripts m of the expressions f m appearing on theright hand side of (27). Then M n = n − + r . Take ( R , P ) : = ( X ( k ) / I , P k ) in Example 2.1, and f : = ( f ℓ ) ∈ ( X ( k ) / I ) N with f ℓ : = δ ℓ, M r z = δ ℓ, r − z . For each n ∈ N + with n ≤ r , (27) becomes b P ω n ( f ) = r X i = a i δ n − + i , r − z = a r δ n − + r , r − z = a r z , if n = r , , if 1 ≤ n < r . Also by b P ω ( f ) = P ( f ) =
0, we have(29) b P ω r ( f ) = a r z , b P ω n ( f ) = n ∈ N with n < r . Let g : = ( g k ) ∈ ( X ( k ) / I ) N with g k : = δ k , z , i.e., g is the identity element of ( X ( k ) / I ) N .Then (28) and (29) give(30) b P ω ( b P ω ( f ) g ) = b P ω ( b P ω ( f )) = r X i = a i b P ω i ( f ) = a r b P ω r ( f ) = a r z . Since r ≤ r −
1, we have f i = δ i , r − z = i < r . By (28), b P ω ( f ) = r P i = a i f i = a r f r . Thenwe obtain(31) b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) = a r f r b P ω ( g ) . By (28), b P ω ( f b P ω ( g )) = r P i = a i ( f b P ω ( g )) i . So applying (9), we have b P ω ( f b P ω ( g )) = r X i = a i i X j = ij ! f j b P ω i − j ( g ) . Applying f j = j < r again, we obtain(32) b P ω ( f b P ω ( g )) = a r f r b P ω ( g ) . LASSIFICATION OF EXTENSIONS, LIFTINGS AND DISTRIBUTIVE LAWS 13
Combining (30), (31) and (32), we obtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = ( a r z + a r f r b P ω ( g )) − a r f r b P ω ( g ) = a r z , . Thus by Corollary 2.10, b P ω is not a Rota-Baxter operator on R N .(ii) = ⇒ (i). By Proposition 2.9.(ii), we need to show that for any Rota-Baxter algebra ( R , P ), andall f , g ∈ R N , n ∈ N , n X k = nk !b P ω k ( f ) b P ω n − k ( g ) = b P ω n ( b P ω ( f ) g ) + b P ω n ( f b P ω ( g ))holds. Applying Proposition 2.9.(i), we have b P ω ( f ) b P ω ( g ) = b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) . Since ω = xy − a , (27) gives(33) b P ω n ( h ) = a h n − for all h ∈ R N , n ∈ N + . Then b P ω n ( b P ω ( f ) g ) = a ( b P ω ( f ) g ) n − (by (33)) = b P ω ( f )( a g n − ) + n − X k = n − k !b P ω k ( f )( a g n − k − ) (by (9)) = b P ω ( f ) b P ω n ( g ) + n − X k = n − k !b P ω k ( f ) b P ω n − k ( g ) . (by (33))(34)Exchanging f and g , and then applying the commutativity of the multiplication, we obtain b P ω n ( f b P ω ( g )) = b P ω n ( f ) b P ω ( g ) + n − X k = n − k !b P ω n − k ( f ) b P ω k ( g ) = b P ω n ( f ) b P ω ( g ) + n − X k = n − n − k !b P ω k ( f ) b P ω n − k ( g )(35)by exchanging k and n − k . Combining (34) and (35), and (cid:16) nk (cid:17) = (cid:16) n − k (cid:17) + (cid:16) n − n − k (cid:17) , we obtain n X k = nk !b P ω k ( f ) b P ω n − k ( g ) = b P ω ( f ) b P ω n ( g ) + b P ω n ( f ) b P ω ( g ) + n − X k = nk !b P ω k ( f ) b P ω n − k ( g ) = b P ω n ( b P ω ( f ) g ) + b P ω n ( f b P ω ( g )) , as required. (cid:3) Proof of Theorem 3.1.(i): Case 2.
In this case ω : = xy − y ψ ( x ) ∈ Ω with ψ ∈ k [ x ]. Thus toprove the Case 2 of Theorem 3.1.(i), we only need to prove the following strengthened form. Proposition 3.5.
Let ω : = xy − y ψ ( x ) with ψ ( x ) : = s P j = b j x j , . The following statements areequivalent. (i) For every Rota-Baxter algebra ( R , P ) of weight , the cover b P ω of P on the di ff erentialalgebra ( R N , ∂ R ) of weight is again a Rota-Baxter operator of weight ; (ii) deg ψ = and b = , that is, ω = xy − ( b y + yx ) ; (iii) For every Rota-Baxter algebra ( R , P ) of weight and each f ∈ R N , we have b P ω ( f ) = ( b P ω ( f ) , b P ω ( f ) , · · · , b P ω n ( f ) , · · · ) , where b P ω ( f ) = P ( f ) and for each n ∈ N + , b P ω n ( f ) is given recursively by (36) b P ω n ( f ) = b b P ω n − ( f ) + b P ω n − ( ∂ R f ) . In particular, if b = , then b P ω n ( f ) = b P ω n − ( ∂ R f ) = · · · = b P ω ( ∂ nR f ) = P ( f n ) . That is, b P ω ( f ) = ( P ( f ) , P ( f ) , P ( f ) , · · · ) . Proof.
Recall from (25) that the cover b P ω is given by(37) b P ω n ( f ) = s X j = b j b P ω n − ( ∂ jR f ) for all f ∈ R N , n ∈ N + . In particular,(38) b P ω ( f ) = s X j = b j b P ω ( ∂ jR f ) = s X j = b j P ( f j ) for all f ∈ R N . In general, by iterating (37), we obtain(39) b P ω n ( f ) = s X j = b j s X j = b j · · · s X j n = b j n b P ω ( ∂ R j + j + ··· + j n f ) = s X j , j , ··· , j n = b j b j · · · b j n P ( f j + j + ··· + j n ) . (ii) = ⇒ (iii). For any f ∈ R N , b P ω ( f ) = P ( f ) follows from the definition of a cover. By ω : = xy − ( b y + yx ) and (37), we obtain b P ω n ( f ) = b b P ω n − ( f ) + b P ω n − ( ∂ R f ) for all n ∈ N + . (iii) = ⇒ (ii). Assume that Item (iii) holds. Suppose s : = deg ψ ≥
2. Take ( R , P ) : = ( X ( k ) / I , P k )in Example 2.1. Let f : = ( f k ) ∈ ( X ( k ) / I ) N with f k : = δ k , s z . Then (36) gives b P ω ( f ) = b b P ω ( f ) + b P ω ( ∂ R f ) = b P ( f ) + P ( f ) = b P ( δ , s z ) + P ( δ , s z ) = b P ω ( f ) = s P j = b j P ( δ j , s z ) = b s z , s = deg ψ ≤ f : = ( f ℓ ) ∈ ( X ( k ) / I ) N with f ℓ : = δ ℓ, z . Then (36) gives b P ω ( f ) = b b P ω ( f ) + b P ω ( ∂ R f ) = b P ( f ) + P ( f ) = b P ( δ , z ) + P ( δ , z ) = z while (38) gives b P ω ( f ) = s X j = b j P ( δ j , z ) = b P ( z ) = b z , if s = , , if s = . Thus we obtain s = z = b z . Then b = z is one of the basis elements. Therefore, ω = xy − ( b y + yx ). LASSIFICATION OF EXTENSIONS, LIFTINGS AND DISTRIBUTIVE LAWS 15 (i) = ⇒ (ii). Let s : = deg ψ . Consider ( R , P ) : = ( X ( k ) / I , P k ) and take g : = ( g k ) ∈ ( X ( k ) / I ) N with g k : = δ k , z , i.e., g is the identity element of ( X ( k ) / I ) N . Then b P ω ( b P ω ( f ) g ) = b P ω ( b P ω ( f )).So applying (38), we obtain(40) b P ω ( b P ω ( f ) g ) = s X j = b j P ( b P ω j ( f )) . Suppose s =
0, i.e., ψ = b . Then (40) becomes b P ω ( b P ω ( f ) g ) = b P ( b P ω ( f )). Now let f : = ( f ℓ ) ∈ ( X ( k ) / I ) N with f ℓ : = δ ℓ, z . Applying f = g and the commutativity of the multiplication,we have(41) b P ω ( f b P ω ( g )) = b P ω ( b P ω ( f ) g ) = b P ( P ( f )) = b z . Applying (7) and (38), we obtain(42) b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) = P ( f ) b P ( g ) + b P ( f ) P ( g ) = b z = b z . Combining (41) and (42) gives (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) − (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) = b z − b z = b z , . So b P ω is not a Rota-Baxter operator on R N by Corollary 2.10. So we must have s ≥ s ≥ b s ,
0. Let M n denote the maximum of the subscripts m of theexpressions f m appearing on the right hand side of (39): M n : = max { j + j + · · · + j n | ≤ j , j , · · · , j n ≤ s } = ns . Take f : = ( f ℓ ) ∈ ( X ( k ) / I ) N with f ℓ : = δ ℓ, M s z = δ ℓ, s z . For each n ∈ N + with n ≤ s , (39)becomes b P ω n ( f ) = s X j , j , ··· , j n = b j b j · · · b j n P ( δ j + j + ··· + j n , s z ) = b ns P ( δ ns , s z ) = b ss P ( z ) = b ss z , if n = s , , if 1 ≤ n < s . Together with b P ω ( f ) = P ( f ) = P ( δ , s z ) = s >
0, we obtain(43) b P ω s ( f ) = b ss z , b P ω n ( f ) = n ∈ N with n < s . Then (40) gives(44) b P ω ( b P ω ( f ) g ) = b s P ( b P ω s ( f )) = b s P ( b ss z ) = b s + s z . Also by (38) and (9), we have(45) b P ω ( f b P ω ( g )) = s X j = b j P (( f b P ω ( g )) j ) = s X j = b j P j X i = ji ! f i b P ω j − i ( g ) = ( , if s ≥ , b z , if s = . Here the last equation follows from f i = δ i , s z since 0 ≤ i ≤ j ≤ s ≤ s with equality holding inthe last inequality if and only if s =
1. Further applying (43), we have(46) b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) = ( , if s ≥ , b z b P ω ( g ) = b z = b z , if s = . Combining (44), (45) and (46), we obtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = ( b s + s z , , if s ≥ , b ( b − z if s = . Then by Corollary 2.10, when s ≥ b P ω is not a Rota-Baxter operator. When s =
1, we obtain b − =
0, i.e., b = s = b = = ⇒ (i). Let ( R , P ) be an arbitrary Rota-Baxter algebra ( R , P ). We will prove that b P ω is aRota-Baxter operator on R N by verifying the componentwise formulation(47) ( b P ω ( f ) b P ω ( g )) n = b P ω n ( b P ω ( f ) g ) + b P ω n ( f b P ω ( g )) for all f , g ∈ R N , n ∈ N , of the Rota-Baxter relation in (2). We will carry out the verification by induction on n .First by Proposition 2.9.(i), we have( b P ω ( f ) b P ω ( g )) = b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) . Assume that for a given k ∈ N , (47) holds. Then we derive b P ω k + ( b P ω ( f ) g ) + b P ω k + ( f b P ω ( g )) = ( b b P ω k + b P ω k ∂ R )( b P ω ( f ) g + f b P ω ( g )) (by (37)) = b b P ω k (cid:16)b P ω ( f ) g + f b P ω ( g ) (cid:17) + b P ω k (cid:16) ( ∂ R b P ω )( f ) g + b P ω ( f ) ∂ R ( g ) (cid:17) + b P ω k (cid:16) ∂ R ( f ) b P ω ( g ) + f ( ∂ R b P ω )( g ) (cid:17) (by (1)) = b b P ω k (cid:16)b P ω ( f ) g + f b P ω ( g ) (cid:17) + b P ω k (cid:16) ( b b P ω + b P ω ∂ R )( f ) g + b P ω ( f ) ∂ R ( g ) (cid:17) + b P ω k (cid:16) ∂ R ( f ) b P ω ( g ) + f ( b b P ω + b P ω ∂ R )( g ) (cid:17) (by (15)) = b (cid:16)b P ω ( f ) b P ω ( g ) (cid:17) k + (cid:16)b P ω ( ∂ R f ) b P ω ( g ) (cid:17) k + (cid:16)b P ω ( f ) b P ω ( ∂ R g ) (cid:17) k (by the induction hypothesis) = (cid:16) ( b b P ω + b P ω ∂ R )( f ) b P ω ( g ) (cid:17) k + (cid:16)b P ω ( f )( b b P ω + b P ω ∂ R )( g ) (cid:17) k = (cid:16) ( ∂ R b P ω )( f ) b P ω ( g ) + b P ω ( f )( ∂ R b P ω )( g ) (cid:17) k (by (15)) = (cid:16)b P ω ( f ) b P ω ( g ) (cid:17) k + (by (10)) . This completes the induction. (cid:3)
Proof of Theorem 3.1.(i): Case 3.
In this case, ω : = xy − ( φ ( x ) + y ψ ( x )) ∈ Ω , where φ, ψ ∈ k [ x ] are nonzero with r : = deg φ, s : = deg ψ ∈ N . To prove Theorem 3.1.(i) in this case,we will apply the same idea as in the previous two cases, namely by taking the maximum of thesubscripts. But in order for the idea to work, we need to partition N into eight subsets beforecarrying out the proof in Proposition 3.7.Let ( R , P ) denote an arbitrary Rota-Baxter algebra, and φ ( x ) : = r P i = a i x i and ψ ( x ) : = s P j = b j x j . Asin (25), we have(48) b P ω n ( f ) = r X i = a i f n − + i + s X j = b j b P ω n − ( ∂ jR f ) for all f ∈ R N , n ∈ N + . In particular, if n =
1, then (48) becomes(49) b P ω ( f ) = r X i = a i f i + s X j = b j P ( f j ) . LASSIFICATION OF EXTENSIONS, LIFTINGS AND DISTRIBUTIVE LAWS 17
Expanding the recursion in (48), we obtain b P ω n ( f ) = r X i = a i f n − + i + s X j = b j b P ω n − ( ∂ j R f ) (by (48)) = r X i = a i f n − + i + s X j = b j r X i = a i f n − + i + j + s X j = b j b P ω n − ( ∂ j + j R f ) (by (48)) = r X i = a i f n − + i + r X i = s X j = a i b j f n − + i + j + s X j , j = b j b j b P ω n − ( ∂ j + j R f ) . Repeating this process leads to(50) b P ω n ( f ) = r X i = n − X k = s X j , ··· , j k = a i b j · · · b j k f n − − k + i + j + ··· + j k + s X j , ··· , j n = b j · · · b j n P ( f j + ··· + j n )for all f ∈ R N , n ∈ N + .Let M n denote the maximum of the subscripts of the expressions f m appearing on the right handside of (50): M n : = max { n − − k + i + j + · · · + j k , j + · · · + j n | ≤ k ≤ n − , ≤ i ≤ r , ≤ j , · · · , j n ≤ s } . By first partitioning s ∈ N into s > s = s < s =
0) and then partitioning eachof the three cases into the subcases of r > s , r = s and r < s (the latter subcase is valid only when s > s = r , s ) ∈ N into eight cases in the following lemma. Lemma 3.6.
Let n ∈ N + and f : = ( f ℓ ) ∈ R N with f ℓ : = δ ℓ, M n u, where u is a given nonzero elementin R. The possibilities of M n and b P ωσ ( f ) for all σ ≤ n are as follows. (i) If s > and r > s, then M n = r + ( n − s, b P ω n ( f ) = a r b n − s u and b P ωσ ( f ) = for σ < n; (ii) If s > and r = s, then M n = ns, b P ω n ( f ) = a r b n − s u + b ns P ( u ) and b P ωσ ( f ) = for σ < n; (iii) If s > and r < s, then M n = ns, b P ω n ( f ) = b ns P ( u ) and b P ωσ ( f ) = for σ < n; (iv) If s = and r > s, then M n = n − + r, b P ω n ( f ) = n − P k = a r b k u and b P ωσ ( f ) = for σ < n; (v) If s = and r = s, then M n = n, b P ω n ( f ) = n − P k = a r b k u + b n P ( u ) and b P ωσ ( f ) = for σ < n; (vi) If s = and r < s, then M n = n, b P ω n ( f ) = b n P ( u ) and b P ωσ ( f ) = for σ < n; (vii) If s = and r > s, then M n = n − + r, b P ω n ( f ) = a r u and b P ωσ ( f ) = for σ < n; (viii) If s = and r = s, then M n = n − , b P ω n ( f ) = a r u + δ n , b P ( u ) , b P ωσ ( f ) = δ n , P ( u ) for σ < n.Proof. By the choice of f , (50) becomes b P ω n ( f ) = r X i = n − X k = s X j , ··· , j k = a i b j · · · b j k δ n − − k + i + j + ··· + j k , M n u + s X j , ··· , j n = b j · · · b j n P ( δ j + ··· + j n , M n u ) . Since the two indices of the Kronecker deltas are possibly equal only when i and j , · · · , j n aremaximized, we have b P ω n ( f ) = n − X k = a r b ks δ n − − k + r + ks , M n u + b ns P ( δ ns , M n u ) = n − X k = a r b ks δ n − + r + k ( s − , M n u + b ns δ n − + s + ( n − s − , M n P ( u ) . (51)We first prove the first and second equations in all the cases of the lemma.When s >
1, namely s − >
0, by maximizing k , (51) becomes b P ω n ( f ) = a r b n − s δ n − + r + ( n − s − , M n u + b ns δ n − + s + ( n − s − , M n P ( u ) = a r b n − s δ r + ( n − s , M n u + b ns δ s + ( n − s , M n P ( u ) . Thus when r > s (resp., r = s , resp., r < s ), we obtain M n = r + ( n − s (resp., M n = ns , resp., M n = ns ) and b P ω n ( f ) = a r b n − s u (resp., b P ω n ( f ) = a r b n − s u + b ns P ( u ), resp., b P ω n ( f ) = b ns P ( u )), provingthe first and second equations in cases (i) – (iii) of the lemma.When s =
1, namely s − =
0, (51) becomes b P ω n ( f ) = n − X k = a r b k δ n − + r , M n u + b n δ n − + s , M n P ( u ) . Thus when r > s = r = s =
1, resp., r < s = M n = n − + r (resp., M n = n ,resp., M n = n ) and b P ω n ( f ) = n − P k = a r b k u (resp., b P ω n ( f ) = n − P k = a r b k u + b n P ( u ), resp., b P ω n ( f ) = b n P ( u )),proving the first and second equations in Item (iv) – (vi) of the lemma.When s <
1, namely s = s − = −
1, by minimizing k , (51) becomes b P ω n ( f ) = a r δ n − + r , M n u + b n δ , M n P ( u ) . Thus when r > s = r = s = M n = n − + r (resp., M n = n −
1) and b P ω n ( f ) = a r u (resp., b P ω n ( f ) = a r u + δ n , b P ( u )), proving the first and second equations in Item (vii)– (viii) of the lemma.Now we prove the third equations in all the cases of the lemma. In each of the cases (i) – (vii),since M n >
0, we have b P ω ( f ) = P ( f ) = P ( δ , M n ) =
0. For case (viii), b P ω ( f ) = P ( f ) = P ( δ , M n u ) = P ( δ , n − u ) = δ n , P ( u ) . This proves the third equations when σ = σ with 1 ≤ σ < n . Then n > M σ < M n . Thus theexpressions f τ appearing in b P ωσ ( f ) all vanish since the subscripts of the expressions are strictlysmaller than M n . Therefore, b P ωσ ( f ) =
0. This completes the proof of Lemma 3.6. (cid:3)
We also need the following facts to proceed.For a Rota-Baxter algebra ( R , P ), take f : = ( f ℓ ) and g : = ( g k ) in R N . Then b P ω ( b P ω ( f ) g ) = r X i = a i ( b P ω ( f ) g ) i + s X j = b j P (( b P ω ( f ) g ) j ) (by (49)) = r X i = a i i X σ = i σ !b P ωσ ( f ) g i − σ + s X j = b j P j X τ = j τ !b P ωτ ( f ) g j − τ (by (9))(52)and b P ω ( f b P ω ( g )) = r X i = a i ( f b P ω ( g )) i + s X j = b j P (( f b P ω ( g )) j ) (by (49)) LASSIFICATION OF EXTENSIONS, LIFTINGS AND DISTRIBUTIVE LAWS 19 = r X i = a i i X σ = i σ ! f σ b P ω i − σ ( g ) + s X j = b j P j X τ = j τ ! f τ b P ω j − τ ( g ) . (by (9))(53)Let N f denote the maximal subscript of expressions f m appearing in the right hand side of (53):(54) N f : = max { σ, τ | σ ≤ i ≤ r , τ ≤ j ≤ s } = max { r , s } . Now take ( R , P ) : = ( X ( k ) / I m , P k ) from Example 2.1. Let g = ( g k ) with g k : = δ k , z , i.e., g isthe identity element of ( X ( k ) / I m ) N . Then by (49),(55) b P ω ( b P ω ( f ) g ) = b P ω ( b P ω ( f )) = r X i = a i b P ω i ( f ) + s X j = b j P ( b P ω j ( f )) . (49) also gives b P ω ( g ) = a g + b P ( g ) = a z + b z . Then(56) b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) = b P ω ( f )( a z + b z ) + b P ω ( f ) z . Now we are ready to prove Case 3 of Theorem 3.1.(i).
Proposition 3.7.
For each ω : = xy − ( φ ( x ) + y ψ ( x )) ∈ Ω with nonzero φ, ψ ∈ k [ x ] , there is a Rota-Baxter algebra ( R , P ) of weight such that the cover b P ω of P on ( R N , ∂ R ) is not a Rota-Baxteroperator of weight .Proof. By Corollary 2.10, we only need to prove that, for each given ω as in the proposition, thereis a Rota-Baxter algebra ( R , P ) and f , g ∈ R N such that(57) (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) , . We will divide the proof into the eight cases of r : = deg φ and s : = deg ψ as in Lemma 3.6.Denote φ ( x ) : = r P i = a i x i and ψ ( x ) : = s P j = b j x j . So a r , b s , Case (i). s > , r > s . In Lemma 3.6.(i), take ( R , P ) : = ( X ( k ) / I , P k ), n : = r and u : = z . Thenthe lemma gives M r = r + ( r − s , b P ω r ( f ) = a r b r − s z and b P ωσ ( f ) = σ < r . Let g ∈ ( X ( k ) / I ) N be the identity element. Since b P ωσ ( f ) = σ < r and r > b P ω ( b P ω ( f ) g ) = a r b P ω r ( f ) and b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) = , respectively. Also in this case, N f = max { r , s } = r in (54). So we have N f < M r . Then by f ℓ = δ ℓ, M r z , (53) gives b P ω ( f b P ω ( g )) =
0. Thus we obtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = a r b P ω r ( f ) = a r b r − s z , . This is what we need.We use the similar argument as in Case (i) to prove other cases as follows.
Case (ii). s > , r = s . In Lemma 3.6.(ii), taking ( R , P ) : = ( X ( k ) / I , P k ), n : = s and u : = z gives M s = s , b P ω s ( f ) = a r b s − s z + b ss P ( z ) = a r b s − s z and b P ωσ ( f ) = σ < s . Let g ∈ ( X ( k ) / I ) N bethe identity. Then by (55) and (56), we have b P ω ( b P ω ( f ) g ) = a r b P ω r ( f ) + b s P ( b P ω s ( f )) = a r b s − s z and b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) = , respectively. Further by N f = s < M s and f ℓ = δ ℓ, M s z , (53) becomes b P ω ( f b P ω ( g )) =
0. Thus weobtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = a r b s − s z , . Case (iii). s > , r < s . In Lemma 3.6.(iii), take ( R , P ) : = ( X ( k ) / I , P k ), n : = s and u : = z . Then M s = s , b P ω s ( f ) = b ss P ( z ) = b ss z and b P ωσ ( f ) = σ < s . Let g ∈ ( X ( k ) / I ) N be the identity.By (55) and (56), we have b P ω ( b P ω ( f ) g ) = b s P ( b P ω s ( f )) and b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) = , respectively. Since N f = s < M s and f ℓ = δ ℓ, M s z , (53) becomes b P ω ( f b P ω ( g )) =
0. Thus we obtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = b s P ( b P ω s ( f )) = b s + s z , . Case (iv). s = , r > s . We consider ( R , P ) : = ( X ( k ) / I , P k ) and divide the proof into twosubcases depending on whether or not r − P k = b k is zero.First assume r − P k = b k ,
0. In Lemma 3.6.(iv), take n : = r and u : = z . Then M r = r − b P ω r ( f ) = r − P k = a r b k z and b P ωσ ( f ) = σ < r . Let g : = ( g k ) ∈ ( X ( k ) / I ) N be the identity.Then (55) and (56) give b P ω ( b P ω ( f ) g ) = a r b P ω r ( f ) and b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) = , respectively. Further, by N f = r < M r and f ℓ = δ ℓ, M r z , (53) gives b P ω ( f b P ω ( g )) =
0. Thus weobtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = a r b P ω r ( f ) = a r (cid:16) r − X k = b k (cid:17) z , . Next assume r − P k = b k =
0. Then r − P k = b k = − b r − . In Lemma 3.6.(iv), we can take n : = r − u : = z . Then M r − = r − b P ω r − ( f ) = r − X k = a r b k z = a r ( r − X k = b k ) z = − a r b r − z and b P ωσ ( f ) = σ < r −
1. Let g : = ( g k ) ∈ ( X ( k ) / I ) N with g k : = δ k , z . Then by (52), we have b P ω ( b P ω ( f ) g ) = a r rr − !b P ω r − ( f ) g = − ra r b r − z . Further, by N f = r ≤ M r − and f ℓ = δ ℓ, M r − z , (53) gives b P ω ( f b P ω ( g )) = a r f r b P ω ( g ) = a r f r P ( g ) = . By b P ω ( f ) = b P ω ( g ) =
0, we have b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) =
0. Thus we obtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = − ra r b r − z , . LASSIFICATION OF EXTENSIONS, LIFTINGS AND DISTRIBUTIVE LAWS 21
Case (v). s = , r = s . In Lemma 3.6.(v), take ( R , P ) : = ( X ( k ) / I , P k ), n : = u : = z .Then M = b P ω ( f ) = a z + b P ( z ) = a z and b P ω ( f ) =
0. Let g : = ( g k ) ∈ ( X ( k ) / I ) N with g k : = δ k , z . So g is the identity. By (55) and b P ω ( f ) =
0, we have b P ω ( b P ω ( f ) g ) = a b P ω ( f ) + b P ( b P ω ( f )) . Since f ℓ = δ ℓ, z and b P ω ( g ) = P ( g ) =
0, (53) gives b P ω ( f b P ω ( g )) =
0. By (56) and b P ω ( f ) =
0, wehave b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) =
0. Thus we obtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = a b P ω ( f ) + b P ( b P ω ( f )) = a z , . Case (vi). s = , r < s . We consider ( R , P ) : = ( X ( k ) / I , P k ) and divide the proof into twosubcases depending on whether or not b = b ,
1. In Lemma 3.6.(vi), take n : = u : = z . Then M = b P ω ( f ) = b P ω ( f ) = b P ( z ) = b z . Let g : = ( g k ) ∈ ( X ( k ) / I ) N be the identity, so g k : = δ k , z . Then (53)and (55) give b P ω ( f b P ω ( g )) = b P ( f b P ω ( g )) and b P ω ( b P ω ( f ) g ) = b P ( b P ω ( f )) , respectively. By b P ω ( f ) =
0, (56) becomes b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) = b P ω ( f ) z . Thus we obtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = b z + b z − b z = b ( b − z , . Next assume b =
1. Let both f : = ( f ℓ ) and g : = ( g k ) be the identity element of ( X ( k ) / I ) N .Then b P ω ( f ) = b P ω ( g ) = z . By (49), we have b P ω ( f ) = b P ω ( g ) = a z + b z . Then applying thecommutativity of the multiplication and (55), we have b P ω ( f b P ω ( g )) = b P ω ( b P ω ( f ) g ) = a b P ω ( f ) + b P ( b P ω ( f )) + P ( b P ω ( f )) = a z + b z . Further, b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) = z ( a z + b z ) = a z + b z . Thus we obtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = a z , . Case (vii). s = , r > s . In Lemma 3.6.(vii), take ( R , P ) : = ( X ( k ) / I , P k ), n : = r and u : = z .Then M r = r − b P ω r ( f ) = a r z and b P ωσ ( f ) = σ < r . Let g : = ( g k ) ∈ ( X ( k ) / I ) N be theidentity with g k : = δ k , z . Since b P ωσ ( f ) = σ < r , (55) gives b P ω ( b P ω ( f ) g ) = a r b P ω r ( f ). By N f = r ≤ M r and f ℓ = δ ℓ, M r z , (53) gives b P ω ( f b P ω ( g )) = a r f r b P ω ( g ) =
0. By b P ω ( f ) =
0, (56)becomes b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) =
0. Thus we obtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = a r b P ω r ( f ) = a r z , . Case (viii). s = , r = s . In Lemma 3.6.(viii), take ( R , P ) : = ( X ( k ) / I , P k ), n : = u : = z .Then M = b P ω ( f ) = a z + b P ( z ) = a z + b z and b P ω ( f ) = P ( z ) = z . Let g = f , i.e., g : = ( g k ) ∈ ( X ( k ) / I ) N with g k : = δ k , z . Then applying the commutativity of the multiplicationand (55), we have b P ω ( f b P ω ( g )) = b P ω ( b P ω ( f ) g ) = a b P ω ( f ) + b P ( b P ω ( f )) . Thus we obtain (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) (cid:17) − (cid:16)b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) (cid:17) = a z + b z ) − z ( a z + b z ) = − b z , . To recapitulate, applying Corollary 2.10, we obtain that for each ( s , r ) ∈ N × N , the givencover b P ω of the chosen P on ( R N , ∂ R ) is not a Rota-Baxter operator, completing the proof ofProposition 3.7. (cid:3) Proof of Theorem 3.1.(ii).
Finally we prove Theorem 3.1.(ii).Let ( R , P ) be an arbitrary Rota-Baxter algebra of arbitrary weight λ . Recall from Proposi-tion 2.9.(ii) that the cover b P ω of P on ( R N , ∂ R ) is again a Rota-Baxter operator of weight λ if andonly if for all f , g ∈ R N , and n ∈ N ,(58) n X k = n − k X j = nk ! n − kj ! λ k b P ω n − j ( f ) b P ω k + j ( g ) − (cid:16)b P ω n ( b P ω ( f ) g ) + b P ω n ( f b P ω ( g )) + λ b P ω n ( f g ) (cid:17) = . (iia) = ⇒ (iib). If Item (iia) holds, then as a special case, for every Rota-Baxter algebra ( R , P ) ofweight 0, the cover b P ω of P is still a Rota-Baxter operator of weight 0. So by Theorem 3.1.(i), ω is in Ω , that is, ω = xy − a or ω = xy − ( b y + yx ).First consider ω = xy − a . Then (25) gives b P ω n ( f ) = a f n − for all f ∈ R N , n ∈ N + . Togetherwith b P ω ( f ) = P ( f ), we obtain(59) b P ω ( f ) = ( P ( f ) , a f , a f , · · · ) . We take ( R , P ) : = ( X ( k ) / I , P k ) of weight λ , and f = g ∈ ( X ( k ) / I ) N with f ℓ : = δ ℓ, z . Apply-ing (59), we have(60) b P ω ( f ) = b P ω ( g ) = a z , b P ω ( f ) = b P ω ( g ) = P ( z ) = . Then we obtain X k = − k X j = k ! − kj ! λ k b P ω − j ( f ) b P ω k + j ( g ) = b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) + λ b P ω ( f ) b P ω ( g ) = λ a z (by (60))and b P ω ( b P ω ( f ) g ) + b P ω ( f b P ( g )) + λ b P ω ( f g ) = a b P ω ( f ) g + a f b P ω ( g ) + λ a f g (by (59)) = λ a z . (by (60))Then X k = − k X j = k ! − kj ! λ k b P ω − j ( f ) b P ω k + j ( g ) − (cid:16)b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) + λ b P ω ( f g ) (cid:17) = λ a ( a − z . Thus for a given nonzero λ , (58) holds in the case of n = X ( k ) / I , P k ) and f , g ∈ X ( k ) / I if and only if a = a =
1, i.e., ω = xy or ω = xy −
1. Next consider ω = xy − ( b y + yx ). Then applying (25) gives(61) b P ω ( f ) = b P ( f ) + P ( f ) for all f ∈ R N . We take ( R , P ) : = ( X ( k ) / I , P k ), and f , g ∈ ( X ( k ) / I ) N with f ℓ : = δ ℓ, z , g k : = δ k , z . Then weobtain(62) b P ω ( f ) = , b P ω ( g ) = z , b P ω ( f ) = P ( f ) = z , b P ω ( g ) = b P ( g ) = b z . LASSIFICATION OF EXTENSIONS, LIFTINGS AND DISTRIBUTIVE LAWS 23
Thus X k = − k X j = k ! − kj ! λ k b P ω − j ( f ) b P ω k + j ( g ) = b P ω ( f ) b P ω ( g ) + b P ω ( f ) b P ω ( g ) + λ b P ω ( f ) b P ω ( g ) = z + λ b z = λ z + λ b z and b P ω ( b P ω ( f ) g ) + b P ω ( f b P ω ( g )) + λ b P ω ( f g ) = b P ω ( b P ω ( f )) + b P ω ( f b P ω ( g )) + λ b P ω ( f ) (since g is the identity element) = b P ( b P ω ( f )) + P ( b P ω ( f )) + b P (( f b P ω ( g )) ) + P (( f b P ω ( g )) ) + λ ( b P ( f ) + P ( f )) (by (61)) = b P ( b P ω ( f )) + P ( b P ω ( f )) + b P ( f b P ω ( g )) + P (cid:16) f b P ω ( g ) + f b P ω ( g ) + λ f b P ω ( g ) (cid:17) + λ ( b P ( f ) + P ( f )) (by (8)) = λ z (by (62)) . Thus for a nonzero λ ∈ k , (58) holds in the case of n = f , g if and only if λ z + λ b z − λ z = λ b z =
0, which holds if and only if b = ω = xy − yx .Therefore, ω must be in Ω k = { xy , xy − , xy − yx } . (iib) = ⇒ (iia). For ω = xy −
1, the cover b P ω of P on ( R N , ∂ R ) is again a Rota-Baxter operator ofweight λ by [26, Proposition 3.8]. When ω = xy or ω = xy − yx , we need to show that the cover b P ω of P on ( R N , ∂ R ) satisfies (58).For ω = xy , applying (25), we have(63) b P ω n ( f ) = n ∈ N + , f ∈ R N . By Proposition 2.9.(i), (58) holds for n =
0. If n ∈ N + , then the maximum of the subscripts m ofthe expressions b P ω m appearing in each term of the left side of (58) is strictly larger than 0 and thenby (63), each term in the left side of (58) is 0. Thus (58) holds for all n ∈ N + .For ω = xy − yx , by (25), the cover b P ω of P is given by(64) b P ω n ( f ) = b P ω n − ( ∂ R f ) = b P ω ( ∂ nR f ) = P ( f n ) for all n ∈ N , f ∈ R N . Furthermore, for all f , g ∈ R N , and n ∈ N , n X k = n − k X j = nk ! n − kj ! λ k b P ω n − j ( f ) b P ω k + j ( g ) = n X k = n − k X j = nk ! n − kj ! λ k P ( f n − j ) P ( g k + j ) (by (64)) = n X k = n − k X j = nk ! n − kj ! λ k (cid:16) P ( P ( f n − j ) g k + j ) + P ( f n − j P ( g k + j )) + λ P ( f n − j g k + j ) (cid:17) (by (2)) = n X k = n − k X j = nk ! n − kj ! λ k (cid:16) P ( b P ω n − j ( f ) g k + j ) + P ( f n − j b P ω k + j ( g )) + λ P ( f n − j g k + j ) (cid:17) (by (64)) = P (( b P ω ( f ) g ) n ) + P (( f b P ω ( g )) n ) + λ P (( f g ) n ) (by (8)) = b P ω n ( b P ω ( f ) g ) + b P ω n ( f b P ω ( g )) + λ b P ω n ( f g ) . (by (64))Then (58) holds.Now we have completed the proof of Theorem 3.1. Acknowledgements : This work is supported by the National Natural Science Foundation ofChina (Grant No. 11771190) and the China Scholarship Council (Grant No. 201606180084).Shilong Zhang thanks Rutgers University-Newark for its hospitality during his visit from August2016 to August 2017. R eferences [1] C. Bai, A unified algebraic approach to the classical Yang-Baxter equations,
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