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permutations
Samuele Giraudo
To cite this version:
Samuele Giraudo. Algebraic and combinatorial structures on Baxter permutations. 23rd International
Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik,
Iceland. pp.387-398. �hal-00790742v2�
Algebraic and combinatorial structures on Baxter permutations
Samuele Giraudo
11Institut Gaspard Monge, Universit´e Paris-Est Marne-la-Vall´ee, Marne-la-Vall´ee, France
Abstract.We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees,etc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like insertion algorithm. The algebraic properties of this Hopf algebra are studied. This Hopf algebra appeared for the first time in the work of Reading [Lattice congruences, fans and Hopf algebras,Journal of Combinatorial Theory Series A, 110:237–273, 2005].
R´esum´e.Nous proposons une nouvelle construction d’une sous-alg`ebre de Hopf de l’alg`ebre de Hopf des fonctions quasi-sym´etriques libres dont les bases sont index´ees par les objets de la famille combinatoire de Baxter (i.e.permu- tations de Baxter, couples d’arbres binaires jumeaux,etc.). Cette construction repose sur la d´efinition du mono¨ıde de Baxter, analogue du mono¨ıde plaxique et du mono¨ıde sylvestre, et d’un algorithme d’insertion analogue `a l’algorithme de Robinson-Schensted. Les propri´et´es alg´ebriques de cette alg`ebre de Hopf sont ´etudi´ees. Cette alg`ebre de Hopf est apparue pour la premi`ere fois dans le travail de Reading [Lattice congruences, fans and Hopf algebras,Journal of Combinatorial Theory Series A, 110:237–273, 2005].
Keywords:Hopf algebras, Robinson-Schensted algorithm, quotient monoid, Baxter permutations
1 Introduction
In the recent years, many combinatorial Hopf algebras, whose bases are indexed by combinatorial ob- jects, have been intensively studied. For example, the Malvenuto-Reutenauer Hopf algebraFQSymof Free quasi-symmetric functions [19, 7] has bases indexed by permutations. This Hopf algebra admits several Hopf subalgebras: The Hopf algebra of Free symmetric functionsFSym [21, 7], whose bases are indexed by standard Young tableaux, the Hopf algebraBell[23] whose bases are indexed by set partitions, the Loday-Ronco Hopf algebraPBT[18, 12] whose bases are indexed by planar binary trees and the Hopf algebraSymof non-commutative symmetric functions [10] whose bases are indexed by integer compositions. An unifying approach to construct all these structures relies on a definition of a congruence on words leading to the definition of monoids on combinatorial objects. Indeed,FSymis directly obtained from the plactic monoid [15],Bellfrom the Bell monoid [23],PBTfrom the sylvester monoid [11, 12], andSymfrom the hypoplactic monoid [20]. The richness of these constructions relies on the fact that, in addition to construct Hopf algebras, the definition of such monoids often brings partial orders, combinatorial algorithms and Robinson-Schensted-like algorithms, of independent interest.
In this paper, we propose to enrich this collection of Hopf algebras by providing a construction of a Hopf algebra whose bases are indexed by objects belonging to the Baxter combinatorial family. This com- binatorial family admits various representations as Baxter permutations [4], pairs of twin binary trees [8],
1365–8050 c2011 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
quadrangulations [1], plane bipolar orientations [5],etc. In [22], Reading defines first a Hopf algebra on Baxter permutations in the context of lattice congruences; Moreover, very recently, Law and Reading [16]
have studied and detailed their construction of this Hopf algebra. However, even if both points of view lead to the same general theory, their paths are different and provide different ways of understanding this Hopf algebra, one centered, as in Law and Reading’s work, on lattice theory, the other, as in our work, centered on combinatorics on words. Moreover, a large part of the results of each paper does not appear in the other.
We begin by recalling in Section 2 the preliminary notions used thereafter. In Section 3, we define the Baxter congruence. This congruence allows to define a quotient of the free monoid, the Baxter monoid, which has a number of properties required for the Hopf algebraic construction which follows. We show that the Baxter monoid is intimately linked to the sylvester monoid. Next, in Section 4, we develop a Robinson-Schensted-like insertion algorithm that allows to decide if two words are equivalent according to the Baxter congruence. Given a word, this algorithm computes a pair of twin binary trees. Section 5 is devoted to the study of some properties of the equivalence classes of permutations under the Baxter congruence. This leads to the definition of a lattice structure on pairs of twin binary trees. Finally, in Section 6, we define the Hopf algebraBaxterand study it. Using the order structure on pairs of twin binary trees, we provide multiplicative bases and show thatBaxter is free as an algebra. Using the results of Foissy on bidendriform bialgebras [9], we show thatBaxteris also self-dual and that the Lie algebra of its primitive elements is free.
Acknowledgements
The author would like to thank Florent Hivert and Jean-Christophe Novelli for their advice and help during all stages of the preparation of this paper. The computations of this work have been done with the open-source mathematical software Sage [25].
2 Preliminaries
2.1 Words
In the sequel,A := {a1 < a2 < . . .} is a totally ordered infinite alphabet andA∗ is the free monoid spanned byA. Letu∈A∗. ForS⊆A, we denote byu|Stherestrictionofuon the alphabetS, that is the longest subword ofumade of letters ofS. Theevaluationeval(u)of the worduis the non-negative integer vector such that itsi-th entry is the number of occurrences of the letteraiinu. Letmax(u)be the maximal letter ofu. TheSch¨utzenberger transformation#is defined byu#:= max(u)+1−u|u|. . .max(u)+1−u1; For example,(a5a3a1a1a5a2)#=a4a1a5a5a3a1. Note that it is an involution ifuhas an occurrence of a1. Letv∈A∗anda,b∈A. Theshuffle productis defined onZhAirecursively byu:=u:=u andaubv:=a(ubv) +b(auv).
2.2 Permutations
Denote bySnthe set of permutations of sizenandS:=∪n≥0Sn. We shall call(i, j)aco-inversionof σ∈Sifi < jandσi−1> σj−1. Let us recall that the(right) permutohedron orderis the partial order≤P
defined onSn whereσis covered byν ifσ=uabvandν =ubavwherea< b. Letσ, ν ∈S. The permutationσνis obtained by concatenatingσand the letters ofνincremented by|σ|; In the same way, the permutationσνis obtained by concatenating the letters ofνincremented by|σ|andσ; For example, 3122314 = 3125647and3122314 = 5647312. The permutationσisconnectedifσ = νπ impliesν = σorπ = σ. The shifted shuffle productof two permutations is defined byσν :=
σ(ν1+|σ|. . . ν|ν|+|σ|); For example,1221 =1243 =1243+1423+1432+4123+4132+4312.
Thestandardized wordstd(u)ofu∈A∗is the unique permutationσsatisfyingσi < σjiffui ≤ujfor all1≤i < j≤ |u|; For example,std(a3a1a4a2a5a7a4a2a3) = 416289735.
2.3 Binary trees
Denote byBTn the set of binary trees withninternal nodes andBT :=∪n≥0BTn. We use in the sequel the standard terminology (i.e.,child,ancestor, . . . ) about binary trees [2]. The only element ofBT0is the leaf orempty tree, denoted by⊥. Let us recall that theTamari order[14] is the partial order≤Tdefined onBTnwhereT0∈ BTnis covered byT1∈ BTnif it is possible to transformT0intoT1by performing a right rotation (see Figure 1).
y x
A B
C T0=
y x A
B C
=T1
Figure 1:The right rotation of rooty.
LetT0, T1∈ BT. The binary treeT0T1is obtained by graftingT0from its root on the leftmost leaf of T1; In the same way, the binary treeT0T1is obtained by graftingT1from its root on the rightmost leaf ofT0. Thecanopy(see [18] and [26])cnp(T)ofT ∈ BT is the word on the alphabet{0,1}obtained by browsing the leaves ofTfrom left to right except the first and the last one, writing0if the considered leaf is oriented to the right,1otherwise (see Figure 2). Note that the orientation of the leaves in a binary tree is determined only by its nodes so that we can omit to draw the leaves in our next graphical representations.
0
1 0 0 1 0 1
Figure 2:The canopy of this binary tree is0100101.
AnA-labeled binary treeT is aleft(resp.right)binary search treeif for any nodexlabeled byb, each labelaof a node in the left subtree ofxand each labelcof a node in the right subtree ofx, the inequality a<b≤c(resp.a≤b<c) holds. A binary treeT ∈ BTnis adecreasing binary treeif it is bijectively labeled on{1, . . . , n}and, for all nodeyofT, ifxis a child ofy, then the label ofxis smaller than the label ofy. Theshapeof a labeled binary tree is the unlabeled binary tree obtained by forgetting its labels.
2.4 Baxter permutations and pairs of twin binary trees
A permutationσis aBaxter permutationif for any subword u = u1u2u3u4 of σsuch that the letters u2 andu3 are adjacent in σ,std(u) ∈ {2413,/ 3142}. In other words, σis a Baxter permutation if it avoids thegeneralized permutation patterns2−41−3and3−14−2(see [3] for an introduction on generalized permutation patterns). For example, 42173856is not a Baxter permutation; On the other hand436975128is a Baxter permutation. Let us denote bySBn the set of Baxter permutations of sizen andSB:=∪n≥0SBn.
Apair of twin binary trees(TL, TR)is made of two binary treesTL, TR∈ BTnsuch that the canopies ofTLandTRare complementary, that iscnp(TL)i6= cnp(TR)ifor all1≤i≤n−1. Denote byT BTn
the set of pairs of twin binary trees where each binary tree hasnnodes andT BT :=∪n≥0T BTn. In [8], Dulucq and Guibert have highlighted a bijection between Baxter permutations and pairs of twin binary trees. In the sequel, we shall make use of a very similar bijection.
3 The Baxter monoid
3.1 Definition and first properties
Recall that an equivalence relation≡defined onA∗is acongruenceif for allu, u0, v, v0 ∈ A∗,u≡u0 andv≡v0implyu.v≡u0.v0.
Definition 3.1 TheBaxter monoidis the quotient of the free monoidA∗by the congruence ≡B that is the transitive closure of theadjacency relations B and B defined foru, v∈A∗anda,b,c,d∈Aby:
cuadvbBcudavb where a≤b<c≤d, (1) budavcBbuadvc where a<b≤c<d. (2) Foru∈A∗, denote bybuthe ≡B-equivalence class ofu; For example, the ≡B-equivalence class of 5273641is{5237641,5273641,5276341,5723641,5726341,5762341}.
An equivalence relation≡defined onA∗iscompatible with the restriction of alphabet intervalsif for all intervalIofAand for allu, v ∈A∗,u≡vimpliesu|I ≡v|I.
Proposition 3.2 The Baxter monoid is compatible with the restriction of alphabet intervals.
Proof:We only have to check the property on adjacency relations. 2 An equivalence relation≡defined onA∗ iscompatible with the destandardization processif for all u, v∈A∗,u≡viffstd(u)≡std(v)andeval(u) = eval(v).
Proposition 3.3 The Baxter monoid is compatible with the destandardization process.
An equivalence relation≡defined onA∗ iscompatible with the Sch¨utzenberger involutionif for all u, v∈A∗,u≡vimpliesu#≡v#.
Proposition 3.4 The Baxter monoid is compatible with the Sch¨utzenberger involution.
3.2 Connection with the sylvester monoid
Thesylvester monoid[11, 12] is the quotient of the free monoidA∗by the congruence ≡S that is the transitive closure of the adjacency relation S defined foru∈A∗anda,b,c∈Aby:
acubScaub where a≤b<c. (3)
In the same way, let us define the#-sylvester monoidby the congruence ≡S# that is the transitive closure of the adjacency relation S# defined foru∈A∗anda,b,c∈Aby:
buacS#buca where a<b≤c. (4)
Note that this adjacency relation is defined by taking the images by the Sch¨utzenberger involution of the sylvester adjacency relation. Indeed, for allu, v ∈ A∗,u≡S#viffu#≡Sv#. The Baxter monoid and the sylvester monoid are related in the following way:
Proposition 3.5 Letu, v∈A∗. Then,u≡Bviffu≡Svandu≡S#v.
Proposition 3.5 shows that the ≡B-equivalence classes are the intersection of ≡S-equivalence classes and ≡S#-equivalence classes.
4 A Robinson-Schensted-like algorithm
We shall describe here an insertion algorithmu 7→ (P(u),Q(u)), such that, given a wordu ∈ A∗, it computes itsP-symbol, that is a pair ofA-labeled twin binary trees(TL, TR)whereTL (resp. TR) is a left (resp. right) binary search tree, and itsQ-symbol, a decreasing binary tree.
4.1 Definition of the insertion algorithm
LetT be anA-labeled right binary search tree andba letter ofA. Thelower restricted binary treeofT compared tob, namelyT≤b, is the right binary search tree uniquely made of the nodesxofT labeled by a letterasatisfyinga≤band such that for all nodesxandyofT≤b, ifxis ancestor ofyinT≤b, then xis ancestor ofyinT. In the same way, we define thehigher restricted binary treeofT compared tob, namelyT>b(see Figure 3).
1 1
2 3
3 4
5 1
1
2 3
3 4
5
Figure 3:A right binary search treeT,T≤2andT>2.
LetTbe anA-labeled right binary search tree andaa letter ofA. Theroot insertionofaintoTconsists in modifyingTso that the root ofTis a new node labeled bya, its left subtree isT≤aand its right subtree isT>a.
LetT be anA-labeled left (resp. right) binary search tree andaa letter ofA. Theleaf insertionofa intoT is recursively defined by: IfT =⊥, the result is the one-node binary tree labeled bya; Else, if the labelbof the root ofT satisfiesa<b(resp.a≤b), make a leaf insertion ofainto the left subtree ofT, else, make a leaf insertion ofainto the right subtree ofT.
Given a pair ofA-labeled twin binary trees(TL, TR)whereTL(resp.TR) is a left (resp. right) binary search tree, theinsertionof the letteraofAinto(TL, TR)consists in making a leaf insertion ofaintoTL
and a root insertion ofaintoTR.
TheP-symbol(TL, TR)of a wordu∈A∗is computed by iteratively inserting the letters ofu, from left to right, into the pair of twin binary trees(⊥,⊥). TheQ-symbol ofuis the decreasing binary tree labeled on{1, . . . ,|u|}, built by recording the dates of creation of each node ofTR(see Figure 4).
⊥⊥ −→5 5 5 −→4 4 5 4 5 −→2 2
4 5 2
4 5
−5
→
2 4
5 5
2 4
5 5
−4
→ 2
4 4
5 5 2
4 4
5
5 −→2
2 2
4 4
5 5 2
2 4
4 5
5
−4
→ 2 2
4 4
4 5
5 2
2 4
4 4
5 5 =
P(u);
3 6
2 5
7 1
4 = Q(u)
Figure 4:Steps of computation of theP-symbol and theQ-symbol ofu:= 5425424.
4.2 Validity of the insertion algorithm
Lemma 4.1 Letu∈A∗. LetT be the right binary search tree obtained by root insertions of the letters ofu, from left to right. LetT0be the right binary search tree obtained by leaf insertions of the letters of u, from right to left. Then,T =T0.
Lemma 4.2 Letσ∈SandT ∈ BT|σ|be the binary search tree obtained by leaf insertions of the letters ofσ, from left to right. Then, for1 ≤i ≤ |σ|−1, thei+1-st leaf ofT is right-oriented iff(i, i+1)is a co-inversion ofσ.
If(TL, TR)is a pair of labeled twin binary trees, define its shape, that is the pair of unlabeled twin binary trees(TL0, TR0)whereTL0 (resp.TR0) is the shape ofTL(resp.TR).
Proposition 4.3 For all wordu∈A∗, the shape of theP-symbol ofuis a pair of twin binary trees.
Proposition 4.4 Letu, v∈A∗. Then,u≡BviffP(u) =P(v).
In particular, we haveP(σ) =P(ν)iff the permutationsσandν are ≡B-equivalent. Moreover, each
≡B-equivalence class of permutations can be encoded by a pair of unlabeled twin binary trees because there is one unique way to bijectively label a binary tree withnnodes on{1, . . . , n}such that it is a binary search tree.
Remark 4.5 Letu, v ∈A∗and(TL, TR) :=P(u). We haveu≡Bviff the following two assertions are satisfied:
(i) vis a linear extension ofTLseen as a poset in which the smallest element is its root;
(ii) v is a linear extension ofTR seen as a poset in which minimal elements are the nodes with no descents.
5 The Baxter lattice
5.1 Some properties of the ≡
B-equivalence classes of permutations
Theorem 5.1 For alln≥0, each equivalence class ofSn/≡B contains exactly one Baxter permutation.
Proposition 5.2 For alln≥0, each equivalence class ofSn/≡B is an interval of the permutohedron.
For all permutationσ, let us defineσ ↑(resp. σ ↓) the maximal (resp. minimal) permutation of the
≡B-equivalence class ofσfor the permutohedron order.
Proposition 5.3 Letσ, ν ∈Snsuch thatσ≤Pν. Then,σ↑≤Pν ↑andσ↓≤Pν ↓.
5.2 A lattice structure on the set of pairs of twin binary trees
Definition 5.4 For alln ≥ 0, define the order relation≤B on the setT BTn settingJ0 ≤B J1, where J0, J1∈ T BTn, if there existsσ0, σ1∈Snsuch thatP(σ0) =J0,P(σ1) =J1andσ0≤Pσ1.
Propositions 5.2 and 5.3 ensure that this order is well-defined, and in particular that the relation≤B is transitive and antisymmetric.
The pair of twin binary trees(TL, TR)is covered by(TL0, TR0)∈ T BT if one of the three following conditions is satisfied:
1. TR0 =TRandTL0 is obtained fromTLby performing a left rotation intoTLsuch thatcnp(TL) = cnp(TL0);
2. TL0 =TLandTR0 is obtained fromTRby performing a right rotation intoTRsuch thatcnp(TR) = cnp(TR0);
3. TL0 (resp. TR0) is obtained by performing a left (resp. right) rotation intoTL (resp. TR) such that cnp(TL)6= cnp(TL0)(resp.cnp(TR)6= cnp(TR0)).
Moreover, it is possible to compare two pairs of twin binary treesJ0 := (TL0, TR0)andJ1:= (TL1, TR1) very easily by computing theTamari vector(see [14]) of each binary tree. Indeed, we haveJ0≤B J1iff the Tamari vector ofTL0 (resp. TR0) is greater (resp. smaller) component by component than the Tamari vector ofTL1(resp.TR1).
Propositions 5.2 and 5.3 implies that that ≡B is also a lattice congruence [6, 22]. Thus, since the permutohedron is a lattice,
Proposition 5.5 For alln≥0, the poset(T BTn,≤B)is a lattice.
6 The Baxter Hopf Algebra
In the sequel, all the algebraic structures have a field of characteristic zeroKas ground field.
6.1 The Hopf algebra FQSym
Recall that the family{Fσ}σ∈Sform thefundamentalbasis ofFQSym[7]. Its product and its coproduct are defined by:
Fσ·Fν := X
π∈σν
Fπ, ∆ (Fσ) := X
0≤i≤|σ|
Fstd(σ1...σi)⊗Fstd(σi+1...σ|σ|). (5)
The following theorem due to Hivert and Nzeutchap [13] shows that an equivalence relation on A∗ satisfying some properties can be used to define Hopf subalgebras ofFQSym:
Theorem 6.1 Let≡be an equivalence relation defined on A∗. If≡is a congruence, compatible with the restriction of alphabet intervals and compatible with the destandardization process, then, the family {Pbσ}
bσ∈S/≡defined by:
Pbσ:=X
σ∈bσ
Fσ (6)
spans a Hopf subalgebra ofFQSym.
6.2 The Hopf algebra Baxter
By definition, ≡B is a congruence, and, by Proposition 3.2 and 3.3, ≡B checks the conditions of Theo- rem 6.1. Moreover, by Proposition 4.4, the ≡B-equivalence classes of permutations can be encoded by pairs of unlabeled twin binary trees. Hence, we have the following theorem:
Theorem 6.2 The family{PJ}J∈T BT defined by:
PJ:= X
σ∈S P(σ)=J
Fσ (7)
spans a Hopf subalgebra ofFQSym, namely the Hopf algebraBaxter.
The Hilbert series ofBaxterisB(z) := 1 +z+ 2z2+ 6z3+ 22z4+ 92z5+ 422z6+ 2074z7+ 10754z8+ 58202z9+ 326240z10 + 1882960z11+. . ., the generating series of Baxter permutations (sequenceA001181of [24]).
One has for example,
P =F12, P =F2143+F2413, P =F542163+F542613+F546213. (8) By Theorem 6.1, the product ofBaxteris well-defined. We deduce it from the product ofFQSym and obtain
PJ0·PJ1 = X
P(σ)=J0,P(ν)=J1 π∈σν ∩SB
PP(π). (9)
For example,
P ·P =P +P +P
+P +P +P . (10)
In the same way, we deduce the coproduct ofBaxterfrom the coproduct ofFQSymand obtain
∆(PJ) = X
P(π)=J π=u.v
σ:=std(u), ν:=std(v)∈SB
PP(σ)⊗PP(ν). (11)
For example,
∆P = 1⊗P +P ⊗P +P ⊗P +P ⊗P
+P ⊗P +P ⊗P +P ⊗P +P ⊗1. (12)
Remark 6.3 It is well-known that the Hopf algebraPBT[18, 12] is a Hopf subalgebra ofFQSym.
Besides, we have the following sequence of injective Hopf maps:
PBT,→ρ Baxter,→FQSym. (13) Indeed, by Proposition 3.5, every ≡S-equivalence class is an union of some ≡B-equivalence classes.
Denoting by{PT}T∈BT the basis ofPBTdefined in accordance with (6) by the sylvester equivalence relation ≡S, we have
ρ(PT) = X
T0∈BT J:=(T0,T)∈T BT
PJ. (14)
For example,
ρ
P
=P +P +P . (15)
6.3 Multiplicative bases
Define theelementaryfamily{EJ}J∈T BT and thehomogeneousfamily{HJ}J∈T BT respectively by:
EJ:= X
J≤BJ0
PJ0 and HJ:= X
J0≤BJ
PJ0. (16) These families are bases ofBaxtersince they are defined by triangularity.
LetJ0:= (TL0, TR0)andJ1:= (TL1, TR1)be two pairs of twin binary trees. Let us define the pair of twin binary treesJ0J1byJ0J1 := (TL0TL1, TR0TR1). In the same way, the pair of twin binary trees J0J1is defined byJ0J1:= (TL0TL1, TR0TR1).
Using the multiplicative bases ofFQSym, we establish the following proposition:
Proposition 6.4 For allJ0, J1∈ T BT, we have
EJ0·EJ1 =EJ0J1 and HJ0·HJ1 =HJ0J1. (17) Lemma 6.5 LetCbe an equivalence class ofSn/≡B. The Baxter permutation belonging toCis con- nected iff all the permutations ofCare connected.
Let us say that a pair of twin binary treesJisconnectedif the unique Baxter permutationσsatisfying P(σ) =Jis connected.
Proposition 6.6 The Hopf algebraBaxteris free on the elementsEJ whereJ is a connected pair of twin binary trees.
The generating seriesBC(z)of connected Baxter permutations isBC(z) = 1−B(z)−1. First di- mensions of algebraic generators ofBaxter are1,1,1, 3, 11,47, 221, 1113, 5903,32607,186143, 1092015.
6.4 Bidendriform bialgebra structure
A Hopf algebra(H,·,∆) can be fitted into a bidendriform bialgebra structure [9] if (H+,≺,)is a dendriform algebra [17] and(H+,∆≺,∆)a codendriform coalgebra, whereH+ is the augmentation ideal ofH. The operators≺,,∆≺and∆have to fulfil some compatibility relations. In particular, for allx, y ∈H+, the product·ofH is retrieved byx·y =x≺y+xyand the coproduct∆ofH is retrieved by∆(x) = 1⊗x+ ∆≺(x) + ∆(x) +x⊗1.
The Hopf algebraFQSymadmits a bidendriform bialgebra structure [9]. Indeed, for allσ, ν∈Sset Fσ≺Fν := X
π∈σν π|π|=σ|σ|
Fπ, FσFν := X
π∈σν π|π|=ν|ν|+|σ|
Fπ, (18)
∆≺(Fσ) := X
σ−1|σ|≤i≤|σ|−1
Fstd(σ1...σi)⊗Fstd(σi+1...σ|σ|), (19)
∆(Fσ) := X
1≤i≤σ−1|σ|−1
Fstd(σ1...σi)⊗Fstd(σi+1...σ|σ|). (20) Proposition 6.7 If≡is an equivalence relation defined onA∗satisfying the conditions of Theorem 6.1 and additionally, for allu, v ∈ A∗, the relationu ≡ vimpliesu|u| = v|v|, then, the family defined in (6) spans a bidendriform sub-bialgebra ofFQSym, and is free as an algebra, cofree as a coalgebra, self-dual, and the Lie algebra of its primitive elements is free.
The equivalence relation ≡B satisfies the premises of Proposition 6.7 so thatBaxter is free as an algebra, cofree as a coalgebra, self-dual, and the Lie algebra of its primitive elements is free.
6.5 The dual Hopf algebra Baxter
?Let {P?J}J∈T BT be the dual basis of the basis {PJ}J∈T BT. The Hopf algebra Baxter?, dual of Baxter, is a quotient Hopf algebra ofFQSym?. More precisely,
Baxter?=FQSym?/I (21) whereIis the Hopf ideal ofFQSym?spanned by the relationsF?σ=F?νwheneverσ≡Bν.
Letφ:FQSym? →Baxter?be the canonical projection, mappingF?σonP?JwheneverP(σ) =J. By definition, the product ofBaxter?is
P?J0·P?J1 =φ(F?σ·F?ν) (22) whereσandνare any permutations such thatP(σ) =J0andP(ν) =J1. For example,
P? ·P? =P? +P? +P? +P? +P?
+P? +P? +P? +P? +P? .
(23) In the same way, the coproduct ofBaxter?is
∆(PJ) = (φ⊗φ)(∆ (F?σ)) (24) whereσis any permutation such thatP(σ) =J. For example,
∆P? = 1⊗P? +P? ⊗P? +P? ⊗P? +P? ⊗P? +P? ⊗1.
(25) Remark 6.8 By Proposition 6.7, the Hopf algebrasBaxterandBaxter?are isomorphic. However, de- noting byθ:Baxter,→FQSymthe injection fromBaxtertoFQSym,ψ:FQSym↔FQSym? the isomorphism from FQSym to FQSym? defined by ψ(Fσ) := F?σ−1, and φ : FQSym? Baxter? the surjection fromFQSym? toBaxter?, the map φ◦ψ◦θ : Baxter → Baxter? is not an isomorphism. Indeed:
φ◦ψ◦θP =φ◦ψ(F2143+F2413) =φ(F?2143+F?3142) =P? +P? , (26) φ◦ψ◦θP =φ◦ψ(F3142+F3412) =φ(F?2413+F?3412) =P? +P? , (27) showing thatφ◦ψ◦θis not injective.
6.6 Primitive and totally primitive elements 6.6.1 Primitive elements
Since the family{EJ}J∈C, whereC is the set of connected pairs of twin binary trees, are indecompos- able elements ofBaxter, its dual family{E?J}J∈Cforms a basis of the Lie algebrap?of the primitive elements ofBaxter?. By Proposition 6.7, the Lie algebrap?is free.
6.6.2 Totally primitive elements
An elementxof a bidendriform bialgebra istotally primitiveif∆≺(x) = 0 = ∆(x).
Following [9], the generating seriesBT(z)of the totally primitive elements ofBaxterisBT(z) =
B(z)−1
B(z)2 . First dimensions of totally primitive elements ofBaxterare0,1,0,1,4,19,96,511,2832, 16215,95374,573837. Here follows a basis of the totally primitive elements ofBaxterof order1,3 and4:
t1,1=P , (28)
t3,1=P −P , (29)
t4,1=P +P +P +P (30)
−P −P −P ,
t4,2=P −P , (31)
t4,3=P −P , (32)
t4,4=P −P . (33)
Baxteris free as dendriform algebra on its totally primitive elements.
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