Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gröbner-Shirshov bases
aa r X i v : . [ m a t h . R A ] F e b HOPF ALGEBRA OF MULTI-DECORATED ROOTED FORESTS, FREEMATCHING ROTA-BAXTER ALGEBRAS AND GR ¨OBNER-SHIRSHOV BASES
XING GAO, LI GUO, AND YI ZHANGA bstract . We first use rooted forests with multiple decoration sets to construct free Hopf algebraswith multiple Hochschild 1-cocycle conditions. Applying the universal property of the underlyingoperated algebras and the method of Gr¨obner-Shirshov bases, we then construct free objects inthe category of matching Rota-Baxter algebras which is a generalization of Rota-Baxter algebrasto allow multiple Rota-Baxter operators, inspired from the recent work [6, 13] on stochastic PDEsHopf algebras of typed trees and integral equations. Finally the free matching Rota-Baxter algebrasare equipped with a cocycle Hopf algebra structure. C ontents
1. Introduction 11.1. Rooted tree Hopf algebras and Rota-Baxter algebras 11.2. Matching Rota-Baxter algebras and outline of the paper 22. Ω -operated Hopf algebras of decorated rooted forests 32.1. Noncommutative Connes-Kreimer Hopf algebras 32.2. Multi-decorated planar rooted trees and forests 52.3. Free Ω -cocycle Hopf algebras of decorated planar rooted forests 72.4. Free Ω -operated monoids and algebras 103. Gr¨obner-Shirshov bases and free matching Rota-Baxter algebras 123.1. Matching Rota-Baxter algebras 123.2. Gr¨obner-Shirshov bases of free matching Rota-Baxter algebras 133.3. Construction of free matching Rota-Baxter algebras on a set 173.4. Free matching Rota-Baxter algebras on decorated rooted forests 204. Ω -cocycle Hopf algebras and free matching Rota-Baxter algebras 21References 251. I ntroduction This paper applies rooted forests to obtain free objects for generalized cocycle Hopf algebrasand then for matching Rota-Baxter algebras. To pass from the former to the latter, we interpretrooted forests as bracketed words and utilize the method of Gr¨obner-Shirshov bases.1.1.
Rooted tree Hopf algebras and Rota-Baxter algebras.
The Hopf algebra of rooted forestsarose from the study of Connes and Kreimer [9, 26] on renormalization of quantum field theorywhere the rooted forests serves as a baby model of Feynman diagrams. The importance of thisHopf algebra and its noncommutative analog, the Foissy-Holtkamp Hopf algebra [11, 25], hasbeen investigated from various points of view. From the algebraic viewpoint, this importance is
Date : February 10, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Rooted forest; Hopf algebra; Rota-Baxter algebra; 1-cocycle condition; Gr¨obner-Shirshov bases. revealed by its universal property in the category of cocycle Hopf algebras [11, 27, 30], in termsof a Hopf algebra with a grafting operator tied together by the Hochschild 1-cocycle condition.It is also characterized as the free operated algebra [19, 39], namely an algebra equipped with alinear operator, and thus are interpreted in terms of bracketed words and Motzkin paths. Recentlysuch a universal property was generalized to braided Hopf algebras of rooted forests `a la Connes-Kreimer [12, 22].Another algebraic structure playing a key role in the work of Connes and Kreimer, and in thecontext of operated algebras, is the Rota-Baxter algebra which has its origin in the work of G.Baxter [4] in fluctuation theory in probability. Connections of Rota-Baxter algebras have been es-tablished with broad areas in mathematics and mathematical physics, including quasi-symmetricfunctions, operads, integrable system and renormalization methods. See for example [1, 23, 37].Free Rota-Baxter algebras can also be realized on rooted forests [10, 39]. In fact, the realiza-tion can be obtained from certain generalization of the noncommutative Connes-Kreimer Hopfalgebra. This connection of free Rota-Baxter algebras with generalized Connes-Kreimer Hopfalgebras not only highlights the combinatorial nature of Rota-Baxter algebras which attractedthe attentions of outstanding combinatorists such as Cartier and Rota [7, 34], but also provides anatural Hopf algebra structure on free Rota-Baxter algebras from that on the rooted forests.Further various Hochschild 1-cocycle conditions have been applied to establish or characterizeother Hopf like algebraic structures, such as Hopf algebras on free commutative modified Rota-Baxter algebras [42], left counital Hopf algebras on free (commutative) Nijenhuis algebras [18,44] and free Rota-Baxter systems [32, 33], as well as the Loday-Ronco Hopf algebra of binaryrooted trees [41] and infinitesimal bialgebras of rooted forests [40].1.2.
Matching Rota-Baxter algebras and outline of the paper.
As a multi-operator general-ization of the Rota-Baxter algebra, the recent notion of a matching Rota-Baxter algebra [16] hasits motivation from the study of multiple pre-Lie algebras [13] originated in the important work ofBruned, Hairer and Zambotti [6] on algebraic renormalization of regularity structures and furthermotivated by the studies of associative Yang-Baxter equations, Volterra integral equations andlinear structure of Rota-Baxter operators [17, 21]. See Section 3.1 for a summary of the broadconnections of matching Rota-Baxter algebras.Our purpose of this paper is to construct free matching Rota-Baxter algebras from rootedforests by generalizing the construction of free Rota-Baxter algebras mentioned above from thecocycle Hopf algebra of rooted forests, from one operator to multiple operators. Thus we firstintroduce a class of decorated rooted forests which will serve as the carrier of the free Hopf alge-bra with multiple Hochschild 1-cocycle conditions. Free matching Rota-Baxter algebras will bea quotient of this free cocycle Hopf algebra modulo the operated ideal generated by the matchingRota-Baxter algebra relations. We next display a basis of this quotient for which we apply themethod of Gr¨obner-Shirshov bases. As this method works better with the algebraic notion ofbracketed words, we utilize the dictionary between rooted forests and bracketed words providedin [19] which allows us to take advantage of both the combinatorial structure of rooted forestsfor their ease description of the coproduct and the algebraic structure of bracketed words for theiramenability for detailed computations.In the process, we also take advantage of the two gradings and filtrations on rooted forests (andthe corresponding bracketed words), one by the number of vertices and one by the depths of theforests.To provide more details, our first step is to construct free multiple cocycle Hopf algebras fromrooted forests Section 2. Thus we will need to work with a set X of generators and a set Ω of OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 3 operators that satisfy the cocycle condition. For this purpose, we introduce rooted forests withtwo decoration sets X and Ω , with X only allowed to decorate the leaf vertices. We then show thatthe resulting space H RT ( X , Ω ) meets our needs for the free Ω -cocycle Hopf algebra on X (Theo-rem 2.10). Then the free matching Rota-Baxter algebra on X is the quotient of H RT ( X , Ω ) modulothe matching Rota-Baxter algebra relation. In order to apply the method of Gr¨obner-Shirshovbases to give an explicit construction of the free matching Rota-Baxter algebra in the next sec-tion, we utilize the one-one correspondence between rooted forests and bracketed words [19] torephrase Theorem 2.10 in terms of (multiple) bracketed words (Corollary 2.14).Our goal in Section 3 is to give an explicit construction of free matching Rota-Baxter algebrasby applying the method of Gr¨obner-Shirshov bases to obtain a canonical linear basis of the quo-tient of the free Ω -cocycle Hopf algebra H RT ( X , Ω ) (reinterpreted in terms of bracketed words)modulo the matching Rota-Baxter algebra relations. First the notion of matching Rota-Baxteralgebras is recalled together with a list of their properties. Then with a suitable monomial order,it is established in Theorem 3.5 that the matching Rota-Baxter algebra relations form a Gr¨obner-Shirshov basis and thus give rise to a linear basis of the free matching Rota-Baxter algebra as theaforementioned quotient. The operations of the matching Rota-Baxter algebra are given in termsof this linear basis. In view of establishing a cocycle Hopf algebra structure on the free matchingRota-Baxter algebra in the next section, we apply the one-one correspondence between rootedforests and bracketed words again and rephrase the free matching Rota-Baxter algebra in termsof decorated rooted forests.In Section 4, we equip the free matching Rota-Baxter algebra with an Ω -cocycle Hopf algebrastructure descending from the one on H RT ( X , Ω ), by first establishing an Ω -cocycle bialgebrastructure and then verifying the needed connected condition in order to obtain the Hopf algebrastructure in Theorem 4.7. Notations.
Throughout this paper, we fix a unitary commutative ring k which will be the basering of all modules, algebras, coalgebras, bialgebras, tensor products, as well as linear maps. Byan algebra, we mean a unitary associative algebra unless otherwise specified.2. Ω - operated H opf algebras of decorated rooted forests In this section, we construct free operated algebras with multiple operators by decorated rootedforests, as well as by bracketed words.The rooted forests we consider have di ff erent decorations on their leafs and (internal) vertices,but can still be obtained as a suitable subset of the classical noncommutative Connes-KreimerHopf algebra H RT ( Ω ), that is, Foissy-Holtkamp Hopf algebra [11, 25], of rooted forests with alltheir vertices (leafs and internal vertices) decorated by the same set. Thus we first recall thenotions for H RT ( Ω ) for later applications.2.1. Noncommutative Connes-Kreimer Hopf algebras.
Various Hopf algebras of decoratedplanar rooted trees and forests are commonly studied in combinatorics, algebra and other fields.We recall the needed notions and results to be applied in our constructions in this paper.A rooted tree is a connected and simply connected set of vertices and oriented edges such thatthere is precisely one distinguished vertex, called the root . A planar rooted tree is a rooted treewith a fixed embedding into the plane.Let T denote the set of planar rooted trees and F the set of planar rooted forests , expressedalgebraically as the free monoid F : = M ( T ) generated by T with the concatenation product m RT which is usually suppressed for brevity. The empty tree in F is denoted by 1, the unit of M ( T ). XING GAO, LI GUO, AND YI ZHANG
Thus a planar rooted forest is a noncommutative concatenation of planar rooted trees, denoted by F = T · · · T n with T , . . . , T n ∈ T , with the convention that F = n = Ω , let T ( Ω ) (resp. F ( Ω ) : = M ( T ( Ω ))) denote the set of planar rootedtrees (resp. forests) whose vertices, including the leaves and internal vertices, are decorated byelements of Ω . Define the free k -module spanned by the set F ( Ω ): H RT ( Ω ) : = k F ( Ω ) = k M ( T ( Ω )) = k h T ( Ω ) i , which is also the noncommutative polynomial algebra on the set T ( Ω ) with the concatenation.The noncommutative Connes-Kreimer Hopf algebra H RT ( Ω ) introduced by Foissy [11] andHoltkamp [25] is the above algebra equipped with a coproduct which can be defined in severalways, by subforests, by admissible cuts and by a 1-cocycle condition. A subforest of a planarrooted forest F ∈ F ( Ω ) is the forest consisting of a set of vertices of F together with their descen-dants and edges connecting all these vertices. Let F F be the set of subforests of F , including theempty tree 1 and the full subforest F . Define(1) ∆ RT ( F ) : = X G ∈ F F G ⊗ ( F / G ) , where F / G is obtained by removing the subforest G and edges connecting G to the rest of thetree [20]. Here we use the convention that F / G = F = G , and F / G = F when G = ∆ RT is given by the grafting opera-tors . For ω ∈ Ω , define B + ω : H RT ( Ω ) → H RT ( Ω )to be the linear grafting operation by sending a rooted forest in F ( Ω ) to its grafting with the newroot decorated by ω and sending 1 to • ω .Then a recursive description of ∆ RT is the for T ∈ T (2) ∆ RT B + ω ( T ) : = B + ω ( T ) ⊗ + (id ⊗ B + ω ) ∆ RT ( T )with the convention ∆ RT (1) = ⊗
1. In particular, we have ∆ RT ( • ω ) = • ω ⊗ + ⊗ • ω , ω ∈ Ω . (3)For a decorated rooted forest F = T · · · T m ∈ F ( Ω ) with m ≥
2, we have(4) ∆ RT ( F ) = ∆ RT ( T ) · · · ∆ RT ( T m ) . Also define ε RT : k F ( Ω ) → k by taking ε RT ( F ) = F ∈ T ( Ω ), and ε RT (1) =
1. Let u RT : k → H RT ( Ω ) be the linear map given by 1 k H , m , u , ∆ , ε ) is called graded if there are k -submodules H ( n ) , n ≥
0, of H such that H = ∞ M n ≥ H ( n ) , H ( p ) H ( q ) ⊆ H ( p + q ) , ∆ ( H ( n ) ) ⊆ M p + q = n H ( p ) ⊗ H ( q ) , n , p , q ≥ . The bialgebra H is called connected graded if in addition H (0) = im u ( = k ) and ker ε = L n ≥ H ( n ) . It is well known that a connected graded bialgebra is a Hopf algebra. Theorem 2.1. [11, 25]
With the degree of a rooted forest defined by its number of vertices, thequintuple ( H RT ( Ω ) , m RT , u RT , ∆ RT , ε RT ) is a connected graded bialgebra and hence a Hopf alge-bra. OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 5
Multi-decorated planar rooted trees and forests.
We now give a generalization of theHopf algebra H RT ( Ω ) of decorated rooted forests by allowing the leaf vertices and internal verticesdecorated by di ff erent sets. We then show that it gives a realization of the free object in thecategory of algebras with multiple operators, thus automatically equipping the free object with aHopf algebra structure given by a cocycle condition.Let X be a set and let Ω be a nonempty set disjoint from X . Replacing Ω by X ⊔ Ω in H RT ( Ω ),we obtain the Hopf algebra H RT ( X ⊔ Ω ) = k F ( X ⊔ Ω ) as in Theorem 2.1.Let T ( X , Ω ) (resp. F ( X , Ω )) denote the subset of T ( X ⊔ Ω ) (resp. F ( X ⊔ Ω )) consisting ofvertex decorated planar rooted trees (resp. forests) with the property that elements of only Ω candecorate the internal vertices, namely vertices which are not leafs. The unique vertex of the tree • is regarded as a leaf vertex. In other words, elements of X can only be used to decorate theleaf vertices. Of course, some of the leaf vertices can also be decorated by elements from Ω . Forexample,1 , q α , q x , qq αβ , qq α x , q ∨ qq αβγ , q ∨ qq α x γ , q ∨ qq α xy , q ∨ qq q αββ γ , q ∨ qq q αβ x γ , q ∨ qq q αβ y x , x , y , ∈ X , α, β, γ ∈ Ω , are in T ( X , Ω ) whereas, the following are not in T ( X , Ω ): qq x α , qq xy , q ∨ qq x βα , q ∨ qq q α x β γ , x , y ∈ X , α, β, γ ∈ Ω . Remark 2.2.
Now we give some special cases of our decorated planar rooted forests.(a) If X = ∅ , then F ( X , Ω ) = F ( Ω ) is the linear basis in the decorated Foissy-Holtkamp Hopfalgebra H RT ( Ω ) [11].(b) If Ω is a singleton, then F ( X , Ω ) was introduced and studied in [39] to construct a cocycleHopf algebra on decorated planar rooted forests.(c) The subset of F ( X , Ω ) consisting of rooted forests whose (all) leaf vertices are decorated byelements of X and whose internal vertices are decorated by elements of Ω , are introducedin [19] to construct free operated nonunitary semigroups and free operated nonunitaryalgebras.Define H RT ( X , Ω ) : = k F ( X , Ω ) = k M ( T ( X , Ω ))to be the free k -module spanned by F ( X , Ω ).We define the degree deg( F ) of F ∈ F ( X , Ω ) to be its number of vertices. For n ≥
0, let F ( n ) denote the set of F ∈ F ( X , Ω ) with degree n and let H RT ( X , Ω ) ( n ) : = H ( n ) RT : = k F ( n ) . For F = T · · · T k ∈ F ( X , Ω ) with k ≥ T , · · · , T k ∈ T ( X , Ω ), define bre( F ) : = k to be the breadth of F with the convention that bre(1) = k = Theorem 2.3.
The quintuple ( H RT ( X , Ω ) , m RT , , ∆ RT , ε RT ) is a connected graded subbialgebraof H RT ( X ⊔ Ω ) with the grading H RT ( X , Ω ) = ⊕ n ≥ H ( n ) RT and hence is a Hopf algebra.Proof. Note that H RT ( X , Ω ) is a subspace of H RT ( X ⊔ Ω ). It is su ffi cient to show that H RT ( X , Ω )is closed under the multiplication m RT and the coproduct ∆ RT . Recall that forests in H RT ( X , Ω )are characterized by the condition that their internal vertices are decorated by elements from Ω only. If two forests have this condition, then their concatenation also has this condition, since aninternal vertex of the concatenated forest is an internal vertex of one of the two forests. For thesame reason, for any forest F with this condition, any of its subforest G and the quotient forest XING GAO, LI GUO, AND YI ZHANG F / G still have this condition. Thus the subspace H RT ( X , Ω ) is closed under the concatenationproduct and the coproduct defined in Eq. (1). (cid:3) Combining the degree and the grafting operators, we can equip F ( X , Ω ) with other structures. Definition 2.4. An Ω -operated algebra ( R , P Ω ) with a grading R = ⊕ n ≥ R n (resp. an increasingfiltration { R n } n ≥ ) is called an Ω -operated graded algebra (resp. Ω -operated filtered algebra )if ( R , ⊕ n ≥ R n ) is a graded algebra (resp. ( R , { R n } n ≥ ) is a filtered algebra) and(5) P ω ( R n ) ⊆ R n + (resp. P ω ( R n ) ⊆ R n + ) . As is well-known, a graded algebra ( R , ⊕ n ≥ R n ) is a filtered algebra with the filtration R n : = ⊕ k ≤ n R k , n ≥
0. Further by definition, the grafting operator B + ω increases the degree of a rootedforest by one. Thus we have Lemma 2.5.
The Ω -operated algebra H RT ( X , Ω ) with its grading ⊕ n ≥ H nRT and the associated fil-tration H RT , ( n ) : = ⊕ k ≤ n H kRT , is an Ω -operated graded algebra and an Ω -operated filtered algebra. For our later applications to matching Rota-Baxter algebras, we introduce another increasingfiltration on F ( X , Ω ) which leads to the notion of the depth of a decorated rooted forest. Fordistinction, we use F ( n ) and H RT , ( n ) for the previous filtration defined by degree and F n and H RT , n for the new filtration defined by depth.Denote • X : = {• x | x ∈ X } and set F : = M ( • X ) = S ( • X ) ⊔ { } , where M ( • X ) (resp. S ( • X )) is the submonoid (resp. subsemigroup) of F ( X , Ω ) generated by • X .Here the using of the notations M and S are justified since M ( • X ) (resp. S ( • X )) is indeed isomor-phic to the free monoid (resp. semigroup) generated by • X . Suppose that F n has been defined foran n ≥
0. Then define F n + : = M ( • X ⊔ ( ⊔ ω ∈ Ω B + ω ( F n ))) . (6)Thus we obtain F n ⊆ F n + and F ( X , Ω ) = lim −→ F n = ∞ [ n = F n . (7)Elements F ∈ F n \ F n − are said to have depth n , denoted by dep( F ) = n . Here are some examples:dep(1) = dep( • x ) = , dep( • ω ) = dep( B + ω (1)) = , dep( qq ωα ) = dep( B + ω ( B + α (1))) = , dep( q x qq ω y q y ) = dep( qq ω y ) = dep( B + ω ( • y )) = , dep( q ∨ qq ω x α ) = dep( B + ω ( B + α (1) • x )) = , where α, ω ∈ Ω and x , y ∈ X .Note the subtle di ff erence of this depth from the usual notion of depth of a rooted tree, definedto be the length of the longest path from a root of F to its leafs. The advantage of this new depth isthat it is consistent with the natural depth of bracketed words to be introduced in Section 2.4. Thisdi ff erence is most easily seen in dep( • x ) = • ω ) =
1. In general our notion dep( F ) ofdepth for F ∈ F is the same as the usual depth when all the longest paths from a root of F to theleafs end in vertices with decorations from X ; while dep( F ) is the usual depth of F plus one ifone of these longest paths ends in a vertex with decoration from Ω . OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 7
Free Ω -cocycle Hopf algebras of decorated planar rooted forests. In this subsection, wewill combine the notions of Ω -operated algebras and Hopf algebras to define an Ω -(operated)Hopf algebra and Ω -(operated) cocycle Hopf algebra. We then show that H RT ( X , Ω ) is a free Ω -cocycle Hopf algebra on the set X . For this purpose, we recall the following concepts. Definition 2.6. [19] Let Ω be a nonempty set.(a) An Ω -operated algebra is an algebra A together with a family of linear operators P ω : A → A , ω ∈ Ω .(b) Let ( A , ( P ω ) ω ∈ Ω ) and ( A ′ , ( P ′ ω ) ω ∈ Ω ) be Ω -operated algebras. A linear map φ : A → A ′ iscalled an Ω -operated algebra homomorphism if φ is an algebra homomorphism suchthat φ P ω = P ′ ω φ for ω ∈ Ω .(c) A free Ω -operated algebra on a set X is an Ω -operated algebra ( A , ( P ω ) ω ∈ Ω ) together witha set map j X : X → A with the property that, for any Ω -operated algebra ( A ′ , ( P ′ ω ) ω ∈ Ω ) andany set map f : X → A ′ , there is a unique homomorphism ¯ f : A → A ′ of Ω -operatedalgebras such that ¯ f j X = f .Now we enrich these notions by adding the bialgebra structures. Definition 2.7. (a) An Ω -operated bialgebra is a bialgebra ( H , m , H , ∆ , ε ) which is also an Ω -operated algebra ( H , ( P ω ) ω ∈ Ω ).(b) Let ( H , ( P ω ) ω ∈ Ω ) and ( H ′ , ( P ′ ω ) ω ∈ Ω ) be Ω -operated bialgebras. A linear map φ : H → H ′ iscalled an Ω -operated bialgebra homomorphism if φ is a bialgebra homomorphism suchthat φ P ω = P ′ ω φ for ω ∈ Ω .(c) An Ω -cocycle bialgebra is an Ω -operated bialgebra ( H , m , H , ∆ , ε, ( P ω ) ω ∈ Ω ) which satis-fies the cocycle condition:(8) ∆ P ω = P ω ⊗ H + (id ⊗ P ω ) ∆ for ω ∈ Ω . If the bialgebra in an Ω -cocycle bialgebra is a Hopf algebra, then it is called an Ω -cocycleHopf algebra .(d) The free Ω -cocycle bialgebra on a set X is an Ω -cocycle bialgebra ( H X , m X , X , ∆ X , ε X , ( P ω ) ω ∈ Ω ) together with a set map j X : X → H X with the property that, for any cocyclebialgebra ( H , m , H , ∆ , ε, ( P ′ ω ) ω ∈ Ω ) and set map f : X → H whose images are primitive(that is, ∆ ( f ( x )) = f ( x ) ⊗ H + H ⊗ f ( x )), there is a unique homomorphism ¯ f : H X → H of Ω -operated bialgebras such that ¯ f j X = f . The concept of a free Ω -cocycle Hopf algebra is defined in the same way.We are indebt to Foissy for the following result. Lemma 2.8.
Let ( H , m , H , ∆ , ε, ( P ω ) ω ∈ Ω ) be an Ω -cocycle bialgebra. Then for each ω ∈ Ω ,P ω ( H ) : = { P ω ( h ) | h ∈ H } is a coideal of H.Proof. Let ω ∈ Ω . We first show P ω ( h ) ⊆ ker ε . Let h ′ : = P ω ( h ) ∈ H be arbitrary with h ∈ H .Using the Sweedler notation, we can write ε ( h ′ ) = ε (cid:18)X ( h ′ ) h ′ (1) ε ( h ′ (2) ) (cid:19) = X ( h ′ ) ε ( h ′ ) ε ( h ′ ) = ( ε ⊗ ε ) ∆ ( h ′ ) . Thus ε ( h ′ ) = ε P ω ( h ) = ( ε ⊗ ε ) ∆ ( h ′ ) = ( ε ⊗ ε ) ∆ ( P ω ( h )) XING GAO, LI GUO, AND YI ZHANG = ( ε ⊗ ε ) (cid:16) P ω ( h ) ⊗ H + (id ⊗ P ω ) ∆ ( h ) (cid:17) (by Eq. (8)) = ( ε ⊗ ε )( P ω ( h ) ⊗ H ) + ( ε ⊗ ε P ω ) ∆ ( h ) = ε P ω ( h ) + X ( h ) ε ( h (1) ) ε P ω ( h (2) ) = ε P ω ( h ) + ε P ω (cid:16) X ( h ) ε ( h (1) )( h (2) ) (cid:17) = ε P ω ( h ) + ε P ω ( h ) , which implies that ε P ω ( h ) = h ∈ H , ∆ ( P ω ( h )) = P ω ( h ) ⊗ H + (id ⊗ P ω ) ∆ ( h ) (by Eq. (8)) = P ω ( h ) ⊗ H + (id ⊗ P ω ) (cid:18) X ( h ) ( h (1) ⊗ h (2) ) (cid:19) = P ω ( h ) ⊗ H + X ( h ) h (1) ⊗ P ω ( h (2) ) ∈ P ω ( H ) ⊗ H + H ⊗ P ω ( H ) . Thus P ω ( H ) is a coideal. (cid:3) The following result generalizes the universal properties of several related structures studiedin [9, 19, 30, 39]. See [8, Theorem 2.3] for the commutative case.
Lemma 2.9. [40]
Let j X : X ֒ → H RT ( X , Ω ) , x
7→ • x be the nature embedding and m RT be theconcatenation product. The quadruple ( H RT ( X , Ω ) , m RT , , ( B + ω ) ω ∈ Ω ) together with j X is the free Ω -operated algebra on X. We next strengthen Lemma 2.9 to include the bialgebra structure.
Theorem 2.10.
Let j X : X ֒ → H RT ( X , Ω ) , x
7→ • x be the nature embedding and m RT be theconcatenation product. (a) The sextuple ( H RT ( X , Ω ) , m RT , , ∆ RT , ε RT , ( B + ω ) ω ∈ Ω ) together with j X is the free Ω -cocyclebialgebra on X. (b) The Hopf algebra given by the connected graded bialgebra ( H RT ( X , Ω ) , m RT , , ∆ RT , ε RT , ( B + ω ) ω ∈ Ω ) together with j X is the free Ω -cocycle Hopf algebra on X.Proof. (a) By Theorem 2.3, the quintuple ( H RT ( X , Ω ) , m RT , , ∆ RT , ε RT ) is a bialgebra. Further-more, by Eq. (2), the sextuple ( H RT ( X , Ω ) , m RT , , ∆ RT , ε RT , ( B + ω ) ω ∈ Ω ) is an Ω -cocycle bialgebra.To verify the freeness, let ( H , m , H , ∆ , ε, ( P ω ) ω ∈ Ω ) be an Ω -cocycle bialgebra and f : X → H a set map such that ∆ ( f ( x )) = f ( x ) ⊗ H + H ⊗ f ( x ) for all x ∈ X . (9)In particular, ( H , m , H , ( P ω ) ω ∈ Ω ) is an Ω -operated algebra. It follows from Lemma 2.9 that thereexists a unique Ω -operated algebra homomorphism ¯ f : H RT ( X , Ω ) → H such that ¯ f j X = f .It remains to check the following two compatibilities between the coproducts ∆ and ∆ RT , andbetween the counit ε and ε RT . ∆ ¯ f ( F ) = ( ¯ f ⊗ ¯ f ) ∆ RT ( F ) , (10) ε ¯ f ( F ) = ε RT ( F ) for all F ∈ F ( X , Ω ) . (11) OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 9
To verify Eq. (10), we consider the set A : = { F ∈ H RT ( X , Ω ) | ∆ ( ¯ f ( F )) = ( ¯ f ⊗ ¯ f ) ∆ RT ( F ) } . By Lemma 2.9, H RT ( X , Ω ) is generated by X as an Ω -operated algebra. Thus to verify Eq. (10)we just need to show that A is an Ω -operated subalgebra of H RT ( X , Ω ) that contains X .Since ¯ f is an Ω -operated algebra homomorphism, and ∆ RT and ∆ are algebra homomorphismsfrom H RT ( X , Ω ) and H , respectively, we get 1 ∈ A and A is a subalgebra of H RT ( X , Ω ). For any x ∈ X , we have ∆ ( ¯ f ( • x )) = ∆ ( f ( x )) = f ( x ) ⊗ H + H ⊗ f ( x ) (by Eq. (9)) = ¯ f ( • x ) ⊗ ¯ f (1) + ¯ f (1) ⊗ ¯ f ( • x ) = ( ¯ f ⊗ ¯ f )( • x ⊗ + ⊗ • x ) = ( ¯ f ⊗ ¯ f ) ∆ RT ( • x ) . Thus • x ∈ A . Further for any F ∈ A and ω ∈ Ω , we have ∆ ¯ f ( B + ω ( F )) = ∆ P ω ( ¯ f ( F )) (by ¯ f being an Ω -operated algebra homomorphism) = P ω ( ¯ f ( F )) ⊗ H + (id ⊗ P ω ) ∆ ( ¯ f ( F )) (by Eq. (8)) = P ω ( ¯ f ( F )) ⊗ H + (id ⊗ P ω )( ¯ f ⊗ ¯ f ) ∆ RT ( F ) (by F ∈ A ) = P ω ( ¯ f ( F )) ⊗ H + ( ¯ f ⊗ P ω ¯ f ) ∆ RT ( F ) = ¯ f ( B + ω ( F )) ⊗ H + ( ¯ f ⊗ ¯ f B + ω ) ∆ RT ( F ) (by ¯ f being an Ω -operated algebra homomorphism) = ( ¯ f ⊗ ¯ f ) (cid:16) B + ω ( F ) ⊗ + (id ⊗ B + ω ) ∆ RT ( F ) (cid:17) = ( ¯ f ⊗ ¯ f ) ∆ RT ( B + ω ( F )) = ( ¯ f ⊗ ¯ f ) ∆ RT ( F ) . Thus A is stable under B + ω for any ω ∈ Ω and so A = H RT ( X , Ω ).Similarly, to verify Eq. (11), we just need to show that the subset B : = { F ∈ H RT ( X , Ω ) | ε ( ¯ f ( F )) = ε RT ( F ) } ⊆ H RT ( X , Ω ) . is an Ω -operated subalgebra of H RT ( X , Ω ).Since ¯ f is an Ω -operated algebra homomorphism, ε RF and ε are algebra homomorphisms from H RT ( X , Ω ) and H , respectively. So we get 1 ∈ B and B is a subalgebra of H RT ( X , Ω ). For any x ∈ X , by Eq. (9) and the left counicity, we obtain( ε ⊗ id) ∆ ( f ( x )) = ε ( f ( x ))1 H + k ⊗ f ( x ) = β ℓ ( f ( x )) , which implies that ε ( f ( x )) =
0. Then ε ( ¯ f ( • x )) = ε ( f ( x )) = = ε RT ( • x ) , showing • x ∈ B . For F ∈ B and ω ∈ Ω , we have ε ( ¯ f ( B + ω ( F ))) = ε ( P ω ( ¯ f ( F ))) (by ¯ f being an Ω -operated algebra homomorphism) = ε P ω ( ¯ f ( F )) = = ε RT ( B + ω ( F )) . Hence B is stable under B + ω for any ω ∈ Ω and so B = H RT ( X , Ω ). This completes the proof.(b) The proof follows from Item (a) and the well-known fact that any bialgebra homomorphismbetween two Hopf algebras is compatible with the antipodes [36, Lemma 4.04]. (cid:3) If X = ∅ , we obtain the freeness of H RT ( ∅ , Ω ) = H RT ( Ω ), which is the decorated noncommuta-tive Connes-Kreimer Hopf algebra by Remark 2.2 (a). Corollary 2.11.
The sextuple ( H RT ( Ω ) , m RT , , ∆ RT , ε RT , ( B + ω ) ω ∈ Ω ) is the free Ω -cocycle Hopfalgebra on the empty set, that is, the initial object in the category of Ω -cocycle Hopf algebras. Further taking
Ω = { ω } to be a singleton in Corollary 2.11, all decorated planar rooted forestshave the same decoration and hence can be rendered undecorated as in the Foissy-Holtkamp Hopfalgebra [11, 25]. Thus we have, similar to [9, 30, 39], Corollary 2.12.
Let F be the set of planar rooted forests without decorations. Then sextuple ( k F , m RT , , ∆ RT , ε RT , B + ω ) is the free cocycle Hopf algebra on the empty set, that is, the initialobject in the category of cocycle Hopf algebras. Free Ω -operated monoids and algebras. Our next goal is to construct free matching Rota-Baxter algebras by applying the method of Gr¨obner-Shirshov bases which works better in thecontext of bracketed words. Thus in this subsection, we recall the construction of a free Ω -operated monoid and Ω -operated algebra in terms of bracketed words on a set X and identifythem with the free Ω -operated algebra H RT ( X , Ω ) = k F ( X , Ω ). See [19] for more details of thesebracketed words.Given an ω ∈ Ω and a set Y , let ⌊ Y ⌋ ω denote the set (cid:8) ⌊ y ⌋ ω | y ∈ Y (cid:9) , so it is indexed by Y butdisjoint with Y . We also assume that the sets ⌊ Y ⌋ ω to be disjoint with each other as ω varies in Ω .We now define the free Ω -operated monoid over the set X as the limit of a direct system { i n , n + : M n → M n + } ∞ n = of inductively defined free monoids M n , where the transition homomorphisms i n + , n are naturalembeddings. For the initial step of n =
0, we define M : = M ( X ) and then define M : = M (cid:0) X ⊔ ( ⊔ ω ∈ Ω ⌊ M ⌋ ω ) (cid:1) with the natural embedding i , : M = M ( X ) ֒ → M = M ( X ⊔ ( ⊔ ω ∈ Ω ⌊ M ⌋ ω )) . Note that ⌊ M ⌋ ω ⊆ M for each ω ∈ Ω . In particular, 1 ∈ M is sent to 1 ∈ M . Inductivelyassume that, for any given n ≥ M k , k ≥ n with the natural embedding(12) i n − , n : M n − ֒ → M n have been defined. We then define(13) M n + : = M ( X ⊔ ( ⊔ ω ∈ Ω ⌊ M n ⌋ ω )) . The natural embedding in Eq. (12) induces the natural embedding ⌊ M n − ⌋ ω ֒ → ⌊ M n ⌋ ω , yielding a monomorphism of free monoids i n , n + : M n = M ( X ⊔ ( ⊔ ω ∈ Ω ⌊ M n − ⌋ ω )) ֒ → M ( X ⊔ ( ⊔ ω ∈ Ω ⌊ M n ⌋ ω )) = M n + . OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 11
This completes the inductive construction of the direct system. Finally we define the direct limitof monoids M ( X , Ω ) : = lim −→ M n = [ n ≥ M n with identity 1. Elements in M ( X , Ω ) are called Ω -bracketed words in X and elements of M n \ M n − are said to have depth n , denoted by dep M ( w ) = n . Define P ω : M ( X , Ω ) → M ( X , Ω ) , u
7→ ⌊ u ⌋ ω , ω ∈ Ω , and extend it by linearity to a linear operator on k M ( X , Ω ), still denoted by P ω . Then the pair (cid:0) M ( X , Ω ) , ( P ω ) ω ∈ Ω (cid:1) is an Ω -operated monoid and its linear span (cid:0) k M ( X , Ω ) , ( P ω ) ω ∈ Ω (cid:1) is an Ω -operated algebra.Let X be a set and Ω a nonempty set disjoint with X . Taking direct limit in Eq. (13) we obtain(14) M ( X , Ω ) = M ( X ⊔ ( ⊔ ω ∈ Ω ⌊ M ( X , Ω ) ⌋ )) . Thus any 1 , u ∈ M ( Ω , X ) has a unique factorization(15) u = w · · · w k , w i ∈ X ∪ M ( Ω , X ) , ≤ i ≤ k , k ≥ . We call k the breadth of u and denote it by | u | . For u = ∈ M ( Ω , X ) , we define | u | : = . Proposition 2.13. [19, Corollary 3.6]
Let X be a set and Ω a nonempty set. Let j X : X → k M ( X , Ω ) be the natural embedding and let · be the concatenation product. Then the triple ( k M ( X , Ω ) , · , ( P ω ) ω ∈ Ω ) together with j X is the free Ω -operated algebra on X. Lemma 2.9 and the uniqueness of the free objects in the category of Ω -operated algebras thenyield the isomorphism of Ω -operated algebras θ : ( k M ( X , Ω ) , · , ( P ω ) ω ∈ Ω ) (cid:27) ( k F ( X , Ω ) , · , ( B + ω ) ω ∈ Ω ) , (16)sending x ∈ X to θ ( x ) : = • x . Comparing Eqs. (6) and (13), we see that θ preserves the filtrationsof bracketed words in M ( X , Ω ) and forests in F ( X , Ω ) given by depths:(17) θ ( M n ) = F n , n ≥ . Further, for w ∈ M ( X , Ω ), let deg td ( w ), called the total degree of w , denote the total number(counting multiplicities) of the appearances of elements of X and brackets ⌊·⌋ ω , ω ∈ Ω , in w . So ⌊ xy ⌊ x ⌋ α z ⌋ α has deg td ( w ) = n ≥
0, let M ( n ) denote the subset of M ( X , Ω ) with total degree n and let M ( n ) denote theunion ∪ k ≤ n M ( k ) . Then we have a grading and a filtration(18) k M ( X , Ω ) = ⊕ n ≥ k M ( n ) , k M ( n ) ⊆ k M ( n + , n ≥ . Since the map θ sends x ∈ X to • x and ⌊ w ⌋ ω to B + ω ( θ ( w )), it preserves the degrees: deg tw ( w ) = deg( θ ( w )), and the resulting gradings and filtrations. Thus as a consequence of Lemma 2.9, wehave Corollary 2.14.
With the grading and its associated filtration on k M ( X , Ω ) defined by the totaldegree deg td in Eq. (18) , the free Ω -operated algebra k M ( X , Ω ) is an Ω -operated graded algebraand an Ω -operated filtered algebra, isomorphic to the ones for k F ( X , Ω ) in Lemma 2.9.
3. G r ¨ obner -S hirshov bases and free matching R ota -B axter algebras In this section we construct free matching Rota-Baxter algebras from bracketed words anddecorated rooted forests by the method of Gr¨obner-Shirshov bases. We begin with a brief re-view of matching Rota-Baxter algebras emphasizing their many connections. We then recall theComposition-Diamond (CD) Lemma for the Gr¨obner-Shirshov bases of operated algebras. Withthese preparations, the Gr¨obner-Shirshov bases for matching Rota-Baxter algebras is then ob-tained. This gives the desired construction of free matching Rota-Baxter algebras in terms ofbracketed words. We finally apply the isomorphism in Eq. (16) to give a construction of freematching Rota-Baxter algebras in terms of decorated rooted forests.3.1.
Matching Rota-Baxter algebras.
In this subsection, we recall the concept of matchingRota-Baxter algebras, which generalizes that of Rota-Baxter algebras.
Definition 3.1. [16] Let Ω be a nonempty set and let λ Ω : = ( λ ω ) ω ∈ Ω ⊆ k be a parameterizedfamily of scalars with index set Ω . More precisely, λ Ω is a map Ω → k .(a) A matching (multiple) Rota-Baxter algebra of weight λ Ω is a pair ( R , P Ω ) consisting ofan algebra R and a family P Ω : = ( P ω ) ω ∈ Ω of linear operators P ω : R −→ R , ω ∈ Ω , thatsatisfy the matching Rota-Baxter equation P α ( x ) P β ( y ) = P α ( xP β ( y )) + P β ( P α ( x ) y ) + λ β P α ( xy ) for all x , y ∈ R , α, β ∈ Ω . (19) When λ Ω = { λ } in ( R , λ Ω ), that is, when λ Ω : Ω → k is constant, we also call the matchingRota-Baxter algebra to have weight λ .(b) Let ( R , P Ω ) and ( R ′ , P ′ Ω ) be matching Rota-Baxter algebras of the same weight λ Ω . Alinear map φ : R → R ′ is called a matching Rota-Baxter algebra homomorphism if φ isan algebra homomorphism such that φ P ω = P ′ ω ◦ φ for all ω ∈ Ω .To motivate of our study of free matching Rota-Baxter algebras, we list some properties ofmatching Rota-Baxter algebras and refer the reader to [16, 17, 21] for further details.(a) Any Rota-Baxter algebra of weight λ can be viewed as a matching Rota-Baxter algebra ofweight λ by taking Ω to be a singleton.(b) When λ =
0, the matching Rota-Baxter equation is Lie compatible in the sense that[ P α ( x ) , P β ( y )] = P α ([ x , P β ( y )]) + P β ([ P α ( x ) , y ]) . Here the Lie bracket is taking as the commutator. In the case when | Ω | =
2, this has beenstudied in [35].(c) [16, Proposition 2.5] Matching Rota-Baxter algebras provide a solution to the linearity ofthe set of Rota-Baxter operators on an algebra as follows. Let ( R , ( P ω ) ω ∈ Ω ) be a matchingRota-Baxter algebra of weight λ . Then any finite linear combination P : = X ω ∈ Ω k ω P ω , k ω ∈ k , with k ω ∈ k is a Rota-Baxter operator of weight λ P ω k ω . In particular, if P ω k ω =
1, then P is a Rota-Baxter algebra of weight λ . Thus any element in the linear span P ω ∈ Ω k P ω of P Ω is a Rota-Baxter operator of certain weight.(d) [16, Corollary 4.5] Matching Rota-Baxter algebras have a close connection with matching(multiple) pre-Lie algebras introduced by Foissy [13]. Let ( R , ( P ω ) ω ∈ Ω ) be a matchingRota-Baxter algebra of weight λ Ω . Define x ∗ ω y : = P ω ( x ) y − yP ω ( x ) − λ ω yx for x , y , z ∈ R , ω ∈ Ω . OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 13
Then the pair ( R , ( ∗ ω ) ω ∈ Ω ) is a matching (multiple) pre-Lie algebra.(e) [16, Theorem 3.4] A matching Rota-Baxter algebra ( R , ( P ω ) ω ∈ Ω ) of weight λ Ω induces amatching dendriform algebra ( R , ( ≺ ω ) ω ∈ Ω , ( ≻ ω ) ω ∈ Ω ), where x ≺ ω y : = xP ω ( y ) + λ ω xy , x ≻ ω y : = P ω ( x ) y for x , y ∈ R , ω ∈ Ω . (f) [16, Example 2.3], [21] Consider the R -algebra R : = Cont( R ) of continuous functions on R . Let K ω ( x , t ) be a parameterized family of kernels of continuous functions on R and let(20) I ω : R −→ R , f ( x ) Z x K ω ( x , t ) f ( t ) dt , ω ∈ Ω , be the corresponding family of Volterra integral operators [38]. Then when K ω ( x , t ) isindependent of x , the pair ( R , ( I ω ) ω ∈ Ω ) is a matching Rota-Baxter algebra of weight zero.(g) [16, § r , s ∈ R ⊗ R , let r s − r s + r s = − λ s be the polarized associative Yang-Baxter equation of weight λ . Then a solution of thisequation gives a matching Rota-Baxter operator of weight λ .The purpose of this section is to construct free matching Rota-Baxter algebras. Since by Propo-sition 2.13, k M ( X , Ω ) is the free Ω -operated algebra on a set X , the free matching Rota-Baxteralgebra on X is obtained by taking the quotient of k M ( X , Ω ) modulo the operated ideal generatedby the matching Rota-Baxter algebra relations. More precisely, let Id( S ) be the operated ideal of k M ( X , Ω ) generated by the set(21) S : = n ⌊ x ⌋ α ⌊ y ⌋ β − ⌊ x ⌊ y ⌋ β ⌋ α − ⌊⌊ x ⌋ α y ⌋ β − λ β ⌊ xy ⌋ α (cid:12)(cid:12)(cid:12) x , y ∈ M ( Ω , X ) , α, β ∈ Ω o . Then the free matching Rota-Baxter algebra X NC Ω ( X , Ω ) is given by the quotient k M ( X , Ω ) / Id( S ).We will identify a canonical subset of M ( X , Ω ) which gives a linear basis of this quotient andexpress the operation of the matching Rota-Baxter algebra in terms of this basis.Of course the free matching Rota-Baxter algebra is also given by taking the quotient of the otherrealization k F ( X , Ω ) of the free Ω -operated algebra on the set X modulo the matching Rota-Baxteralgebra relations: k F ( X , Ω ) / Id( S ), where Id( S ) is the Ω -operated ideal of k F ( X , Ω ) generated by (22) S : = n B + α ( F ) ⋄ ℓ B + β ( F ) − B + α (cid:16) F ⋄ ℓ B + β ( F ) (cid:17) − B + β (cid:16) B + α ( F ) ⋄ ℓ F (cid:17) − λ β B + α ( F ⋄ ℓ F ) | α, β ∈ Ω o . We choose to work with k M ( X , Ω ) since it provides a simpler context to apply the method of Gr¨obner-Shirshov bases, as we will carry out next.3.2. Gr¨obner-Shirshov bases of free matching Rota-Baxter algebras.
In this subsection, we recall theComposition-Diamond Lemma for the Ω -operated (unitary) algebra k M ( X , Ω ) and apply it to construct alinear basis of the free matching Rota-Baxter algebra on a set.3.2.1. Composition-Diamond Lemma for free Ω -operated algebras. For further details on the notationsand background, we refer the reader to [5, 14, 24].Let X be a set and Ω a nonempty set, ⋆ < X , and X ⋆ : = X ⊔ { ⋆ } . By a ⋆ -bracketed word on X , we meanany bracketed word in M ( Ω , X ) ⋆ : = M ( Ω , X ⋆ ) with exactly one occurrence of ⋆ , counting multiplicities.For q ∈ M ( Ω , X ) ⋆ and u ∈ M ( Ω , X ), we define q | u : = q | ⋆ u to be the bracketed word on X obtained by replacing the unique occurrence of ⋆ in q by u . For q ∈ M ( Ω , X ) ⋆ and s = Σ i c i q | u i ∈ k M ( Ω , X ) , where c i ∈ k and u i ∈ M ( Ω , X ), we define q | s : = Σ i c i q | u i , and extend this notation to any q ∈ k M ⋆ ( Ω , X ) by linearity. Note that the element q | s is usually not abracketed word but a bracketed polynomial.A monomial order on M ( Ω , X ) is a well order ≤ on M ( Ω , X ) such that(23) u < v ⇒ q | u < q | v , for all u , v ∈ M ( Ω , X ) and all q ∈ M ⋆ ( Ω , X ) . Here, as usual, we denote u < v if u ≤ v but u , v . Since ≤ is a well order, it follows from Eq. (23) that1 ≤ u and u < ⌊ u ⌋ ω for all u ∈ M ( Ω , X ) and ω ∈ Ω .Let ≤ be a monomial order on M ( Ω , X ) and let f ∈ k M ( Ω , X ).(a) If f < k , the unique largest monomial ¯ f appearing in f is called the leading bracketed word(monomial) of f .(b) The coe ffi cient of ¯ f in f is called the leading coe ffi cient of f , which is denoted by c ( f ).(c) If f < k and c ( f ) =
1, then f is monic with respect to the monomial order ≤ and a subset S ⊂ k M ( Ω , X ) is monic with respect to ≤ if every s ∈ S is monic with respect to ≤ . The notion of a Gr¨obner-Shirshov basis is given in terms of compositions and triviality of compositions,encoding the notion of critical pairs in a rewriting system [3].
Definition 3.2.
Let f , g ∈ k M ( Ω , X ) be Ω -bracketed polynomials monic with respect to ≤ . Let ¯ f be theleading monomial of f , and let | f | denote its breadth.(a) If there exist u , v , w ∈ M ( Ω , X ) such that w = ¯ f u = v ¯ g with max (cid:8) | ¯ f | , | ¯ g | (cid:9) < w < | ¯ f | + | ¯ g | , then the Ω -bracketed polynomial ( f , g ) w : = ( f , g ) u , v , w : = f u − vg is called the intersection composition of f and g with respect to ( u , v ).(b) If there exist q ∈ M ⋆ ( Ω , X ) and w ∈ M ( Ω , X ) such that w = ¯ f : = ( f , g ) q , w : = q | ¯ g , then the Ω -bracketed polynomial ( f , g ) w : = f − q | g is called the including composition of f and g with respect to q .In both cases, the bracketed word w is called the ambiguity for the compositions.Now we arrive at the key notion of a Gr¨obner-Shirshov basis in which the confluncy of critical pairs iscaptured by a triviality condition. Definition 3.3.
Let S ⊆ k M ( Ω , X ) be a set of Ω -bracketed polynomials that is monic with respect to amonomial order ≤ , and let w ∈ M ( Ω , X ) . (a) An element u in k M ( X , Ω ) is called trivial modulo ( S , w ) if u can be written as a linear combination P i c i q i | s i with 0 , c i ∈ k , q i ∈ M ( Ω , X ) ⋆ , s i ∈ S and q i | ¯ s i < w . Then we denote u ≡ S , W ) . (b) The set S is called a Gr¨obner-Shirshov bases (with respect to ≤ ), if for each pair f , g ∈ S with f , g , every intersection composition and including composition ( f , g ) w of f and g is trivialmodulo ( S , w ) . (c) For u , v ∈ k M ( Ω , X ) , we say u and v are congruent modulo ( S , w ) and denote by u ≡ v mod ( S , w )if u − v is trivial modulo ( S , w ).The following theorem is the Composition-Diamond Lemma for Ω -(unitary) algebras , adapting fromthe case for Ω -nonunitary algebras in [5]. See also [33]. Theorem 3.4. [24, Theorem 3.13]
Let X be a set and Ω a nonempty set, and let ≤ be a monomial orderon M ( Ω , X ) . Let S be a set of Ω -bracketed polynomials in k M ( Ω , X ) which are monic with respect to ≤ and let Id( S ) be the Ω -operated ideal of k M ( Ω , X ) generated by S . Then the following statements areequivalent: (a) S is a Gr¨obner-Shirshov basis in k M ( Ω , X ) . OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 15 (b)
For every non-zero f ∈ Id( S ) , then ¯ f = q | ¯ s for some q ∈ M ( Ω , X ) ⋆ and s ∈ S . (c) Let
Irr( S ) : = n w ∈ M ( Ω , X ) | w , q | ¯ s , q ∈ M ( Ω , X ) ⋆ , s ∈ S o = M ( X , Ω ) \{ q | ¯ s | q ∈ M ( Ω , X ) ⋆ , s ∈ S } . Then there is a linear decomposition k M ( X , Ω ) = Id( S ) ⊕ Irr( S ) . Thus Irr( S ) modulo Id( S ) is a k -linear basis of k M ( Ω , X ) / Id( S ) . Gr¨obner-Shirshov bases for free matching Rota-Baxter algebras.
We now show that the matchingRota-Baxter relations form a Gr¨obner-Shirshov basis of the free Ω -operated algebras k M ( Ω , X ), and hencegives rise to a linear basis of the free matching Rota-Baxter algebra thanks to the Composition-DiamondLemma in Theorem 3.4.Let X and ∅ , Ω be well-ordered sets. For notational convenience, we also denote P ω ( u ) = ⌊ u ⌋ ω . Soone appearance of P ω in a bracketed work w ∈ M ( X , Ω ) means one appearance of a brackets ⌊⌋ ω in w .For u = u · · · u r ∈ M ( X ) with u , · · · , u r ∈ X , define deg X ( u ) = r if u , X (1) =
0. Extend thewell order ≤ on X to the degree lexicographical order ≤ on M ( X ) by taking, for any u = u · · · u r , v = v · · · v s ∈ M ( X ) \{ } , where u , · · · , u r , v , · · · , v s ∈ X ,(24) u ≤ v ⇔ deg X ( u ) < deg X ( v ) , or deg X ( u ) = deg X ( v )( = r ) and ( u , · · · , u r ) ≤ ( v , · · · , v r ) lexicographically , Here we use the convention that the empty word 1 ≤ u for all u ∈ M ( X ). Then ≤ is a well order on M ( X ) [3].Further we extend ≤ to M ( X , Ω ). Applying Eq. (15) and grouping adjacent letters in X together, we findthat every u ∈ M ( X ) may be uniquely written as a product in the form(25) u = u P α ( u ∗ ) u P α ( u ∗ ) u · · · P α r ( u ∗ r ) u r , where u , · · · , u r ∈ M ( X ) , u ∗ , · · · , u ∗ r ∈ M n − ( X ) and α , · · · , α r ∈ Ω . Denote by deg P ( u ) the number of occurrence of P ω = ⌊ ⌋ ω , ω ∈ Ω , and define the P -breadth bre P ( u ) of u tobe r . For example, we have u : = x P α ( x ) x P α ( x P α ( x )) x x = u P α ( u ∗ ) u P α ( u ∗ ) u , x , · · · , x ∈ X , α , α , α ∈ Ω , where u = x , u = x , u = x x , u ∗ = x , u ∗ = x P α ( x ) , deg P ( u ) = P ( u ) = u , v ∈ M ( X ) and write them uniquely in the form of Eq. (25): u = u P α ( u ∗ ) u P α ( u ∗ ) u · · · P α r ( u ∗ r ) u r and v = v P β ( v ∗ ) v P β ( v ∗ ) v · · · P β s ( v ∗ s ) v s . We define u ≤ db v by induction on dep( u ) + dep( v ) ≥
0. For the initial step of dep( u ) + dep( v ) =
0, wehave u , v ∈ M ( X ) and use the degree lexicographical order given in Eq. (24). For the induction step ofdep( u ) + dep( v ) ≥
1, we define u ≤ db v ⇔ deg P ( u ) < deg P ( v ) , or deg P ( u ) = deg P ( v ) and bre P ( u ) < bre P ( v ) , or deg P ( u ) = deg P ( v ) , bre P ( u ) = bre P ( v )( = r ) and( P α , u ∗ , · · · , P α r , u ∗ r , u , · · · , u r ) ≤ ( P β , v ∗ , · · · , P β r , v ∗ r , v , · · · , v r ) lexicographically.Here P α i ≤ P β i is compared by the order on Ω and u ∗ i ≤ db v ∗ i and u i ≤ db v i are compared by the inductionhypothesis. With a similar argument to the case of ≤ db on M ( X ) [43], the above defined ≤ db is a monomialorder on M ( X , Ω ). In fact when Ω is a singleton, the above defined ≤ db is exactly the one given in [43] on M ( X ). See also [33]. Theorem 3.5.
With the order ≤ db on M ( Ω , X ) , the setS = n ⌊ x ⌋ α ⌊ y ⌋ β − ⌊ x ⌊ y ⌋ β ⌋ α − ⌊⌊ x ⌋ α y ⌋ β − λ β ⌊ xy ⌋ α (cid:12)(cid:12)(cid:12) x , y ∈ M ( Ω , X ) , α, β ∈ Ω o is a Gr¨obner-Shirshov basis in k M ( Ω , X ) . Proof.
With the leading terms from S in the form of ⌊ x ⌋ α ⌊ y ⌋ β , all the possible ambiguities for compositionsof Ω -bracketed polynomials in S are of the following three forms. w : = ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ , w : = ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ , w : = ⌊ z ⌋ δ ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α . where x , y , z ∈ M ( Ω , X ), α, β, γ, δ ∈ Ω , u ∈ M ( Ω , X ) ⋆ . We now check that all these compositions aretrivial. Case 1. w = ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ . In this case, we may write f : = f α, β ( x , y ) = ⌊ x ⌋ α ⌊ y ⌋ β − ⌊ x ⌊ y ⌋ β ⌋ α − ⌊⌊ x ⌋ α y ⌋ β − λ β ⌊ xy ⌋ α , g : = g β, γ ( y , z ) = ⌊ y ⌋ β ⌊ z ⌋ γ − ⌊ y ⌊ z ⌋ γ ⌋ β − ⌊⌊ y ⌋ β z ⌋ γ − λ γ ⌊ yz ⌋ β . Then we have ¯ f = ⌊ x ⌋ α ⌊ y ⌋ β and ¯ g = ⌊ y ⌋ β ⌊ z ⌋ γ . Thus ( f , g ) w = f ⌊ z ⌋ γ − ⌊ x ⌋ α g = ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ − ⌊ x ⌊ y ⌋ β ⌋ α ⌊ z ⌋ γ − ⌊⌊ x ⌋ α y ⌋ β ⌊ z ⌋ γ − λ β ⌊ xy ⌋ α ⌊ z ⌋ γ − ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ + ⌊ x ⌋ α ⌊ y ⌊ z ⌋ γ ⌋ β + ⌊ x ⌋ α ⌊⌊ y ⌋ β z ⌋ γ + λ γ ⌊ x ⌋ α ⌊ yz ⌋ β = − ⌊ x ⌊ y ⌋ β ⌋ α ⌊ z ⌋ γ − ⌊⌊ x ⌋ α y ⌋ β ⌊ z ⌋ γ − λ β ⌊ xy ⌋ α ⌊ z ⌋ γ + ⌊ x ⌋ α ⌊ y ⌊ z ⌋ γ ⌋ β + ⌊ x ⌋ α ⌊⌊ y ⌋ β z ⌋ γ + λ γ ⌊ x ⌋ α ⌊ yz ⌋ β = − f α, γ ( x ⌊ y ⌋ β , z ) − ⌊ x ⌊ y ⌋ β ⌊ z ⌋ γ ⌋ α − ⌊⌊ x ⌊ y ⌋ β ⌋ α z ⌋ γ − λ γ ⌊ x ⌊ y ⌋ β z ⌋ α − g β, γ ( ⌊ x ⌋ α y , z ) − ⌊⌊ x ⌋ α y ⌊ z ⌋ γ ⌋ β − ⌊⌊⌊ x ⌋ α y ⌋ β z ⌋ γ − λ γ ⌊⌊ x ⌋ α yz ⌋ β − λ β f α,γ ( xy , z ) − λ β ⌊ xy ⌊ z ⌋ γ ⌋ α − λ β ⌊⌊ xy ⌋ α z ⌋ γ − λ β λ γ ⌊ xyz ⌋ α + f α, β ( x , y ⌊ z ⌋ γ ) + ⌊ x ⌊ y ⌊ z ⌋ γ ⌋ β ⌋ α + ⌊⌊ x ⌋ α y ⌊ z ⌋ γ ⌋ β + λ β ⌊ xy ⌊ z ⌋ γ ⌋ α + f α, γ ( x , ⌊ y ⌋ β z ) + ⌊ x ⌊⌊ y ⌋ β z ⌋ γ ⌋ α + ⌊⌊ x ⌋ α ⌊ y ⌋ β z ⌋ γ + λ β ⌊ x ⌊ y ⌋ β z ⌋ α + λ γ f α, β ( x , yz ) + λ γ ⌊ x ⌊ yz ⌋ β ⌋ α + λ γ ⌊⌊ x ⌋ α yz ⌋ β + λ γ λ β ⌊ xyz ⌋ α = − ⌊ x ⌊ y ⌋ β ⌊ z ⌋ γ ⌋ α + ⌊ x ⌊ y ⌊ z ⌋ γ ⌋ β ⌋ α + ⌊ x ⌊⌊ y ⌋ β z ⌋ γ ⌋ α + λ γ ⌊ x ⌊ yz ⌋ β ⌋ α + ⌊⌊ x ⌋ α ⌊ y ⌋ β z ⌋ γ − ⌊⌊ x ⌊ y ⌋ β ⌋ α z ⌋ γ − ⌊⌊⌊ x ⌋ α y ⌋ β z ⌋ γ − λ β ⌊⌊ xy ⌋ α z ⌋ γ − f α, γ ( x ⌊ y ⌋ β , z ) − g β, γ ( ⌊ x ⌋ α y , z ) − λ β f α,γ ( xy , z ) + f α, β ( x , y ⌊ z ⌋ γ ) + f α, γ ( x , ⌊ y ⌋ β z ) + λ γ f α, β ( x , yz ) = − ⌊ x g β, γ ( y , z ) ⌋ α + ⌊ f α, β ( x , y ) z ⌋ γ − f α, γ ( x ⌊ y ⌋ β , z ) − g β, γ ( ⌊ x ⌋ α y , z ) − λ β f α,γ ( xy , z ) + f α, β ( x , y ⌊ z ⌋ γ ) + f α, γ ( x , ⌊ y ⌋ β z ) + λ γ f α, β ( x , yz ) = − ⌊ x ⋆ | g β,γ ( y , z ) ⌋ α + ⌊ ⋆ | f α,β ( x , y ) z ⌋ γ − ⋆ | f α,γ ( x ⌊ y ⌋ β , z ) − ⋆ | g β,γ ( ⌊ x ⌋ α y , z ) − λ β ⋆ | f α,γ ( xy , z ) + ⋆ | f α,β ( x , y ⌊ z ⌋ γ ) + ⋆ | f α,γ ( x , ⌊ y ⌋ β z ) + λ γ ⋆ | f α,β ( x , yz ) , which is trivial modulo ( S , w ) since ⌊ x ⋆ | g β,γ ( y , z ) ⌋ α = ⌊ x ⋆ | ⌊ y ⌋ β ⌊ z ⌋ γ ⌋ α = ⌊ x ⌊ y ⌋ β ⌊ z ⌋ γ ⌋ α < bd ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ = w ⌊ ⋆ | f α,β ( x , y ) z ⌋ γ = ⌊ ⋆ | ⌊ x ⌋ α ⌊ y ⌋ β z ⌋ γ = ⌊⌊ x ⌋ α ⌊ y ⌋ β z ⌋ γ < bd ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ = w ,⋆ | f α,γ ( x ⌊ y ⌋ β , z ) = ⋆ | ⌊ x ⌊ y ⌋ β ⌋ α ⌊ z ⌋ γ = ⌊ x ⌊ y ⌋ β ⌋ α ⌊ z ⌋ γ < bd ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ = w ,⋆ | g β,γ ( ⌊ x ⌋ α y , z ) = ⋆ | ⌊⌊ x ⌋ α y ⌋ β ⌊ z ⌋ γ = ⌊⌊ x ⌋ α y ⌋ β ⌊ z ⌋ γ < bd ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ = w ,⋆ | f α,γ ( xy , z ) = ⋆ | ⌊ xy ⌋ α ⌊ z ⌋ γ = ⌊ xy ⌋ α ⌊ z ⌋ γ < bd ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ = w ,⋆ | f α,β ( x , y ⌊ z ⌋ γ ) = ⋆ | ⌊ x ⌋ α ⌊ y ⌊ z ⌋ γ ⌋ β = ⌊ x ⌋ α ⌊ y ⌊ z ⌋ γ ⌋ β < bd ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ = w ,⋆ | f α,γ ( x , ⌊ y ⌋ β z ) = ⋆ | ⌊ x ⌋ α ⌊⌊ y ⌋ β z ⌋ γ = ⌊ x ⌋ α ⌊⌊ y ⌋ β z ⌋ γ < bd ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ = w ,⋆ | f α,β ( x , yz ) = ⋆ | ⌊ x ⌋ α ⌊ yz ⌋ β = ⌊ x ⌋ α ⌊ yz ⌋ β < bd ⌊ x ⌋ α ⌊ y ⌋ β ⌊ z ⌋ γ = w . OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 17
Case 2. w = ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ . In this case, we may write q : = ⌊ u ⌋ α ⌊ z ⌋ δ , g : = g β, γ ( x , y ) = ⌊ x ⌋ β ⌊ y ⌋ γ − ⌊ x ⌊ y ⌋ γ ⌋ β − ⌊⌊ x ⌋ β y ⌋ γ − λ γ ⌊ xy ⌋ β , f : = f α, δ ( u | ⌊ x ⌋ β ⌊ y ⌋ γ , z ) = ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ − ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌊ z ⌋ δ ⌋ α − ⌊⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α z ⌋ δ − λ δ ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ z ⌋ α . Then we have ¯ f = ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ , ¯ g = ⌊ x ⌋ β ⌊ y ⌋ γ , and ¯ f = q | ¯ g . Thus ( f , g ) ω = f − q | g = ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ − ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌊ z ⌋ δ ⌋ α − ⌊⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α z ⌋ δ − λ δ ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ z ⌋ α − ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ + ⌊ u | ⌊ x ⌊ y ⌋ γ ⌋ β ⌋ α ⌊ z ⌋ δ + ⌊ u | ⌊⌊ x ⌋ β y ⌋ γ ⌋ α ⌊ z ⌋ δ + λ γ ⌊ u | ⌊ xy ⌋ β ⌋ α ⌊ z ⌋ δ = − ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌊ z ⌋ δ ⌋ α − ⌊⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α z ⌋ δ − λ δ ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ z ⌋ α + ⌊ u | ⌊ x ⌊ y ⌋ γ ⌋ β ⌋ α ⌊ z ⌋ δ + ⌊ u | ⌊⌊ x ⌋ β y ⌋ γ ⌋ α ⌊ z ⌋ δ + λ γ ⌊ u | ⌊ xy ⌋ β ⌋ α ⌊ z ⌋ δ = − ⌊ u | g β,γ ( x , y ) ⌊ z ⌋ δ ⌋ α − ⌊ u | ⌊ x ⌊ y ⌋ γ ⌋ β ⌊ z ⌋ δ ⌋ α − ⌊ u | ⌊⌊ x ⌋ β y ⌋ γ ⌊ z ⌋ δ ⌋ α − λ γ ⌊ u | ⌊ xy ⌋ β ⌊ z ⌋ δ ⌋ α − ⌊⌊ u | g β,γ ( x , y ) ⌋ α z ⌋ δ − ⌊⌊ u | ⌊ x ⌊ y ⌋ γ ⌋ β ⌋ α z ⌋ δ − ⌊⌊ u | ⌊⌊ x ⌋ β y ⌋ γ ⌋ α z ⌋ δ − λ γ ⌊⌊ u | ⌊ xy ⌋ β ⌋ α z ⌋ δ − λ δ ⌊ u | g β,γ ( x , y ) z ⌋ α − λ δ ⌊ u | ⌊ x ⌊ y ⌋ γ ⌋ β z ⌋ α − λ δ ⌊ u | ⌊⌊ x ⌋ β y ⌋ γ z ⌋ α − λ δ λ γ ⌊ u | ⌊ xy ⌋ β z ⌋ α + f α, δ ( u | ⌊ x ⌊ y ⌋ γ ⌋ β , z ) + ⌊ u | ⌊ x ⌊ y ⌋ γ ⌋ β ⌊ z ⌋ δ ⌋ α + ⌊⌊ u | ⌊ x ⌊ y ⌋ γ ⌋ β ⌋ α z ⌋ δ + λ δ ⌊ u | ⌊ x ⌊ y ⌋ γ ⌋ β z ⌋ α + f α, δ ( u | ⌊⌊ x ⌋ β y ⌋ γ , z ) + ⌊ u | ⌊⌊ x ⌋ β y ⌋ γ ⌊ z ⌋ δ ⌋ α + ⌊⌊ u | ⌊⌊ x ⌋ β y ⌋ γ ⌋ α z ⌋ δ + λ δ ⌊ u | ⌊⌊ x ⌋ β y ⌋ γ z ⌋ α + λ γ f α, δ ( u | ⌊ xy ⌋ β , z ) + λ γ ⌊ u | ⌊ xy ⌋ β ⌊ z ⌋ δ ⌋ α + λ γ ⌊⌊ u | ⌊ xy ⌋ β ⌋ α z ⌋ δ + λ γ λ δ ⌊ u | ⌊ xy ⌋ β z ⌋ α = − ⌊ u | g β,γ ( x , y ) ⌊ z ⌋ δ ⌋ α − ⌊⌊ u | g β,γ ( x , y ) ⌋ α z ⌋ δ − λ δ ⌊ u | g β,γ ( x , y ) z ⌋ α + f α, δ ( u | ⌊ x ⌊ y ⌋ γ ⌋ β , z ) + f α, δ ( u | ⌊⌊ x ⌋ β y ⌋ γ , z ) + λ γ f α, δ ( u | ⌊ xy ⌋ β , z ) = − ⋆ | ⌊ u | g β,γ ( x , y ) ⌊ z ⌋ δ ⌋ α − ⋆ | ⌊ u | g β,γ ( x , y ) ⌋ α z ⌋ δ − λ δ ⋆ | ⌊ u | g β,γ ( x , y ) z ⌋ α + ⋆ | f α,δ ( u | ⌊ x ⌊ y ⌋ γ ⌋ β , z ) + ⋆ | f α,δ ( u | ⌊⌊ x ⌋ β y ⌋ γ , z ) + λ γ ⋆ | f α,δ ( u | ⌊ xy ⌋ β , z ) , which is trivial modulo ( S , w ) since ⋆ | ⌊ u | f β,γ ( x , y ) ⌊ z ⌋ δ ⌋ α = ⋆ | ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌊ z ⌋ δ ⌋ α = ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌊ z ⌋ δ ⌋ α < bd ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ = w ,⋆ | ⌊ u | f β,γ ( x , y ) ⌋ α z ⌋ δ = ⋆ | ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ z ⌋ δ = ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ z ⌋ δ < bd ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ = w ,⋆ | ⌊ u | f β,γ ( x , y ) z ⌋ α = ⋆ | ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ z ⌋ α = ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ z ⌋ α < bd ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ = w ,⋆ | f α,δ ( u | ⌊ x ⌊ y ⌋ γ ⌋ β , z ) = ⋆ | ⌊ u | ⌊ x ⌊ y ⌋ γ ⌋ β ⌋ α ⌊ z ⌋ δ = ⌊ u | ⌊ x ⌊ y ⌋ γ ⌋ β ⌋ α ⌊ z ⌋ δ < bd ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ = w ,⋆ | f α,δ ( u | ⌊⌊ x ⌋ β y ⌋ γ , z ) = ⋆ | ⌊ u | ⌊⌊ x ⌋ β y ⌋ γ ⌋ α ⌊ z ⌋ δ = ⌊ u | ⌊⌊ x ⌋ β y ⌋ γ ⌋ α ⌊ z ⌋ δ < bd ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ = w ,⋆ | f α,δ ( u | ⌊ xy ⌋ β , z ) = ⋆ | ⌊ u | ⌊ xy ⌋ β ⌋ α ⌊ z ⌋ δ = ⌊ u | ⌊ xy ⌋ β ⌋ α ⌊ z ⌋ δ < bd ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α ⌊ z ⌋ δ = w . Case 3. w = ⌊ z ⌋ δ ⌊ u | ⌊ x ⌋ β ⌊ y ⌋ γ ⌋ α . The proof is similar to that of Case 2.This completes the proof. (cid:3) Construction of free matching Rota-Baxter algebras on a set.
Now we apply Theorems 3.4 and3.5 to construct free matching Rota-Baxter algebras on a set.Let X be a set. We first describe the set Irr( S ), where S = n ⌊ x ⌋ α ⌊ y ⌋ β − ⌊ x ⌊ y ⌋ β ⌋ α − ⌊⌊ x ⌋ α y ⌋ β − λ β ⌊ xy ⌋ α (cid:12)(cid:12)(cid:12) x , y ∈ M ( Ω , X ) , α, β ∈ Ω o . The set Irr( S ) will give a linear basis of the free matching Rota-Baxter algebras on the set X .By Theorems 3.4 and 3.5, the set consists of bracketed words in M ( X , Ω ) that do not contain subwordsof the form ⌊ u ⌋ α ⌊ v ⌋ β for any u , v ∈ M ( X , Ω ) and α, β ∈ Ω . As in the case of one operator [20], we give a description of Irr( S ) by inclusion conditions, rather than the above exclusive conditions. For subsets U , V ⊆ M ( X , Ω ) and r ≥
1, we use the abbreviations UV : = { uv | u ∈ U , v ∈ V } , U r : = { u · · · u r | u i ∈ U , ≤ i ≤ r } , ⌊ U ⌋ Ω : = {⌊ u ⌋ ω | u ∈ U , ω ∈ Ω } . Definition 3.6.
Let Y , Z be subsets of M ( Ω , X ). Define the alternating products of Y and Z by Λ ( Y , Z ) : = [ r ≥ ( Y ⌊ Z ⌋ Ω ) r [ [ r ≥ ( Y ⌊ Z ⌋ Ω ) r Y [ [ r ≥ ( ⌊ Z ⌋ Ω Y ) r [ [ r ≥ ( ⌊ Z ⌋ Ω Y ) r ⌊ Z ⌋ Ω [ { } , where 1 is the identity in M ( Ω , X ).We observe that Λ ( Y , Z ) ⊆ M ( Ω , X ). Then we recursively define X : = M ( X ) = S ( X ) ∪ { } and X n : = Λ ( S ( X ) , X n − ) , n ≥ . Thus X ⊆ · · · ⊆ X n ⊆ · · · . Finally we define X ∞ : = lim −→ X n = [ n ≥ X n . Elements in X ∞ are called matching Rota-Baxter words (MRBWs). For a MRBW w ∈ X ∞ , we calldep( w ) : = min { n | w ∈ X n } the depth of w , which agrees with the depth of w as an element of M ( X , Ω ) inSection 2.4.The following properties of MRBWs are easy to verify as in the case of one operator [20]. Lemma 3.7.
Every MRBW w , has a unique alternating decomposition: w = w · · · w m , where w i ∈ X ∪ ⌊ X ∞ ⌋ Ω , ≤ i ≤ m, m ≥ , and no consecutive elements in the sequence w , · · · , w m are in ⌊ X ∞ ⌋ Ω . Recall that k M ( Ω , X ) / Id( S ) is a free matching Rota-Baxter algebra on the set X , and has a linear ba-sis Irr( S ) by Theorems 3.4 and 3.5. Thus with the evident identification of Irr( S ) with X ∞ thanks toLemma 3.7, we obtain Theorem 3.8.
The set
Irr( S ) = X ∞ modulo Id( S ) is a linear basis of the free matching Rota-Baxter algebra k M ( Ω , X ) / Id( S ) of weight λ Ω on X. Thus the quotient map ϕ : k M ( Ω , X ) → k M ( Ω , X ) / Id( S ) of Ω -operated algebras becomes the projec-tion(26) ϕ : k M ( Ω , X ) = Id( S ) ⊕ k X ∞ −→ k X ∞ . Let X NC Ω ( X , Ω ) : = k X ∞ be the resulting free matching Rota-Baxter algebra on X with its matchingRota-Baxter operators P ω , ω ∈ Ω and multiplication ⋄ w . We first note the inclusions ⌊ X ∞ ⌋ α ⊆ X ∞ , α ∈ Ω .Thus the matching Rota-Baxter operator P ω on k X ∞ is simply(27) P ω ( w ) = ⌊ w ⌋ ω . For the product ⋄ w in X NC Ω ( X , Ω ), we have the following algorithm in analog to those for the product oftwo Rota-Baxter words (resp. Rota-Baxter system words) in the free Rota-Baxter algebra [20] (resp. freeRota-Baxter system [33]). Algorithm 3.9.
Let w , w ′ ∈ X ∞ . We define the product w ⋄ w w ′ inductively on the sum of depths n : = dep( w ) + dep( w ′ ) ≥ n =
0, then w , w ′ ∈ X = M ( X ) and define w ⋄ w w ′ : = ww ′ .(b) Let k ≥
0. Suppose that w ⋄ w w ′ have been defined for 0 ≤ n ≤ k and consider the case of n = k + Case 1. bre( w ) , bre( w ′ ) ≤
1. We define(28) w ⋄ w w ′ = ( ⌊ w ⋄ w w ′ ⌋ α + ⌊ w ⋄ w w ′ ⌋ β + λ β ⌊ w ⋄ w w ′ ⌋ α , if w = ⌊ w ⌋ α and w ′ = ⌊ w ′ ⌋ β , ww ′ , otherwise , OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 19 where α, β ∈ Ω . Case 2. bre( w ) ≥ w ′ ) ≥
2. Let w = w · · · w m and w ′ = w ′ · · · w ′ m ′ be the alternatingdecompositions of w and w ′ , respectively. Define(29) w ⋄ w w ′ : = w · · · w m − ( w m ⋄ w w ′ ) w ′ · · · w ′ m ′ , where w m ⋄ w w ′ is defined by Eq. (28) and the rest of the products are given by the concatenation.Di ff erent from the filtration defined by the depth, the grading M ( n ) and filtration M ( n ) on M ( X , Ω ) definedby the total degree deg td in Eq. (18) restrict to those on X ∞ :(30) X ( n ) : = M ( n ) ∩ X ∞ , X ( n ) : = M ( n ) ∩ X ∞ , n ≥ . The resulting grading X NC Ω ( X , Ω ) = ⊕ n ≥ k X ( n ) holds only linearly since the multiplication does not pre-serve the grading. However, we have the following compatibility for the filtered structures. Proposition 3.10.
The free matching Rota-Baxter algebra X NC Ω ( X , Ω ) , with the filtration X NC Ω ( X , Ω ) ( n ) : = k X ( n ) , n ≥ , is an Ω -operated filtered algebra as defined in Definition 2.4 :(31) P α ( k X ( p ) ) ⊆ k X ( p + , ( k X ( p ) ) ⋄ w ( k X ( q ) ) ⊆ k X ( p + q ) , p , q ≥ . Moreover, the homomorphism ϕ : k M ( X , Ω ) → X NC Ω ( X , Ω ) of Ω -operated algebras preserves the filtrations :(32) ϕ ( k M ( X , Ω ) ( n ) ) ⊆ k X ( n ) , n ≥ . Proof.
The first inclusion in Eq. (31) follows from the definition of the operators P α in Eq. (27).For the second inclusion in Eq. (31), by linearity we just need to prove X ( p ) ⋄ w X ( q ) ⊆ k X ( p + q ) , p , q ≥ , for which we apply induction on p + q ≥
0, with the general remark that deg td is additive with respect to theconcatenation product. Thus for w ∈ X ( p ) and w ′ ∈ X ( q ) , we have w ⋄ w w ′ ∈ X ( p + q ) as long as w ⋄ w w ′ = ww ′ is the concatenation.When p + q =
0, we have p = q =
0. Since X (0) = X (0) = M ( X ) = M on which the product ⋄ w is the concatenation, the inclusion holds by the above general remark. Let k ≥
0. Assume that theinclusion holds for p + q ≤ k and consider the case when p + q = k +
1. If either p or q is zero, then X p ⋄ w X q is the concatenation and the desired inclusion again follows. If none of p or q is zero and consider w = w · · · w m ∈ X p and w ′ = w ′ · · · w ′ m ′ ∈ X q with their alternating decompositions. Then w ⋄ w w ′ is againthe concatenation and w ⋄ w w ′ is in X ( p + q ) except when w m = ⌊ w m ⌋ α and w ′ = ⌊ w ′ ⌋ β in which case, Eq. (28)gives w ⋄ w w ′ = w · · · w m − ⌊ w ⋄ w w ′ m ′ ⌋ α w ′ · · · w ′ m ′ + w · · · w m − ⌊ w ⋄ w w ′ ⌋ β w ′ · · · w ′ m ′ + λ β w · · · w m − ⌊ w ⋄ w w ′ ⌋ α w ′ · · · w ′ m ′ . Since all the products are the concatenation except the ones in the brackets, by the general remark again,we just need to show that each of the brackets is in X (deg td ( w m ) + deg dt ( w ′ )) . But by the induction hypothesis,the ⋄ w -products inside the three brackets are in k X (deg td ( w m ) − + deg td ( w ′ )) . Hence the three brackets are in k X (deg td ( w m ) + deg td ( w ′ )) by the first inclusion in Eq. (31). This completes the induction.We finally prove Eq. (32) by induction on n ≥
0. The initial case of n = M = M ( X )equals X on which ϕ is the identity. For a given k ≥
0, assume that ϕ ( k M n ) ⊆ k X n for n ≥ k and consider1 , w ∈ M k + . If the width of w is one, then w is either in X or is of the form ⌊ w ⌋ ω , w ∈ M ∞ , ω ∈ Ω . The former case is already proved. For the latter case, we have w ∈ M ( k ) and so ϕ ( w ) is in X ( k ) by theinduction hypothesis and then ϕ ( w ) = P w ( ϕ ( w )) is in k X ( k + by the first inclusion in Eq. (31). If thewidth of w is greater than one, then w = w w with w , w ∈ M ( k ) . Thus by the induction hypothesis, ϕ ( w i ) ∈ k X (deg td ( w i )) , i = , . Since ϕ is an algebra homomorphism, by the second inclusion in Eq. (31), wehave ϕ ( w ) = ϕ ( w ) ⋄ w ϕ ( w ) ∈ k X (deg td ( w )) ⋄ w X (deg td ( w )) ⊆ k X (deg td ( w )) . This completes the induction. (cid:3)
Free matching Rota-Baxter algebras on decorated rooted forests.
In view of the convenience ofworking with rooted forests for Hopf algebra structures in the next section, we apply the isomorphism θ : k M ( X , Ω ) ∼ → k F ( X , Ω ) in Eq. (16) of Ω -operated algebras to reformulate the main results in thissection in terms of rooted forests. We first write L n : = θ ( X n ) , n ≥ L ∞ : = lim −→ L n = lim −→ θ ( X n ) = θ ( X ∞ ) , giving rise to linear isomorphism(33) X NC Ω ( X , Ω ) = k X ∞ θ −→ X NCRT ( X , Ω ) : = k L ∞ , Id( S ) θ −→ Id( S )for the operated ideals generated by S and S defined in Eqs. (21) and (22) respectively. Similar to matchingRota-Baxter bracketed words, elements in L ∞ are called matching Rota-Baxter forests (MRBFs).Further we obtain a homomorphism of Ω -operated algebras(34) ψ : = θϕθ − : ( k F ( X , Ω ) , · , ( B + ω ) ω ∈ Ω ) → ( X NCRT ( X , Ω ) , ⋄ ℓ , ( B + ω ) ω ∈ Ω ) , yielding the commutative diagram k M ( X , Ω ) = Id( S ) ⊕ X NC Ω ( X , Ω ) ϕ / / θ (cid:15) (cid:15) X NC Ω ( X , Ω ) θ (cid:15) (cid:15) k F ( X , Ω ) = Id( S ) ⊕ X NCRT ( X , Ω ) ψ / / X NCRT ( X , Ω )(35)The following result shows an elementary property of ψ . Lemma 3.11.
Let i : X NCRT ( X , Ω ) = k L ∞ → k F ( X , Ω ) be the natural inclusion. Then ψ i = id X NCRT ( X , Ω ) .Consequently, i ψ is idempotent. With this transporting of structures, the free matching Rota-Baxter algebra structure on X NC Ω ( X , Ω ) givesrise to a free matching Rota-Baxter algebra structure on X NCRT ( X , Ω ). More precisely, define a product ⋄ ℓ : X NCRT ( X , Ω ) ⊗ X NCRT ( X , Ω ) → X NCRT ( X , Ω )by taking(36) F ⋄ ℓ F ′ : = θ ( θ − ( F ) ⋄ w θ − ( F ′ )) for F , F ′ ∈ X NCRT ( X , Ω ) . Also define a linear operator on X NCRT ( X , Ω ) by θ P ω θ − which turns out to be just the grafting operator B + ω .Moreover, the degree deg on F ( X , Ω ) and its derived grading F ( n ) and filtration F ( n ) restrict to a gradingand filtration on k L ∞ . By Eq. (30), they are compatible with the ones on X ∞ . More precisely,(37) L ( n ) : = F ( n ) ∩ L ∞ = θ ( X ( n ) ) , L ( n ) : = F ( n ) ∩ L ∞ = θ ( X ( n ) ) , n ≥ . Therefore by Proposition 3.10 we have
Proposition 3.12.
Let j X : X ֒ → X NCRT ( X , Ω ) , x
7→ • x , x ∈ X , be the natural embedding. Then the triple ( X NCRT ( X , Ω ) , ⋄ ℓ , ( B + ω ) ω ∈ Ω ) together with j X is the free matching Rota-Baxter algebra of weight λ Ω on X.Further, X NCRT ( X , Ω ) with k L ( n ) , n ≥ , is an Ω -operated filtered algebra and ψ is a homomorphism of Ω -operated filtered algebras. OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 21 Ω - cocycle H opf algebras and free matching R ota -B axter algebras In this section, we first derive an Ω -cocycle bialgebraic structure on the free matching Rota-Baxteralgebra X NCRT ( X , Ω ), via a construction of a suitable coproduct. We then show that this Ω -cocycle bialgebrais connected cofiltered and so a Hopf algebra.Let u ℓ : k → X NCRT ( X , Ω ) be the linear map given by 1 k
1. By Proposition 3.12, the triple( X NCRT ( X , Ω ) , ⋄ ℓ , u ℓ ) is an algebra. We now define a linear map ∆ ℓ : X NCRT ( X , Ω ) → X NCRT ( X , Ω ) ⊗ X NCRT ( X , Ω )by setting(38) ∆ ℓ ( F ) : = ( ψ ⊗ ψ ) ∆ RT i ( F ) for all F ∈ X NCRT ( X , Ω ) , where i : X NCRT ( X , Ω ) → k F ( X , Ω ) is the natural inclusion. In other words, ∆ ℓ is defined so that the diagram X NCRT ( X , Ω ) ∆ ℓ / / i (cid:15) (cid:15) X NCRT ( X , Ω ) ⊗ X NCRT ( X , Ω ) k F ( X , Ω ) ∆ RT / / k F ( X , Ω ) ⊗ k F ( X , Ω ) ψ ⊗ ψ O O commutes.Define ǫ ℓ : X NCRT ( X , Ω ) → k by setting(39) ǫ ℓ ( F ) = ( , if F , , , if F = . We first verify that the coproduct ∆ ℓ on X NCRT ( X , Ω ) satisfies the Hochschild 1-cocycle condition. Lemma 4.1.
Let F = B + ω ( F ) be in X NCRT ( X , Ω ) . Then (40) ∆ ℓ ( B + ω ( F )) = B + ω ( F ) ⊗ + (id ⊗ B + ω ) ∆ ℓ ( F ) . Proof.
By the linearity, we just need to verify Eq. (40) for F ∈ L ∞ . Then ∆ ℓ ( B + ω ( F )) = ( ψ ⊗ ψ ) ∆ RT i ( B + ω ( F )) (By Eq. (38)) = ( ψ ⊗ ψ ) ∆ RT ( B + ω ( F )) (by i being an inclusion map) = ( ψ ⊗ ψ )( F ⊗ + (id ⊗ B + ω ) ∆ RT ( F )) (By Eq. (2)) = ψ ( F ) ⊗ ψ (1) + ( ψ ⊗ ψ B + ω ) ∆ RT ( F ) = ψ i ( F ) ⊗ ψ (1) + ( ψ ⊗ ψ B + ω ) ∆ RT i ( F ) (by i being an inclusion map) = F ⊗ + ( ψ ⊗ ψ B + ω ) ∆ RT i ( F ) (by Lemma 3.11) = F ⊗ + ( ψ ⊗ B + ω ψ ) ∆ RT i ( F )(by ψ being an operated algebra homomorphism in Eq. (34)) = F ⊗ + (id ⊗ B + ω )( ψ ⊗ ψ ) ∆ RT i ( F ) = F ⊗ + (id ⊗ B + ω ) ∆ ℓ ( F ) (by Eq. (38)) . This completes the proof. (cid:3)
Next we verify the compatibility of ∆ ℓ with ⋄ ℓ , starting with a special case. Lemma 4.2.
Let F , F ′ ∈ L ∞ with F ⋄ ℓ F ′ = FF ′ . Then ∆ ℓ ( F ⋄ ℓ F ′ ) = ∆ ℓ ( F ) ⋄ ℓ ∆ ℓ ( F ′ ) . Proof.
We have ∆ ℓ ( F ⋄ ℓ F ′ ) = ( ψ ⊗ ψ ) ∆ RT i ( FF ′ ) = ( ψ ⊗ ψ ) ∆ RT ( FF ′ ) (by i being an inclusion map) = ( ψ ⊗ ψ ) (cid:18) ∆ RT ( F ) ∆ RT ( F ′ ) (cid:19) (by ∆ RT being an algebra homomorphism) = (cid:18) ( ψ ⊗ ψ ) ∆ RT ( F ) (cid:19) ⋄ ℓ (cid:18) ( ψ ⊗ ψ ) ∆ RT ( F ′ ) (cid:19) (by ψ being an algebra homomorphism) = (cid:18) ( ψ ⊗ ψ ) ∆ RT i ( F ) (cid:19) ⋄ ℓ (cid:18) ( ψ ⊗ ψ ) ∆ RT i ( F ′ ) (cid:19) (by i being an inclusion map) = ∆ ℓ ( F ) ⋄ ℓ ∆ ℓ ( F ′ ) (by Eq. (38)) . This completes the proof. (cid:3)
In general, we have
Lemma 4.3.
Let F , F ′ ∈ X NCRT ( X , Ω ) . Then ∆ ℓ ( F ⋄ ℓ F ′ ) = ∆ ℓ ( F ) ⋄ ℓ ∆ ℓ ( F ′ ) and ǫ ℓ ( F ⋄ ℓ F ′ ) = ǫ ℓ ( F ) ǫ ℓ ( F ′ ) . (41) Proof.
The second equation follows from the definition of ǫ ℓ in Eq. (39).For the first equation, by the linearity, we just need to consider the case when F , F ′ ∈ L ∞ . We applythe induction on the sum of depths s : = dep( F ) + dep( F ′ ) ≥
0. For the initial step of s =
0, we havedep( F ) = dep( F ′ ) = F ⋄ ℓ F ′ = FF ′ . Then Eq. (41) follows from Lemma 4.2.Let t ≥
0. Assume that Eq. (41) holds for s = t and consider the case of s = t +
1. In this case, we firstconsider the case when bre( F ) = bre( F ′ ) =
1. If F ⋄ ℓ F ′ = FF ′ , then Eq. (41) follows from Lemma 4.2. If F ⋄ ℓ F ′ , FF ′ , then we have F = B + α ( F ) and F ′ = B + β ( F ′ ) for some α, β ∈ Ω and F , F ′ ∈ L ∞ . Write(42) ∆ ℓ ( F ) : = X ( F ) F (1) ⊗ F (2) and ∆ ℓ ( F ′ ) : = X ( F ′ ) F ′ (1) ⊗ F ′ (2) . Then ∆ ℓ ( F ⋄ ℓ F ′ ) = ∆ ℓ ( B + α ( F ) ⋄ ℓ B + β ( F ′ )) = ∆ ℓ (cid:16) B + α ( F ⋄ ℓ B + β ( F ′ )) + B + β ( B + α ( F ) ⋄ ℓ F ′ ) + λ β B + α ( F ⋄ ℓ F ′ ) (cid:17) (by Proposition 3.12) = ∆ ℓ B + α ( F ⋄ ℓ B + β ( F ′ )) + ∆ ℓ B + β ( B + α ( F ) ⋄ ℓ F ′ ) + λ β ∆ ℓ B + α ( F ⋄ ℓ F ′ ) = B + α ( F ⋄ ℓ B + β ( F ′ )) ⊗ + (id ⊗ B + α ) ∆ ℓ (cid:16) F ⋄ ℓ B + β ( F ′ ) (cid:17) + B + β ( B + α ( F ) ⋄ ℓ F ′ ) ⊗ + (id ⊗ B + β ) ∆ ℓ (cid:16) B + α ( F ) ⋄ ℓ F ′ (cid:17) + λ β B + α ( F ⋄ ℓ F ′ ) ⊗ + λ β (id ⊗ B + α ) ∆ ℓ ( F ⋄ ℓ F ′ ) (by Eq. (40)) = B + α ( F ) ⋄ ℓ B + β ( F ′ ) ⊗ + (id ⊗ B + α ) ∆ ℓ (cid:16) F ⋄ ℓ B + β ( F ′ ) (cid:17) + (id ⊗ B + β ) ∆ ℓ (cid:16) B + α ( F ) ⋄ ℓ F ′ (cid:17) + λ β (id ⊗ B + α ) ∆ ℓ ( F ⋄ ℓ F ′ ) (by Proposition 3.12) = ( F ⋄ ℓ F ′ ) ⊗ + (id ⊗ B + α ) (cid:16) ∆ ℓ ( F ) ⋄ ℓ ∆ ℓ B + β ( F ′ ) (cid:17) + (id ⊗ B + β ) (cid:16) ∆ ℓ B + α ( F ) ⋄ ℓ ∆ ℓ ( F ′ ) (cid:17) + λ β (id ⊗ B + α ) (cid:16) ∆ ℓ ( F ) ⋄ ℓ ∆ ℓ ( F ′ ) (cid:17) (by the induction hypothesis on s ) = ( F ⋄ ℓ F ′ ) ⊗ + X ( F ) ( F (1) ⋄ ℓ F ′ ) ⊗ B + α ( F (2) ) + X ( F ′ ) ( F ⋄ ℓ F ′ (1) ) ⊗ B + β ( F ′ (2) ) + X ( F ) X ( F ′ ) ( F (1) ⋄ ℓ F ′ (1) ) ⊗ ( B + α ( F (2) ) ⋄ ℓ B + β ( F ′ (2) )) (by Eqs. (40) and (42)) = (cid:18) F ⊗ + X ( F ) F (1) ⊗ B + α ( F (2) ) (cid:19) ⋄ ℓ (cid:18) F ′ ⊗ + X ( F ′ ) F ′ (1) ⊗ B + β ( F ′ (2) ) (cid:19) OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 23 = (cid:18) F ⊗ + (id ⊗ B + α ) ∆ ℓ ( F ) (cid:19) ⋄ ℓ (cid:18) F ′ ⊗ + (id ⊗ B + β ) ∆ ℓ ( F ′ ) (cid:19) = ∆ ℓ ( F ) ⋄ ℓ ∆ ℓ ( F ′ ) . We next consider the general case when bre( F ) > F ′ ) > F = F T or F ′ = T ′ F ′ , for some matchingRota-Baxter forests F , F ′ and some matching Rota-Baxter trees T , T ′ . Since F ( T ⋄ ℓ T ′ ) F ′ = F ⋄ ℓ ( T ⋄ ℓ T ′ ) ⋄ ℓ F ′ , we have ∆ ℓ ( F ⋄ ℓ F ′ ) = ∆ ℓ ( F ( T ⋄ ℓ T ′ ) F ′ ) = ∆ ℓ ( F ) ⋄ ℓ ∆ ℓ ( T ⋄ ℓ T ′ ) ⋄ ℓ ∆ ℓ ( F ′ ) (by Lemma 4.2) = ∆ ℓ ( F ) ⋄ ℓ ∆ ℓ ( T ) ⋄ ℓ ∆ ℓ ( T ′ ) ⋄ ℓ ∆ ℓ ( F ′ ) (by the case when bre( F ) = bre( F ′ ) = = (cid:0) ∆ ℓ ( F ) ⋄ ℓ ∆ ℓ ( T ) (cid:1) ⋄ ℓ (cid:0) ∆ ℓ ( T ′ ) ⋄ ℓ ∆ ℓ ( F ′ ) (cid:1) = ∆ ℓ ( F ⋄ ℓ T ) ⋄ ℓ ∆ ℓ ( T ′ ⋄ ℓ F ′ ) (by Lemma 4.2) = ∆ ℓ ( F T ) ⋄ ℓ ∆ ℓ ( T ′ F ′ ) = ∆ ℓ ( F ) ⋄ ℓ ∆ ℓ ( F ′ ) . This completes the proof. (cid:3)
Theorem 4.4.
The sextuple ( X NCRT ( X , Ω ) , ⋄ ℓ , u ℓ , ∆ ℓ , ǫ ℓ , ( B + ω ) ω ∈ Ω ) is an Ω -cocycle bialgebra.Proof. By Lemmas 4.1 and 4.3, we only need to verify the coassiciativity of ∆ ℓ and the counicity of ǫ ℓ .For the coassociativity of ∆ ℓ , following the idea of the proof of Theorem 2.10, we just need to show thatthe set C : = { F ∈ X NCRT ( X , Ω ) | ( ∆ ℓ ⊗ id) ∆ ℓ ( F ) = (id ⊗ ∆ ℓ ) ∆ ℓ ( F ) } is an Ω -operated subalgebra of X NCRT ( X , Ω ).Note that 1 is in C . By Lemma 4.3, ∆ ℓ is an algebra homomorphism. Then ( ∆ ℓ ⊗ id) ∆ ℓ and (id ⊗ ∆ ℓ ) ∆ ℓ are also algebra homomorphisms, implying that C is a subalgebra of X NCRT ( X , Ω ). For any x ∈ X , we have( ∆ ℓ ⊗ id) ∆ ℓ ( • x ) = ( ∆ ℓ ⊗ id)(1 ⊗ • x + • x ⊗
1) (by Eq. (38)) = ⊗ ⊗ • x + ⊗ • x ⊗ + • x ⊗ ⊗ = (id ⊗ ∆ ℓ ) ∆ ℓ ( • x ) . Thus • x ∈ C . For any F ∈ C and ω ∈ Ω , we have( ∆ ℓ ⊗ id) ∆ ℓ ( B + ω ( F )) = ( ∆ ℓ ⊗ id)( B + ω ( F ) ⊗ + (id ⊗ B + ω ) ∆ ℓ ( F )) (by Lemma 4.1) =∆ ℓ ( B + ω ( F )) ⊗ + ( ∆ ℓ ⊗ B + ω ) ∆ ℓ ( F ) = B + ω ( F ) ⊗ ⊗ + (id ⊗ B + ω ) ∆ ℓ ( F ) ⊗ + (id ⊗ id ⊗ B + ω )( ∆ ℓ ⊗ id) ∆ ℓ ( F ) (by Lemma 4.1) = B + ω ( F ) ⊗ ⊗ + (id ⊗ B + ω ) ∆ ℓ ( F ) ⊗ + (id ⊗ id ⊗ B + ω )(id ⊗ ∆ ℓ ) ∆ ℓ ( F ) (by F ∈ C ) = B + ω ( F ) ⊗ ⊗ + (cid:16) (id ⊗ B + ω ) ⊗ + (id ⊗ id ⊗ B + ω )(id ⊗ ∆ ℓ ) (cid:17) ∆ ℓ ( F ) = B + ω ( F ) ⊗ ⊗ + (id ⊗ ∆ ℓ B + ω ) ∆ ℓ ( F ) (by Lemma 4.1) = (id ⊗ ∆ ℓ )( B + ω ( F ) ⊗ + (id ⊗ B + ω ) ∆ ℓ ( F )) = (id ⊗ ∆ ℓ ) ∆ ℓ ( B + ω ( F )) (by Lemma 4.1) , which means that B + ω ( F ) ∈ C . Thus C is stable under B + ω for any ω ∈ Ω . In summary, C = X NCRT ( X , Ω )and so ∆ ℓ is coassociative. For the counicity, we similarly consider D : = n F ∈ X NCRT ( X , Ω ) | ( ǫ ℓ ⊗ id) ∆ ℓ ( F ) = β l ( F ) and (id ⊗ ǫ ℓ ) ∆ ℓ ( F ) = β r ( F ) o , adapting notations from [20]. Note that 1 ∈ D . By Lemma 4.3, ∆ ℓ and ǫ ℓ are algebra homomorphisms, so D is a subalgebra of X NCRT ( X , Ω ). For any x ∈ X , by Eq. (39), we have( ǫ ℓ ⊗ id) ∆ ℓ ( • x ) = ( ǫ ℓ ⊗ id)(1 ⊗ • x + • x ⊗ = k ⊗ • x = β l ( • x )and (id ⊗ ǫ ℓ ) ∆ ℓ ( • x ) = (id ⊗ ǫ ℓ )(1 ⊗ • x + • x ⊗ = • x ⊗ k = β r ( • x ) . Hence • x ∈ D . For any F ∈ D and ω ∈ Ω , we have( ǫ ℓ ⊗ id) ∆ ℓ ( B + ω ( F )) = ( ǫ ℓ ⊗ id)( B + ω ( F ) ⊗ + (id ⊗ B + ω ) ∆ ℓ ( F )) (by Lemma (4.1)) = ǫ ℓ ( B + ω ( F )) ⊗ + ( ǫ ℓ ⊗ id)(id ⊗ B + ω ) ∆ ℓ ( F ) = ǫ ℓ ( B + ω ( F )) ⊗ + (id ⊗ B + ω )( ǫ ℓ ⊗ id) ∆ ℓ ( F ) = + (id ⊗ B + ω ) β l ( F ) (by Eq. (39) and F ∈ D ) = k ⊗ B + ω ( F ) = β l ( B + ω ( F ))and (id ⊗ ǫ ℓ ) ∆ ℓ ( B + ω ( F )) = (id ⊗ ǫ ℓ )( B + ω ( F ) ⊗ + (id ⊗ B + ω ) ∆ ℓ ( F )) (by Lemma (4.1)) = B + ω ( F ) ⊗ k + (id ⊗ ǫ ℓ )(id ⊗ B + ω ) ∆ ℓ ( F ) = B + ω ( F ) ⊗ k + X ( F ) F (1) ⊗ ǫ ℓ ( B + ω ( F (2) )) = B + ω ( F ) ⊗ k + F ∈ D ) = β r ( B + ω ( F )) . Thus B + ω ( F ) ∈ D and D is stable under B + ω for any ω ∈ Ω . Consequently, D = X NCRT ( X , Ω ) and ǫ ℓ is acounit. This completes the proof. (cid:3) We now introduce the connectedness condition on coalgebras [15].
Definition 4.5. (a) A coalgebra ( C , ∆ , ǫ ) is called coaugmented if there is a linear map u : k → C ,called the coaugmentation , such that ǫ u = id k .(b) A coaugmented coalgebra ( C , u , ∆ , ǫ ) is called cofiltered if there is an exhaustive increasing filtra-tion { C ( n ) } n ≥ of H such that(43) im u ⊆ C ( n ) , ∆ ( C ( n ) ) ⊆ X p + q = n C ( p ) ⊗ C ( q ) , n ≥ , p , q ≥ C ( n ) \ C ( n − are said to have degree n . C is called connected (filtered) if in addition C = im u ( = k ).By the coaugmented condition, we have C = im u ⊕ ker ǫ . Then from im u ⊆ C ( n ) and modularity, wehave C ( n ) = im u ⊕ ( C ( n ) ∩ ker ǫ ), as originally stated in [15].We quote the following condition [15, Theorem 3.4] for Hopf algebras. See also [28, 29, 31]. Lemma 4.6.
Let H = ( H , m , u , ∆ , ǫ ) be a bialgebra such that ( H , ∆ , ǫ ) is a connected cofiltered coaugu-mented coalgebra. Then H is a Hopf algebra. Finally, we arrive at our main result on Hopf algebraic structure on free matching Rota-Baxter algebras.
OPF ALGEBRAS OF ROOTED FORESTS AND FREE MATCHING ROTA-BAXTER ALGEBRAS 25
Theorem 4.7.
The sextuple ( X NCRT ( X , Ω ) , ⋄ ℓ , u ℓ , ∆ ℓ , ǫ ℓ , ( B + ω ) ω ∈ Ω ) is an Ω -cocycle Hopf algebra.Proof. By Theorem 4.4, The quintuple ( X NCRT ( X , Ω ) , ⋄ ℓ , u ℓ , ∆ ℓ , ǫ ℓ , ( B + ω ) ω ∈ Ω ) is an Ω -cocycle bialgebra. Inparticular, ( X NCRT ( X , Ω ) , u ℓ , ∆ ℓ , ǫ ℓ ) is a coaugumented coalgebra. To apply Lemma 4.6, we just need tocheck that ( X NCRT ( X , Ω ) , u ℓ , ∆ ℓ , ǫ ℓ ) is connected and cofiltered, with respect to the filtration k F ( n ) , n ≥ F (0) =
1. Further by Proposition 3.12 and Eq. (38), we have ∆ ℓ ( L n ) = ( ψ ⊗ ψ )( ∆ ℓ ( L n )) ⊆ ( ψ ⊗ ψ ) (cid:16) X p + q = n ( k F ( p ) ) ⊗ ( k F ( q ) ) (cid:17) ⊆ X p + q = n ( k L ( p ) ) ⊗ ( k L ( q ) ) . Now the conclusion follows from Lemma 4.6. (cid:3)
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