COMBINATORIAL HOPF ALGEBRAS FROM PROS

Transcription

1 COMBINATORIAL HOPF ALGEBRAS FROM PROS JEAN-PAUL BULTEL AND SAMUELE GIRAUDO Astrct. We introduce generl construction tht tkes s input so-clled stiff PRO nd tht outputs Hopf lger. Stiff PROs re prticulr PROs tht cn e descried y genertors nd reltions with precise conditions. Our construction generlizes the clssicl construction from operds to Hopf lgers of vn der Ln. We study some of its properties nd review some exmples of ppliction. We get in prticulr Hopf lgers on heps of pieces nd retrieve some deformed versions of the noncommuttive Fà di Bruno lger introduced y Foissy. Contents Introduction 2 1. Algeric structures nd ckground Comintoril Hopf lgers The nturl Hopf lger of n operd PROs nd free PROs 6 2. From PROs to comintoril Hopf lgers The Hopf lger of free PRO Properties of the construction The Hopf lger of stiff PRO Relted constructions Exmples of ppliction of the construction Hopf lgers of forests The Fà di Bruno lger nd its deformtions Hopf lger of forests of itrees Hopf lger of heps of pieces Hopf lger of heps of frile pieces 31 Concluding remrks nd perspectives 33 References 34 Dte: My 12, Key words nd phrses. Operd; PRO; Hopf lger; Noncommuttive symmetric function; Fà di Bruno lger. Phone numer nd emil ddress of the corresponding uthor: , smuele.girudo@univ-mlv.fr. 1

2 2 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO Introduction Operds re lgeric structures introduced in the 70s y Bordmn nd Vogt [BV73] nd y My [My72] in the context of lgeric topology to offer formliztion of the notion of opertors nd their composition (see [Mr08] nd [LV12] for very complete presenttion of the theory of operds). Operds provide therefore unified frmework to study some sorts of priori very different lgers, like ssocitive lgers, Lie lgers, nd commuttive lgers. Besides, the theory of operds is lso eneficil in comintorics [Ch08] since it provides some wys to decompose comintoril ojects into elementry pieces. On the other hnd, the theory of Hopf lgers holds specil plce in lgeric comintorics [JR79]. In recent yers, mny Hopf lgers were defined nd studied nd most of these involve very fmous comintoril ojects like permuttions [MR95, DHT02], stndrd Young tleux [PR95, DHT02], or inry trees [LR98, HNT05]. These two theories operds nd Hopf lgers hve severl interctions. One of these is construction [vdl04] tking n operd O s input nd producing ilger H (O) s output, tht is clled the nturl ilger of O. This construction hs een studied in some recent works: in [CL07], it is shown tht H cn e rephrsed in terms of n incidence Hopf lger of certin fmily of posets, nd in [ML14], generl formul for its ntipode is estlished. Let us lso cite [Fr08] in which this construction is considered to study series of trees. The initil motivtion of our work ws to generlize this H construction with the im of constructing some new nd interesting Hopf lgers. The direction we hve chosen is to strt with PROs, lgeric structures tht generlize operds in the sense tht PROs del with opertors with possily severl outputs. Surprisingly, these structures ppered erlier thn operds in the work of Mc Lne [ML65]. Intuitively, PRO P is set of opertors together with two opertions: n horizontl composition nd verticl composition. The first opertion tkes two opertors x nd y of P nd uilds new one whose inputs (resp. outputs) re, from left to right, those of x nd then those of y. The second opertion tkes two opertors x nd y of P nd produces new one otined y plugging the outputs of y onto the inputs of x. Bsic nd modern references out PROs re [Lei04] nd [Mr08]. Our min contriution consists in the definition of new construction H from PROs to ilgers. Roughly speking, the construction H cn e descried s follows. Given PRO P stisfying some mild properties, the ilger HP hs ses indexed y prticulr suset of elements of P. The product of HP is the horizontl composition of P nd the coproduct of HP is defined from the verticl composition of P, enling to seprte sis element into two smller prts. The properties stisfied y P imply, in nontrivil wy, tht the product nd the coproduct of HP stisfy the required xioms to e ilger. This construction generlizes H nd estlishes new connection etween the theory of PROs nd the theory of Hopf lgers. Our results re orgnized s follows. In Section 1, we recll some generl ckground out Hopf lgers, operds, nd PROs. In prticulr, we give description of free PROs in terms of progrphs, similr to the one of Lfont [Lf11]. We lso recll the nturl ilger construction of n operd. We provide in Section 2 the description of our new construction H. A first version of this construction is presented, ssociting ilger H(P) with free PRO P. We then present n extended version of the construction, tking s input non-necessrily free PROs stisfying some

3 COMBINATORIAL HOPF ALGEBRAS FROM PROS 3 properties, clled stiff PROs. Next, we consider two well-known constructions of PROs [Mr08], one, R, tking s input operds nd the other, B, tking s input monoids. We prove tht under some mild conditions, theses constructions produce stiff PROs. We estlish tht the nturl ilger of n operd cn e reformulted s prticulr cse of our construction H. We conclude y giving some exmples of ppliction of H in Section 3. The Hopf lgers tht we otin re very similr to the Connes-Kreimer Hopf lger [CK98] in the sense tht their coproduct cn e computed y mens of dmissile cuts in vrious comintoril ojects. From very simple stiff PROs, we reconstruct the Hopf lger of noncommuttive symmetric functions Sym [GKL + 95,KLT97] nd the noncommuttive Fà di Bruno lger FdB 1 [BFK06]. Besides, we present wy of using H to reconstruct some of the Hopf lgers FdB γ, γ-deformtion of FdB 1 introduced y Foissy [Foi08]. We lso otin severl other Hopf lgers, tht respectively involve forests of plnr rooted trees, some kind of grphs consisting of nodes with one prent nd severl children or severl prents nd one child tht we cll forests of itrees, heps of pieces (see [Vie86] for generl presenttion of these comintoril ojects), nd prticulr clss of heps of pieces tht we cll heps of frile pieces. All these Hopf lgers depend on nonnegtive integer s prmeter. Acknowledgments. The uthors would like to thnk Jen-Christophe Novelli for his dvice during the preprtion of this pper. The computtions of this work hve een done with the open-source mthemticl softwre Sge [S + 14] nd one of its extensions, Sge-Comint [SCc08]. Nottions. For ny integer n 0, [n] denotes the set {1,, n}. If u is word nd i is positive integer no greter thn the length of u, u i denotes the i-th letter of u. The empty word is denoted y ɛ. 1. Algeric structures nd ckground We recll in this preliminry section some sics out the lgeric structures in ply in ll this work, i.e., Hopf lgers, operds, nd PROs. We lso present some well-known Hopf lgers nd recll the construction ssociting comintoril Hopf lger with n operd Comintoril Hopf lgers. In the sequel, ll vector spces hve C s ground field. By lger we men unitry ssocitive lger nd y colger counitry cossocitive colger. We cll comintoril Hopf lger ny grded ilger H = n 0 H n such tht for ny n 1, the n-th homogeneous component H n of H hs finite dimension nd the dimension of H 0 is 1. The degree of ny element x H n is n nd is denoted y deg(x). Comintoril Hopf lgers re Hopf lgers ecuse the ntipode cn e defined recursively degree y degree. Let us now review some clssicl comintoril Hopf lgers tht ply n importnt role in this work Fà di Bruno lger nd its deformtions. Let FdB e the free commuttive lger generted y elements h n, n 1 with deg(h n ) = n. The ses of FdB re thus indexed y integer prtitions, nd the unit is denoted y h 0. This is the lger of symmetric functions [Mc95]. There re severl wys to endow FdB with coproduct to turn it into Hopf lger. In [Foi08],

4 4 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO Foissy otins, s yproduct of his investigtion of comintoril Dyson-Schwinger equtions in the Connes-Kreimer lger, one-prmeter fmily γ, γ R, of coproducts on FdB, defined y using lphet trnsformtions (see [Mc95]), y n γ (h n ) := h k h n k ((kγ + 1)X), (1.1.1) k=0 ( ) α. where, for ny α R nd n N, h n (αx) is the coefficient of t n in k 0 h kt k In prticulr, 0 (h n ) = n h k h n k. (1.1.2) k=0 The lger FdB with the coproduct 0 is the clssicl Hopf lger of symmetric functions Sym [Mc95]. Moreover, for ll γ 0, ll FdB γ re isomorphic to FdB 1, tht is known s the Fà di Bruno lger [JR79]. The coproduct 0 comes from the interprettion of FdB s the lger of polynomil functions on the multiplictive group (G(t) := {1+ k 1 kt k }, ) of forml power series of constnt term 1, nd 1 comes from its interprettion s the lger of polynomil functions on the group (tg(t), ) of forml diffeomorphisms of the rel line Noncommuttive nlogs. Forml power series in one vrile with coefficients in noncommuttive lger cn e composed (y sustitution of the vrile). This opertion is not ssocitive, so tht they do not form group. For exmple, when nd elongs to noncommuttive lger, one hs ut (t 2 t) t = 2 t 2 t = 2 2 t 2 (1.1.3) t 2 (t t) = t 2 t = t 2. (1.1.4) However, the nlogue of the Fà di Bruno lger still exists in this context, nd is known s the noncommuttive Fà di Bruno lger. It is investigted in [BFK06] in view of pplictions in quntum field theory. In [Foi08], Foissy lso otins n nlogue of the fmily FdB γ in this context. Indeed, considering noncommuttive genertors S n (with deg(s n ) = n) insted of the h n, for ll n 1, leds to free noncommuttive lger FdB whose ses re indexed y integer compositions. This is the lger of noncommuttive symmetric functions [GKL + 95]. The ddition of the coproduct γ defined y n γ (S n ) := S k S n k ((kγ + 1)A), (1.1.5) k=0 ( ) α, where, for ny α R nd n N, S n (αa) is the coefficient of t n in k 0 S kt k forms noncommuttive Hopf lger FdB γ. In prticulr, n 0 (S n ) = S k S n k, (1.1.6) k=0 where S 0 is the unit. In this wy, FdB with the coproduct 0 is the Hopf lger of noncommuttive symmetric functions Sym [GKL + 95,KLT97], nd for ll γ 0, ll the FdB γ re isomorphic to FdB 1, tht is the noncommuttive Fà di Bruno lger.

5 COMBINATORIAL HOPF ALGEBRAS FROM PROS The nturl Hopf lger of n operd. We shll consider in this work only nonsymmetric operds in the ctegory of sets. For this reson, we shll cll these simply operds Operds. In our context, n operd is triple (O, i, 1) where O := n 1 O(n) (1.2.1) is grded set, i : O(n) O(m) O(n + m 1), n, m 1, i [n], (1.2.2) is composition mp, clled prtil composition, nd 1 O(1) is unit. This dt hs to stisfy the reltions (x i y) i+j 1 z = x i (y j z), x O(n), y O(m), z O(k), i [n], j [m], (1.2.3) (x i y) j+m 1 z = (x j z) i y, x O(n), y O(m), z O(k), 1 i < j n, (1.2.4) 1 1 x = x = x i 1, x O(n), i [n]. (1.2.5) Besides, we shll denote y : O(n) O(m 1 ) O(m n ) O(m m n ), n, m 1,, m n 1, (1.2.6) the totl composition mp of O. It is defined for ny x O(n) nd y 1,, y n O y x [y 1,, y n ] := ( ((x n y n ) n 1 y n 1 ) ) 1 y 1. (1.2.7) If x is n element of O(n), we sy tht the rity x of x is n. An operd morphism is mp φ : O 1 O 2 etween two operds O 1 nd O 2 such tht φ commutes with the prtil composition mps nd preserves the rities. A suset S of n operd O is suoperd of O if 1 S nd the composition of O is stle in S. The suoperd of O generted y suset G of O is the smllest suoperd of O contining G The nturl ilger of n operd. Let us recll very simple construction ssociting Hopf lger with n operd. A slightly different version of this construction is considered in [vdl04, CL07, ML14]. Let O e n operd nd denote y O + the set O \ {1}. The nturl ilger of O is the free commuttive lger H (O) spnned y the T x, where the x re elements of O +. The ses of H (O) re thus indexed y finite multisets of elements of O +. The unit of H (O) is denoted y T 1 nd the coproduct of H (O) is the unique lger morphism stisfying, for ny element x of O +, (T x ) = T y T z1 T zl. (1.2.8) y,z 1,...,z l O y [z 1,...,z l ]=x The ilger H (O) cn e grded y deg(t x ) := x 1. Note tht with this grding, when O(1) = {1} nd the O(n) re finite for ll n 1, H (O) ecomes comintoril Hopf lger.

6 6 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO 1.3. PROs nd free PROs. We recll here the definitions of PROs nd free PROs in terms of progrphs nd introduce the notions of reduced nd indecomposle elements, which will e used in the following sections PROs. A PRO is qudruple (P,,, 1 p ) where P is igrded set of the form P := p 0 P(p, q), (1.3.1) q 0 such tht for ny p, q 0, P(p, q) contins elements x with i(x) := p s input rity nd o(x) := q s output rity, is mp of the form : P(p, q) P(p, q ) P(p + p, q + q ), p, p, q, q 0, (1.3.2) clled horizontl composition, is mp of the form : P(q, r) P(p, q) P(p, r), p, q, r 0, (1.3.3) clled verticl composition, nd for ny p 0, 1 p is n element of P(p, p) clled unit of rity p. These dt hve to stisfy for ll x, y, z P the six reltions (x y) z = x (y z), x, y, z P, (1.3.4) (x y) z = x (y z), x, y, z P, i(x) = o(y), i(y) = o(z), (1.3.5) (x y) (x y ) = (x x ) (y y ), x, x, y, y P, i(x) = o(y), i(x ) = o(y ), (1.3.6) 1 p 1 q = 1 p+q, p, q 0, (1.3.7) x 1 0 = x = 1 0 x, x P, (1.3.8) x 1 p = x = 1 q x, x P, p, q 0, i(x) = p, o(x) = q. (1.3.9) A PRO morphism is mp φ : P 1 P 2 etween two PROs P 1 nd P 2 such tht φ commutes with the horizontl nd verticl compositions nd preserves the input nd output rities. A suset S of PRO P is su-pro of P if 1 p S for ny p 0 nd the horizontl nd verticl compositions of P re stle in S. The su-pro of P generted y suset G of P is the smllest su-pro of P contining G. An equivlence reltion on P is congruence of PROs if ll the elements of sme -equivlence clss hve the sme input rity nd the sme output rity, nd is comptile with the horizontl nd the verticl composition. Any congruence of P gives rise to PRO quotient of P denoted y P/ nd defined in the expected wy.

7 COMBINATORIAL HOPF ALGEBRAS FROM PROS Free PROs. Let us now set our terminology out free PROs nd its elements in terms of progrphs. From now, G := G(p, q) (1.3.10) p 1 q 1 is igrded set. An elementry progrph e over G is forml opertor leled y n element of G(p, q). The input (resp. output) rity of e is p (resp. q). We represent e s rectngle leled y g with p incoming edges (elow the rectngle) nd q outgoing edges (ove the rectngle). For instnce, if G(2, 3), the elementry progrph leled y is depicted y (1.3.11) 1 2 A progrph over G is forml opertor defined recursively s follows. A progrph over G cn e either n elementry progrph over G, or specil element, the wire depicted y, (1.3.12) or comintion of two progrphs over G through the following two opertions. The first one, denoted y, consists in plcing two progrphs side y side. For instnce, if x is progrph with p inputs (resp. q outputs) nd y is progrph with p inputs (resp. q outputs), 1 q 1 q 1 q 1 q x 1 p y 1 p = x y 1 p 1 p. (1.3.13) The second one, denoted y, consists in connecting the inputs of first progrph over the outputs of second. For instnce, if x is progrph with p inputs (resp. q outputs) nd y is progrph with r inputs (resp. p outputs), 1 q 1 p 1 q 1 p 1 r = 1 r. (1.3.14) By definition, connecting the input (resp. output) of wire to the output (resp. input) of progrph x does not chnge x. The input (resp. output) rity of progrph x is its numer of inputs i(x) (resp. outputs o(x)). The inputs (resp. outputs) of progrph re numered from left to right from 1 to i(x) (resp. o(x)), possily implicitly in the drwings. The degree deg(x) of progrph x is the numer of

8 8 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO elementry progrphs required to uild it. For instnce, (1.3.15) is progrph over G := G(2, 2) G(3, 1) where G(2, 2) := {} nd G(3, 1) := {}. Its input rity is 7, its output rity is 5, nd its degree is 3. The free PRO generted y G is the PRO Free(G) whose elements re ll the progrphs on G, the horizontl composition eing the opertion on progrphs, nd the verticl composition eing the opertion. Its unit 1 1 is the wire nd for ny p 0, 1 p is the horizontl composition of p occurrences of the wire. Notice tht y (1.3.10), there is no elementry progrph in Free(G) with null input or output rity. Therefore, 1 0 is the only element of Free(G) with null input (resp. output) rity. In this work, we consider only free PROs stisfying this property. Lemm 1.1. Let P e free PRO nd x, y, z, t P such tht x y = z t. Then, there exist four unique elements x 1, x 2, y 1, y 2 of P such tht x = x 1 x 2, y = y 1 y 2, z = x 1 y 1, nd t = x 2 y 2. Proof. Let us prove the uniqueness. Assume tht there re x 1, x 2, y 1, y 2 P nd x 1, x 2, y 1, y 2 P such tht x = x 1 x 2 = x 1 x 2, y = y 1 y 2 = y 1 y 2, z = x 1 y 1 = x 1 y 1, nd t = x 2 y 2 = x 2 y 2. Then, we hve in prticulr o(x 1 ) = o(x 1) nd o(y 1 ) = o(y 1). This, together with the reltion x 1 y 1 = x 1 y 1 nd the fct tht P is free implies x 1 = x 1 nd y 1 = y 1. In the sme wy, we otin x 2 = x 2 nd y 2 = y 2. Let us now give geometricl proof for the existence sed upon the fct tht progrphs re plnr ojects. Since u := x y = z t, u is progrph otined y n horizontl composition of two progrphs. Then, u = x y depicted in plne P cn e split into two regions X nd Y such tht X contins the progrph x, Y contins the progrph y, nd y is t the right of x. On the other hnd, u = z t depicted in the sme plne P cn e split into two regions Z nd T such tht Z contins the progrph z, T contins the progrph t, nd t is elow z, the inputs of z eing connected to the outputs of t. We then otin division of P into four regions X Z, X T, Y Z, nd Y T, respectively contining progrphs x 1, x 2, y 1, nd y 2, nd such tht x 1 y 1 = z, x 2 y 2 = t, x 1 x 2 = x, nd y 1 y 2 = y Reduced nd indecomposle elements. Let P e free PRO. Since P is free, ny element x of P cn e uniquely written s x = x 1 x l where the x i re elements of P different from 1 0 nd l 0 is mximl. We cll the word dec(x) := (x 1,, x l ) the mximl decomposition of x nd the x i the fctors of x. Notice tht the mximl decomposition of 1 0 is the empty word. We hve, for instnce, dec =,,. (1.3.16)

9 COMBINATORIAL HOPF ALGEBRAS FROM PROS 9 An element x of P is reduced if ll its fctors re different from 1 1. For ny element x of P, we denote y red(x) the reduced element of P dmitting s mximl decomposition the longest suword of dec(x) consisting in fctors different from 1 1. We hve, for instnce, red =. (1.3.17) By extension, we denote y red(p) the set of the reduced elements of P. Note tht 1 0 elongs to red(p). Besides, we sy tht n element x of P is indecomposle if its mximl decomposition consists in exctly one fctor. Note tht 1 0 is not indecomposle while 1 1 is. Lemm 1.2. Let P e free PRO nd x, y P, such tht x = red(y). Then, y denoting y (x 1,, x l ) the mximl decomposition of x, there exists unique sequence of nonnegtive integers p 1,, p l, p l+1 such tht y = 1 p1 x 1 1 p2 x 2 x l 1 pl+1. (1.3.18) Proof. The existence comes from the fct tht, since x = red(y), the mximl decomposition of x is otined from the one of y y suppressing the fctors equl to the wire. The uniqueness comes from the fct tht P is free monoid for the horizontl composition. 2. From PROs to comintoril Hopf lgers We introduce in this section the min construction of this work nd review some of its properties. In ll this section, P is free PRO generted y igrded set G. Strting with P, our construction produces ilger H(P) whose ses re indexed y the reduced elements of P. We shll lso extend this construction over clss of non necessrily free PROs The Hopf lger of free PRO. The ses of the vector spce H(P) := Vect(red(P)) (2.1.1) re indexed y the reduced elements of P. The elements S x, x red(p), form thus sis of H(P), clled fundmentl sis. We endow H(P) with product : H(P) H(P) H(P) linerly defined, for ny reduced elements x nd y of P, y S x S y := S x y, (2.1.2) nd with coproduct : H(P) H(P) H(P) linerly defined, for ny reduced elements x of P, y (S x ) := S red(y) S red(z). (2.1.3) y,z P y z=x

10 10 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO Throughout this section, we shll consider some exmples involving the free PRO generted y G := G(2, 2) G(3, 1) where G(2, 2) := {} nd G(3, 1) := {}, denoted y AB. For instnce, we hve in H(AB) S S = S (2.1.4) nd S = S 10 S + S S + S S + S S + S S + S S 10. (2.1.5) Lemm 2.1. Let P e free PRO. Then, the coproduct of H(P) is cossocitive. Proof. The ses of the vector spce Vect(P) re indexed y the (non-necessrily reduced) elements of P. Then, the elements R x, x P, form sis of Vect(P). Let us consider the coproduct defined, for ny x P, y (R x ) := y,z P y z=x R y R z. (2.1.6) The ssocitivity of the verticl composition of P (see (1.3.5)) implies tht is cossocitive nd hence, tht Vect(P) together with form colger. Consider now the mp φ : Vect(P) H(P) defined, for ny x P, y φ(r x ) := S red(x). Let us show tht φ commutes with the coproducts nd, tht is, (φ φ) = φ. Let x P. By Lemm 1.2, y denoting y (x 1,, x l ) the mximl decomposition of red(x), there is unique wy to write x s x = 1 p1 x 1 x l 1 l+1 where the p i re some integers. Then, thnks to

11 COMBINATORIAL HOPF ALGEBRAS FROM PROS 11 the ssocitivity of (see (1.3.4)), y itertively pplying Lemm 1.1, we hve (φ φ) (R x ) = S red(y) S red(z) (2.1.7) = = = y,z P y z=x y 1,...,y l P z 1,...,z l P (1 p1 y 1 y l 1 l+1 ) (1 p1 z 1 z l 1 l+1 )= 1 p1 x 1 x l 1 l+1 y 1,...,y l P z 1,...,z l P (y 1 y l ) (z 1 z l )=x 1 x l y,z P y z=red(x) S red(1p1 y 1 y l 1 l+1 ) S red(1p1 z 1 z l 1 l+1 ) (2.1.8) S red(y1 y l ) S red(z1 z l ) (2.1.9) S red(y) S red(z) (2.1.10) = (φ (R x )). (2.1.11) Now, the cossocitivity of comes from the fct tht φ is surjective mp commuting with nd. In more detils, if x is reduced element of P nd I is the identity mp on H(P), we hve ( I) (S x ) = ( I) (φ (R x )) (2.1.12) = (φ φ φ)( I) (R x ) (2.1.13) = (φ φ φ)(i ) (R x ) (2.1.14) = (I ) (φ (R x )) (2.1.15) = (I ) (S x ). (2.1.16) Lemm 2.2. Let P e free PRO. Then, the coproduct of H(P) is morphism of lgers. Proof. Let x nd y e two reduced elements of P. We hve (S x S y ) = S red(z) S red(t) (2.1.17) nd (S x ) (S y ) = x 1,x 2,y 1,y 2 P x 1 x 2=x y 1 y 2=y z,t P z t=x y S red(x1) red(y 1) S red(x2) red(y 2). (2.1.18) The coproduct is morphism of lgers if (2.1.17) nd (2.1.18) re equl. Let us show tht it is the cse. Assume tht there re two elements z nd t of P such tht z t = x y. Then, the pir (z, t) contriutes to the coefficient of the tensor S red(z) S red(t) in (2.1.17). Moreover, since P is free, y Lemm 1.1, there exist four unique elements x 1, x 2, y 1, nd y 2 of P such tht x = x 1 x 2,

12 12 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO y = y 1 y 2, z = x 1 y 1, nd t = x 2 y 2. Then, since red(x 1 ) red(y 1 ) = red(x 1 y 1 ) = red(z) nd red(x 2 ) red(y 2 ) = red(x 2 y 2 ) = red(t), the qudruple (x 1, x 2, y 1, y 2 ), wholly nd uniquely determined y the pir (z, t), contriutes to the coefficient of the tensor S red(z) S red(t) in (2.1.18). Conversely, ssume tht there re four elements x 1, x 2, y 1, nd y 2 in P such tht x 1 x 2 = x nd y 1 y 2 = y. Then, the qudruple (x 1, x 2, y 1, y 2 ) contriutes to the coefficient of the tensor S red(x1) red(y 1) S red(x2) red(y 2) in (2.1.18). Now, y (1.3.6), we hve x y = (x 1 x 2 ) (y 1 y 2 ) = (x 1 y 1 ) (x 2 y 2 ). (2.1.19) Then, since red(x 1 ) red(y 1 ) = red(x 1 y 1 ) nd red(x 2 ) red(y 2 ) = red(x 2 y 2 ), the pir (x 1 y 1, x 2 y 2 ), wholly nd uniquely determined y the qudruple (x 1, x 2, y 1, y 2 ), contriutes to the coefficient of the tensor S red(x1) red(y 1) S red(x2) red(y 2) in (2.1.17). Hence, the coefficient of ny tensor is the sme in (2.1.17) nd in (2.1.18). expressions re equl. Then, these Theorem 2.3. Let P e free PRO. Then H(P) is ilger. Proof. The ssocitivity of comes directly from the ssocitivity of the horizontl composition of P (see (1.3.4)). Moreover, y Lemms 2.1 nd 2.2, the coproduct of H(P) is cossocitive nd is morphism of lgers. Thus, H(P) is ilger Properties of the construction. Let us now study the generl properties of the ilgers otined y the construction H Algeric genertors nd freeness. Proposition 2.4. Let P e free PRO. Then H(P) is freely generted s n lger y the set of ll S g, where the g re indecomposle nd reduced elements of P. Proof. Any reduced element x of P cn e written s x = x 1 x l where (x 1,, x l ) is the mximl decomposition of x. This implies tht in H(P), we hve S x = S x1 S xl. Since for ll i [l], the S xi re indecomposle nd reduced elements of P, the set of ll S g genertes H(P). The uniqueness of the mximl decomposition of x implies tht H(P) is free on the S g Grdings. There re severl wys to define grdings for H(P) to turn it into comintoril Hopf lger. For this purpose, we sy tht mp ω : red(p) N is grding of P if it stisfies the following four properties: (G1) for ny reduced elements x nd y of P, ω(x y) = ω(x) + ω(y); (G2) for ny reduced elements x of P stisfying x = y z where y, z P, ω(x) = ω(red(y)) + ω(red(z)); (G3) for ny n 0, the fier ω 1 (n) is finite; (G4) ω 1 (0) = {1 0 }.

13 COMBINATORIAL HOPF ALGEBRAS FROM PROS 13 A very generic wy to endow P with grding consists in providing mp ω : G N \ {0} ssociting positive integer with ny genertor of P, nmely its weight; the degree ω(x) of ny element x of P eing the sum of the weights of the occurrences of the genertors used to uild x. For instnce, the mp ω defined y ω() := 3 nd ω() := 2 is grding of AB nd we hve ω = 8. (2.2.1) Proposition 2.5. Let P e free PRO nd ω e grding of P. Then, with the grding H(P) = n 0 Vect (S x : x red(p) nd ω(x) = n), (2.2.2) H(P) is comintoril Hopf lger. Proof. Notice first tht S 10 is the neutrl element of the product of H(P) nd, for ny reduced element x of P of degree different from 0, the coproduct (S x ) contins the tensors S 10 S x nd S x S 10. Then, together with the fct tht y (G4), S 10 is n element of the homogeneous component of degree 0 of H(P), H(P) dmits unit nd counit. Moreover, (G3) implies tht for ll n 0, the homogeneous components of degree n of H(P) re finite, nd (G4) implies tht H(P) is connected. Besides, respectively y (G1) nd y (G2), the grding provided y ω is comptile with the product nd the coproduct of H(P). Hence, together with the fct tht, y Theorem 2.3, H(P) is ilger, it is lso comintoril Hopf lger Antipode. Since the ntipode of comintoril Hopf lger cn e computed y induction on the degree, we otin n expression for the one of H(P) when P dmits grding. This expression is n instnce of the Tkeuchi formul [Tk71] nd is prticulrly simple since the product of H(P) is multiplictive. Proposition 2.6. Let P e free PRO dmitting grding. For ny reduced element x of P different from 1 0, the ntipode ν of H(P) stisfies ν(s x ) = ( 1) l S red(x1 x l ). (2.2.3) x 1,...,x l P,l 1 x 1 x l =x red(x i) 1 0,i [l] Proof. By Proposition 2.5, H(P) is comintoril Hopf lger nd hence, the ntipode ν, which is the inverse of the identity morphism for the convolution product, exists nd is unique. Then, for ny reduced element x of P different from 1 0, ν(s x ) = S x ν ( ) S red(y) Sred(z). (2.2.4) y,z P\{x} y z=x Expression (2.2.3) for ν follows now y induction on the degree of x in P.

14 14 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO We hve for instnce in H(AB), νs = S + S + S S. (2.2.5) Dulity. When P dmits grding, let us denote y H(P) the grded dul of H(P). By definition, the djoint sis of the fundmentl sis of H(P) consists in the elements S x, x red(p). Proposition 2.7. Let P e free PRO dmitting grding. Then, for ny reduced elements x nd y of P, the product nd the coproduct of H(P) stisfy S x S y = x,y P x y red(p) red(x )=x,red(y )=y S x y (2.2.6) nd (S x) = y,z P y z=x S y S z. (2.2.7) Proof. Let us denote y, : H(P) H(P) C the dulity rcket etween H(P) nd its grded dul. By dulity, we hve S x S y = z red(p) (Sz ), S x S y S z. (2.2.8) Expression (2.2.6) follows from the fct tht for ny reduced element z of P, S x S y ppers in (S z ) if nd only if there exist x, y P such tht x y = z, red(x ) = x nd red(y ) = y. Besides, gin y dulity, we hve (S x) = y,z red(p) S y S z, S x S y S z. (2.2.9) Expression (2.2.7) follows from the fct tht for ny reduced elements y nd z of P, S x ppers in S y S z if nd only if y z = x.

15 COMBINATORIAL HOPF ALGEBRAS FROM PROS 15 For instnce, we hve in H(AB) S S = S + S + S + S + S + S + S + 2 S (2.2.10) nd S = S 1 0 S + S S + S S + S S 1 0. (2.2.11) Quotient ilgers. Proposition 2.8. Let G nd G e two igrded sets such tht G G. Then, the mp φ : H(Free(G)) H(Free(G )) linerly defined, for ny reduced element x of Free(G), y { Sx if x Free(G ), φ(s x ) := (2.2.12) 0 otherwise, is surjective ilger morphism. Moreover, H(Free(G )) is quotient ilger of H(Free(G)). Proof. Let V e the liner spn of the S x where the x re reduced elements of Free(G)\Free(G ). Immeditely from the definitions of the product nd the coproduct of H(Free(G)), we oserve tht V is ilger idel of H(Free(G)). The mp φ is the cnonicl projection from H(Free(G)) to H(Free(G))/ V H(Free(G )), whence the result The Hopf lger of stiff PRO. We now extend the construction H to clss nonnecessrily free PROs. Still in this section, P is free PRO. Let e congruence of P. For ny element x of P, we denote y [x] (or y [x] if the context is cler) the -equivlence clss of x. We sy tht is stiff congruence if the following three properties re stisfied: (C1) for ny reduced element x of P, the set [x] is finite; (C2) for ny reduced element x of P, [x] contins reduced elements only;

16 16 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO (C3) for ny two elements x nd x of P such tht x x, the mximl decompositions of x nd x re respectively of the form (x 1,, x l ) nd (x 1,, x l ) for some l 0, nd for ny i [l], x i x i. We sy tht PRO is stiff PRO if it is the quotient of free PRO y stiff congruence. For ny -equivlence clss [x] of reduced elements of P, set T [x] := S x. (2.3.1) x [x] Notice tht thnks to (C1) nd (C2), T [x] is well-defined element of H(P). For instnce, if P is the quotient of the free PRO generted y G := G(1, 1) G(2, 2) where G(1, 1) := {} nd G(2, 2) := {} y the finest congruence stisfying, (2.3.2) one hs T = S + S + S + S. (2.3.3) Moreover, we cn oserve tht is stiff congruence. If is stiff congruence of P, (C2) nd (C3) imply tht ll the elements of sme - equivlence clss [x] hve the sme numer of fctors nd re ll reduced or ll nonreduced. Then, y extension, we shll sy tht -equivlence clss [x] of P/ is indecomposle (resp. reduced) if ll its elements re indecomposle (resp. reduced) in P. In the sme wy, the wire of P/ is the -equivlence clss of the wire of P. We shll now study how the product nd the coproduct of H(P) ehve on the T [x] Product. Let us show tht the liner spn of the T [x], where the [x] re -equivlence clsses of reduced elements of P, forms sulger of H(P). The product on the T [x] is multiplictive nd dmits the following simple description. Proposition 2.9. Let P e free PRO nd e stiff congruence of P. Then, for ny - equivlence clsses [x] nd [y], where x (resp. y) is ny element of [x] (resp. [y]). T [x] T [y] = T [x y], (2.3.4) Proof. We hve T [x] T [y] = S x y (2.3.5) x [x] y [y]

17 COMBINATORIAL HOPF ALGEBRAS FROM PROS 17 nd T [x y] = z [x y] S z. (2.3.6) Let us show tht (2.3.5) nd (2.3.6) re equl. It is enough to check tht these sums hve the sme support. Indeed, (2.3.6) is y definition multiplicity free nd (2.3.5) is multiplicity free ecuse P is free nd ll elements of -equivlence clss hve the sme input rity nd the sme output rity. Assume tht there is reduced element t of P such tht S t ppers in (2.3.5). Then, one hs t = x y for two reduced elements x nd y of P such tht x [x] nd y [y]. Since is congruence of PROs, we hve [x y ] = [x y] nd thus, t [x y]. This shows tht S t lso ppers in (2.3.6). Conversely, ssume tht there is reduced element z of P such tht S z ppers in (2.3.6). Then, one hs z [x y]. Since stisfies (C3), the mximl decomposition of z stisfies dec(z) = (x 1,, x k, y 1,, y l ) where dec(x) = (x 1,, x k ), dec(y) = (y 1,, y l ), x i x i, nd y j y j for ll i [k] nd j [l]. Moreover, s is congruence of PROs, x := x 1 x k x nd y := y 1 y l y. We then hve z = x y with x [x] nd y [y]. This shows tht S z lso ppers in (2.3.5) Coproduct. To prove tht the liner spn of the T [x], where the [x] re -equivlence clsses of reduced elements of P, forms sucolger of H(P) nd provide the description of the coproduct of T [x], we need the following nottion. For ny element x of P, red ([x]) := {red (x ) : x [x]}. (2.3.7) Lemm Let P e free PRO nd e stiff congruence of P. For ny element x of P, red ([x]) = [red(x)]. (2.3.8) Proof. Let us denote y y the element red(x) nd let y red([x]). Let us show tht y [red(x)]. By Definition (2.3.7), there is n element x of P such tht x [x] nd red(x ) = y. Since stisfies (C3), dec(x ) nd dec(x) hve the sme length l nd dec(x ) i dec(x) i for ll i [l]. Moreover, since stisfies (C2), for ll i [l], dec(x ) i nd dec(x) i re oth reduced elements or re oth wires. Hence, since dec(y ) nd dec(y) re respectively suwords of dec(x ) nd dec(x), they hve the sme length k nd dec(y ) j dec(y) j for ll j [k]. Finlly, since is congruence of PROs, y y. This shows tht y [red(x)] nd hence, red([x]) [red(x)]. Agin, let us denote y y the element red(x) nd let y [red(x)]. Let us show tht y red([x]). Since y y nd stisfies (C3), dec(y ) nd dec(y) hve the sme length l nd dec(y ) i dec(y) i for ll i [l]. Moreover, since y = red(x), y Lemm 1.2, for some p 1,, p l+1 0, we hve x = 1 p1 dec(y) 1 1 pl dec(y) l 1 pl+1. Now, y setting x := 1 p1 dec(y ) 1 1 pl dec(y ) l 1 pl+1, the fct tht is congruence of PROs implies x x. Since y = red(x ), this shows tht y red([x]) nd hence, [red(x)] red([x]).

18 18 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO Lemm Let P e free PRO, e stiff congruence of P, nd y nd z e two elements of P such tht y z. Then, red(y) = red(z) implies y = z. Proof. By contrposition, ssume tht y z. Since y z nd stisfies (C3), dec(y) nd dec(z) hve the sme length l, nd dec(y) i dec(z) i for ll i [l]. Moreover, s y z, there exists j [l] such tht dec(y) j dec(z) j. Now, since stisfies (C2), dec(y) j nd dec(z) j re oth reduced elements. Moreover, for ll i [l], dec(y) i nd dec(z) i re oth reduced elements or re oth wires. Hence, there is j 1 such tht dec(red(y)) j = dec(y) j nd dec(red(z)) j = dec(z) j. Since P is free, this implies tht red(y) red(z). Proposition Let P e free PRO nd e stiff congruence of P. Then, for ny -equivlence clss [x], ( ) T [x] = T red([y]) T red([z]). (2.3.9) [y],[z] P/ [y] [z]=[x] Proof. We hve ( T [x] ) = y,z P y z [x] = = = = y,z P [y] [z]=[x] [y],[z] P/ [y] [z]=[x] [y],[z] P/ [y] [z]=[x] S red(y) S red(z) (2.3.10) S red(y) S red(z) (2.3.11) y [y] z [z] y red([y]) z red([z]) S red(y ) S red(z ) (2.3.12) S y S z (2.3.13) T red([y]) T red([z]). (2.3.14) [y],[z] P/ [y] [z]=[x] Let us comment the non-ovious equlities ppering in this computtion. The equlity etween (2.3.10) nd (2.3.11) comes from the fct tht is congruence of PROs. The equlity etween (2.3.12) nd (2.3.13) is consequence of Lemm Finlly, (2.3.13) is, thnks to Lemm 2.10, equl to (2.3.14) Su-ilger. The description of the product nd the coproduct on the T [x] leds to the following result. Theorem Let P e free PRO nd e stiff congruence of P. Then, the liner spn of the T [x], where the [x] re -equivlence clsses of reduced elements of P, forms su-ilger of H(P).

19 COMBINATORIAL HOPF ALGEBRAS FROM PROS 19 Proof. By Propositions 2.9 nd 2.12, the product nd the coproduct of H(P) re still well-defined on the T [x]. Then, since the T [x] re y (2.3.1) sums of some S x, this implies the sttement of the theorem. We shll denote, y slight use of nottion, y H(P/ ) the su-ilger of H(P) spnned y the T [x], where the [x] re -equivlence clsses of reduced elements of P. Notice tht the construction H s it ws presented in Section 2.1 is specil cse of this ltter when is the most refined congruence of PROs. Note tht this construction of su-ilgers of H(P) y tking n equivlence reltion stisfying some precise properties nd y considering the elements otined y summing over its equivlence clsses is nlog to the construction of certin su-ilgers of the Mlvenuto- Reutenuer Hopf lger [MR95]. Indeed, some fmous Hopf lgers re otined in this wy, s the Lody-Ronco Hopf lger [LR98] y using the sylvester monoid congruence [HNT05], or the Poirier-Reutenuer Hopf lger [PR95] y using the plctic monoid congruence [DHT02,HNT05] The importnce of the stiff congruence condition. Let us now explin why the stiff congruence condition required s premise of Theorem 2.13 is importnt y providing n exmple of non-stiff congruence of PROs filing to produce ilger. Consider the PRO P quotient of the free PRO generted y G := G(1, 1) G(2, 2) where G(1, 1) := {} nd G(2, 2) := {} y the finest congruence stisfying (2.3.15) Here, is not stiff congruence since it stisfies (C2) ut not (C3). We hve T [ ] T [ S = S (2.3.16) ] = S ut this lst element cnnot e expressed on the T [x].

20 20 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO Besides, y strightforwrd computtion, we hve T = S + S + S = T [10] T + T T [10] + 2 T T + 2 T T + S S + S S, (2.3.17) showing tht the coproduct is neither well-defined on the T [x] Properties. By using similr rguments s those used to estlish Proposition 2.4 together with the fct tht stisfies (C3) nd the product formul of Proposition 2.9, we otin tht H(P/ ) is freely generted s n lger y the T [x] where the [x] re -equivlence clsses of indecomposle nd reduced elements of P. Moreover, when ω is grding of P so tht ll elements of sme -equivlence clss hve the sme degree, the ilger H(P/ ) is grded y the grding inherited from the one of H(P) nd forms hence comintoril Hopf lger. Proposition Let P e free PRO nd 1 nd 2 e two stiff congruences of P such tht 1 is finer thn 2. Then, H (P/ 2 ) is su-ilger of H (P/ 1 ). Proof. Since 1 is finer tht 2, ny T [x] 2, where [x] 2 is 2 -equivlence clss of reduced elements of P, is sum of some T [x ] 1. More precisely, we hve T [x] 2 = [x ] 1 [x] 2 T [x ] 1, (2.3.18) implying the result Relted constructions. In this section, we first descrie two constructions llowing to uild stiff PROs. The min interest of these constructions is tht the otined stiff PROs cn e plced t the input of the construction H. We next present wy to recover the nturl Hopf lger of n operd through the construction H nd the previous constructions of stiff PROs.

21 COMBINATORIAL HOPF ALGEBRAS FROM PROS From operds to stiff PROs. Any operd O gives nturlly rise to PRO R(O) whose elements re sequences of elements of O (see [Mr08]). We recll here this construction. Let us set R(O) := p 0 q 0 R(O)(p, q) where R(O)(p, q) := {x 1 x q : x i O(p i ) for ll i [q] nd p p q = p}. (2.4.1) The horizontl composition of R(O) is the conctention of sequences nd the verticl composition of R(O) comes directly from the composition mp of O. More precisely, for ny x 1 x r R(O)(q, r) nd y 11 y 1q1 y r1 y rqr R(O)(p, q), we hve x 1 x r y 11 y 1q1 y r1 y rqr := x 1 [y 11,, y 1q1 ] x r [y r1,, y rqr ], (2.4.2) where for ny i [r], x i O(q i ) nd the occurrences of in the right-memer of (2.4.2) refer to the totl composition mp of O. For instnce, if O is the free operd generted y genertor of rity 2, O is n operd involving inry trees. Then, the elements of the PRO R(O) re forests of inry trees. The horizontl composition of R(O) is the conctention of forests nd the verticl composition F 1 F 2 in R(O), defined only etween two forests F 1 nd F 2 such tht the numer of leves of F 1 is the sme s the numer of trees in F 2, consists in the forest otined y grfting, from left to right, the roots of the trees of F 2 on the leves of F 1. Proposition Let O e n operd such tht the monoid (O(1), 1 ) does not contin ny nontrivil sugroup. Then, R(O) is stiff PRO. Proof. As ny operd, O is the quotient y certin operdic congruence of the free operd generted y certin set of genertors G. It follows directly from the definition of the construction R tht the PRO R(O) is the quotient y the congruence of PROs of the free PRO generted y G where { G(p) if p 1 nd q = 1, G (p, q) := (2.4.3) otherwise, nd is the finest congruence of PROs stisfying x y for ny reltion x y etween elements x nd y of the free operd generted y G. Since y hypothesis (O(1), 1 ) does not contin ny nontrivil sugroup, for ll elements x nd y of O(1) \ {1}, x 1 y 1. Then, stisfies (C2). Moreover, y definition of R, stisfies (C3). Hence, R(O) is stiff PRO From monoids to stiff PROs. Any monoid M cn e seen s n operd concentrted in rity one. Then, strting from monoid M, one cn construct PRO B(M) y pplying the construction R on M seen s n operd. This construction cn e rephrsed s follows. We hve B(M) = p 0 q 0 B(M)(p, q) where B(M)(p, q) = { {x1 x p : x i M for ll i [p]} if p = q, otherwise. (2.4.4)

22 22 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO The horizontl composition of B(M) is the conctention of sequences nd the verticl composition : B(M)(p, p) B(M)(p, p) B(M)(p, p) of B(M) stisfies, for ny x 1 x p B(M)(p, p) nd y 1 y q B(M)(q, q), where is the product of M. x 1 x p y 1 y p = (x 1 y 1 ) (x p y p ), (2.4.5) For instnce, if M is the dditive monoid of nturl numers, the PRO B(M) contins ll words over N. The horizontl composition of B(M) is the conctention of words nd the verticl composition of B(M), defined only on words with sme length, is the componentwise ddition of their letters. Proposition Let M e monoid tht does not contin ny nontrivil sugroup. Then, B(M) is stiff PRO. Proof. Since M does not contins ny nontrivil sugroup, seen s n operd, the elements of rity one of M does not contins ny nontrivil sugroup. Hence, y definition of the construction B pssing y R nd y Proposition 2.15, B(M) is stiff PRO The nturl Hopf lger of n operd. We cll eliniztion of ilger H the quotient of H y the ilger idel spnned y the x y y x for ll x, y H. Here is the link etween our construction H nd the construction H. Proposition Let O e n operd such tht the monoid (O(1), 1 ) does not contin ny nontrivil sugroup. Then, the ilger H (O) is the eliniztion of H(R(O)). Proof. By Proposition 2.15, R(O) is stiff PRO nd then, y Theorem 2.13, H(R(O)) is ilger. By construction, this ilger is freely generted y the T x where x O. Hence, the mp φ : H(R(O)) H (O) defined for ny x O y φ(t x ) := T x cn e uniquely extended into ilger morphism, which we denote lso y φ. Since H (O) is generted y the T x where x O, φ is surjective. Directly from the definition of the construction H, we oserve tht the kernel of φ is the ilger idel I spnned y the T x y T y x for ll x, y R(O). Then, the ssocited mp φ I : H(R(O))/ I H (O) is ilger isomorphism. 3. Exmples of ppliction of the construction We conclude this pper y presenting exmples of ppliction of the construction H. The PROs considered in this section fit into the digrm represented y Figure 1 nd the otined Hopf lgers fit into the digrm represented y Figure Hopf lgers of forests. We present here the construction of two Hopf lgers of forests, one depending on nonnegtive integer γ, nd with different grdings. The PRO we shll define in this section will intervene in the next exmples.

23 COMBINATORIAL HOPF ALGEBRAS FROM PROS 23 FBT γ Hep γ PRF γ BAs γ As γ FHep γ Figure 1. Digrm of PROs where rrows (resp. ) re injective (resp. surjective) PRO morphisms. The prmeter γ is positive integer. When γ = 0, PRF 0 = As 0 = Hep 0 = FHep 0 nd FBT 0 = BAs 0. H(FBT γ ) H(Hep γ ) H(PRF γ ) H(BAs γ ) H(As γ ) H(FHep γ ) Figure 2. Digrm of comintoril Hopf lgers where rrows (resp. ) re injective (resp. surjective) PRO morphisms. The prmeter γ is positive integer. When γ = 0, H(PRF 0 ) = H(As 0 ) = H(Hep 0 ) = H(FHep 0 ) nd H(FBT 0 ) = H(BAs 0 ) PRO of forests with fixed rity. Let γ e nonnegtive integer nd PRF γ e the free PRO generted y G := G(γ + 1, 1) := {}, with the grding ω defined y ω() := 1. Any progrph x of PRF γ cn e seen s plnr forest of plnr rooted trees with only internl nodes of rity γ + 1. Since the reduced elements of PRF γ hve no wire, they re encoded y forests of nonempty trees Hopf lger. By Theorem 2.3 nd Proposition 2.5, H(PRF γ ) is comintoril Hopf lger. By Proposition 2.4, s n lger, H(PRF γ ) is freely generted y the S T, where the T re nonempty plnr rooted trees with only internl nodes of rity γ + 1. Its ses re indexed y plnr forests of such trees where the degree of sis element S F is the numer of internl nodes of F. Notice tht the ses of H(PRF 0 ) re indexed y forests of liner trees nd tht H(PRF 0 ) nd Sym re trivilly isomorphic s comintoril Hopf lgers Coproduct. By definition of the construction H, the coproduct of H(PRF γ ) is given on genertor S T y (S T ) = S T S T/T, (3.1.1) T Adm(T ) where Adm(T ) is the set of dmissile cuts of T, tht is the empty tree or the sutrees of T contining the root of T nd where T/ T denotes the forest consisting in the mximl sutrees

24 24 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO of T whose roots re leves of T, y respecting the order of these leves in T nd y removing the empty trees. For instnce, we hve S = S S + S S + S S + S S + S S + S S + S S. (3.1.2) This coproduct is similr to the one of the noncommuttive Connes-Kreimer Hopf lger CK [CK98]. The min difference etween H(PRF γ ) nd CK lies in the fct tht in coproduct of CK, the dmissile cuts cn chnge the rity of some internl nodes; it is not the cse in H(PRF γ ) ecuse for ny T Adm(T ), ny internl node x of T hs the sme rity s it hs in T Dimensions. The series of the lgeric genertors of H(PRF γ ) is G(t) := n 1 1 nγ + 1 ( n(γ + 1) n ) t n (3.1.3) since its coefficients re the Fuss-Ctln numers, counting plnr rooted trees with n internl nodes of rity γ + 1. Since H(PRF γ ) is free s n lger, its Hilert series is H(t) := 1 1 G(t). The first dimensions of H(PRF 1 ) re nd those of H(PRF 2 ) re 1, 1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, (3.1.4) 1, 4, 19, 98, 531, 2974, 17060, 99658, , , (3.1.5) The first sequence is listed in [Slo] s Sequence A nd the second, s Sequence A PRO of generl forests. We denote y PRF the free PRO generted y G := n 1 G(n, 1) := n 1 { n }. Any progrph x of PRF cn e seen s plnr forest of plnr rooted trees. Since the reduced elements of PRF hve no wire, they re encoded y forests of nonempty trees. Oserve tht for ny nonnegtive integer γ, PRF γ is su-pro of PRF.

25 COMBINATORIAL HOPF ALGEBRAS FROM PROS Hopf lger. By Theorem 2.3, H(PRF ) is ilger. By Proposition 2.4, s n lger, H(PRF ) is freely generted y the S T, where the T re nonempty plnr rooted trees. Its ses re indexed y plnr forests of such trees. Besides, y Proposition 2.8, since PRF γ is generted y suset of the genertors of PRF, H(PRF γ ) is ilger quotient of H(PRF ). Moreover, the coproduct of H(PRF ) stisfies (3.1.1). To turn H(PRF ) into comintoril Hopf lger, we cnnot consider the grding ω defined y ω( n ) := 1 ecuse there would e infinitely mny elements of degree 1. Therefore, we consider on H(PRF ) the grding ω defined y ω( n ) := n. In this wy, the degree of sis element S F is the numer of edges of the forest F. By Proposition 2.5, H(PRF ) is comintoril Hopf lger Dimensions. The series of the lgeric genertors of H(PRF ) is G(t) := ( ) 1 2n t n (3.1.6) n + 1 n n 1 since its coefficients re the Ctln numers, counting plnr rooted trees with n edges. As H(PRF ) is free s n lger, its Hilert series is 1 H(t) := 1 G(t) = 1 + ( ) 1 2n t n. (3.1.7) 2 n n 1 The dimensions of H(PRF ) re then the sme s the dimensions of H(PRF 1 ) (see (3.1.4)) The Fà di Bruno lger nd its deformtions. We shll give here method to construct the Hopf lgers FdB γ of Foissy [Foi08] from our construction H in the cse where γ is nonnegtive integer Associtive PRO. Let γ e nonnegtive integer nd As γ e the quotient of PRF γ y the finest congruence stisfying, k 1 + k 2 = γ, l 1 + l 2 = γ. (3.2.1) k1 k2 l1 l2 We cn oserve tht As γ is stiff PRO ecuse stisfies (C2) nd (C3) nd tht As 0 = PRF 0. Moreover, oserve tht, when γ 1, there is in As γ exctly one indecomposle element of rity nγ + 1 for ny n 0. We denote y α n this element. We consider on As γ the grding ω inherited from the one of PRT γ. This grding is still well-defined in As γ since ny -equivlence clss contins progrphs of sme degree nd stisfies, for ll n 0, ω(α n ) = n. Any element of As γ is then word α k1 α kl nd cn e encoded y word of nonnegtive integers k 1 k l. Since the reduced elements of As γ hve no wire, they re encoded y words of positive integers.

26 26 JEAN-PAUL BULTEL AND SAMUELE GIRAUDO Hopf lger. By Theorem 2.13 nd Proposition 2.5, H(As γ ) is comintoril Hopf lger. As n lger, H(As γ ) is freely generted y the T n, n 1, nd its ses re indexed y words of positive integers where the degree of sis element T k1...k l is k k l Coproduct. Since ny element α n of As γ decomposes into α n = x y if nd only if x = α k nd y = α i1 α ikγ+1 with i i kγ+1 = n k, y Proposition 2.12, for ny n 1, the coproduct of H(As γ ) expresses s n (T n ) = T k T i1 T ikγ+1, (3.2.2) k=0 i 1+ +i kγ+1 =n k where T 0 is identified with the unity T ɛ of H(As γ ). For instnce, in H(As 1 ), we hve (T 3 ) = T 0 T 3 + T 1 (T 0 T 2 + T 1 T 1 + T 2 T 0 ) + T 2 (T 0 T 0 T 1 + T 0 T 1 T 0 + T 1 T 0 T 0 ) + T 3 (T 0 T 0 T 0 T 0 ) = T ɛ T T 1 T 2 + T 1 T T 2 T 1 + T 3 T ɛ, (3.2.3) nd in H(As 2 ), we hve (T 3 ) = T 0 T 3 + T 1 (T 0 T 0 T 2 + T 0 T 2 T 0 + T 2 T 0 T 0 + T 0 T 1 T 1 + T 1 T 0 T 1 + T 1 T 1 T 0 ) + T 2 (T 0 T 0 T 0 T 0 T 1 + T 0 T 0 T 0 T 1 T 0 + T 0 T 0 T 1 T 0 T 0 + T 0 T 1 T 0 T 0 T 0 + T 1 T 0 T 0 T 0 T 0 ) + T 3 T 0 T 0 T 0 T 0 T 0 T 0 T 0 = T ɛ T T 1 T T 1 T T 2 T 1 + T 3 T ɛ. (3.2.4) Isomorphism with the deformtion of the noncommuttive Fà di Bruno Hopf lger. Theorem 3.1. For ny nonnegtive integer γ, the Hopf lger H(As γ ) is the deformtion of the noncommuttive Fà di Bruno Hopf lger FdB γ. Proof. Let us set σ 1 := n 0 T n. By (3.2.2), we hve (σ 1 ) = n T k T i1 T ikγ+1 (3.2.5) n 0 k=0 i 1+ +i kγ+1 =n k = T k T i1 T ikγ+1 (3.2.6) k 0 n k i 1+ +i kγ+1 =n k = T k T i1 T ikγ+1 (3.2.7) k 0 i 1+ +i kγ+1 0 = T k T i k 0 i 0 = k 0 kγ+1 (3.2.8) T k σ kγ+1 1, (3.2.9)