(Pure) transcendence bases in φ-deformed shuffle bialgebras

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(Pure) transcendence bases in ϕ-deformed shuffle bialgebras

van Chiên Bui, Gérard H.E. Duchamp, Quoc Hoan Ngô, Vincel Hoang Ngoc Minh, Christophe Tollu

To cite this version:

van Chiên Bui, Gérard H.E. Duchamp, Quoc Hoan Ngô, Vincel Hoang Ngoc Minh, Christophe Tollu.

(Pure) transcendence bases inϕ-deformed shuffle bialgebras. Seminaire Lotharingien de Combinatoire, Université Louis Pasteur, 2015, 74, pp.1-31. �hal-01171515v2�

(PURE) TRANSCENDENCE BASES IN ϕ-DEFORMED SHUFFLE BIALGEBRAS¹

V.C. BUI^‡, G.H.E. DUCHAMP^{† ‡}, Q.H. NGÔ^‡, V. HOANG NGOC MINH^{∗ ‡}, AND C. TOLLU^{† ‡}

†Université Paris XIII, 1, 93430 Villetaneuse, France

‡LIPN - UMR 7030, CNRS, 93430 Villetaneuse, France

∗Université Lille II, 1, Place Déliot, 59024 Lille, France

ABSTRACT. Computations with integro-differential operators are often carried out in an asso- ciative algebra with unit, and they are essentially non-commutative computations. By adjoin- ing a cocommutative co-product, one can have those operators act on a bialgebra isomorphic to an enveloping algebra. That gives an adequate framework for a computer-algebra im- plementation via monoidal factorization, (pure) transcendence bases and Poincaré–Birkhoff–

Witt bases.

In this paper, we systematically study these deformations, obtaining necessary and suf- ficient conditions for the operators to exist, and we give the most general cocommutative deformations of the shuffle co-product and an effective construction of pairs of bases in dual- ity. The paper ends by the combinatorial setting of local systems of coordinates on the group of group-like series.

CONTENTS

1. Introduction

2. A survey of shuffle products 3

3. Algebraic aspects ofϕ-shuffle bialgebras 5

3.1. First properties 5

3.2. Structural properties 9

3.3. Local coordinates byϕ-extended Schützenberger factorization 19

4. Conclusion 20

References 20

1. INTRODUCTION

The shuffle product first appeared in 1953 in the work of Eilenberg and Mac Lane [20].

As soon as 1954, Chen used it to express the product of iterated (path) integrals [9], and Ree, building on Friedrichs’ criterion, proved that the non-commutative generating series of iterated integrals are exponentials of Lie polynomials, thus connecting the Lie polynomials with the shuffle product [40]. In 1956, Radford proved that the Lyndon words form a (pure)

1The present work is part of a series of papers devoted to the study of the renormalization of divergent polyze- tas (at positive and at non-positive indices) via the factorization of the non-commutative generating series of polylogarithms and of harmonic sums, and via the effective construction of pairs of dual bases in duality inϕ- deformed shuffle algebras. It is a sequel to [14], and its content was presented in several seminars and meetings, including the 74th Séminaire Lotharingien de Combinatoire.

transcendence basis of the shuffle algebra [39]. The latter result is now well understood through the duality between bialgebras and enveloping algebras (see for example [41]), of which the construction in 1958 of the Poincaré–Birkhoff–Witt²–Lyndon basis by Chen, Fox and Lyndon [11] and of its dual basis by Schützenberger, via monoidal factorization [42, 41], gave a striking illustration. This pair of dual bases enabled one to factorize the diagonal series in the shuffle bialgebra and, consequently, to proceed combinatorially with the Dyson series [24] or the transport operator [23], which play a leading role in the relations between special functions involved in the theory of quantum groups [29] and in number theory [7].

In 1973, that is, within twenty years of the introduction of the shuffle product, Knutson defined the quasi-shuffle in [30], where it shows up as the inner product of functions on the symmetric groups³. This product is very similar to the Rota–Baxter operator introduced by Cartier in 1972, in his study of the so-called Baxter algebras [8]. Although the analogue of Radford’s theorem was pointed out by Malvenuto and Reutenauer [33], the factorization of the diagonal series in the quasi-shuffle bialgebra, initiated in [26, 27], has not yet been carried over to more general bialgebras.

Schützenberger’s factorization⁴[41] and its extensions have since been applied to the renor- malization of the associators [26, 27], to which matter they turned out to be central⁵.

The coefficients of these power series are polynomial functions of positive integral multi- indices of Riemann’s zeta function⁶ [31, 45], and they satisfy quadratic relations [7] which Lyndon words help explicit and explain. The latter relations can be obtained by identifying the local coordinates on a bridge equation connecting the Cauchy and the Hadamard algebras of polylogarithmic functions, and by using the factorization of the non-commutative gener- ating series of polylogarithms [25] and of harmonic sums [26, 27]. This bridge equation is mainly a consequence of the isomorphisms between the algebra of non-commutative generat- ing series of polylogarithms and the shuffle algebra on the one hand, and between the algebra of non-commutative generating series of harmonic sums and the quasi-shuffle algebra on the other hand.

As for the generalization of Schützenberger’s factorization to more general bialgebras, the key step, and the main difficulty thereof, is to decompose orthogonally such bialgebras into the Lie algebra generated by its primitive elements and the associated orthogonal ideal, as Ree was able to achieve in the case of the shuffle bialgebra [40], and to construct, whenever possible, the respective bases. In favorable cases, it is to be hoped that those bialgebras are enveloping algebras, so that the Eulerian projectors are convergent and other analytic computations can be performed.

To make that decomposition possible, one first needs to determine the Eulerian projectors by taking the logarithm of the diagonal series and second to insure their convergence. A

2From now on, Poincaré–Birkhoff–Witt will be abbreviated to PBW.

3In the present paper, that product will be referred to as the quasi-shuffle or as the stuffle product, indifferently.

4Also called MSR factorization after the names of Mélançon, Schützenberger and Reutenauer.

5These associators, which are formal power series in non-commutative variables, were introduced in quantum field theory by Drinfel’d [13]. The explicit coefficients of the universal associatorΦ_KZ are polyzetas and regularized polyzetas [31].

6These values are usually referred to as MZV’s by Zagier [45] and as polyzetas by Cartier [7].

key ingredient is the fact that the diagonal series are group-like and give a host of group- like elements by specialization, so one can use the exponential-logarithm correspondence to compute within a combinatorial Hausdorff group.

To that effect, the present work generalizes the recursive definitions of the shuffle and quasi-shuffle products given by Fliess [22] and Hoffman [28], respectively, to introduce the ϕ-deformed shuffle product, where ϕstands for an arbitrary algebra law. Recent studies on these structures can be found in [16, 35, 36].

Theseϕ-shuffle products interpolate between the classical shuffle and quasi-shuffle prod- ucts (forϕ≡ 0andϕ≡ 1, respectively), and allow a classification of the associated bialge- bras.

This paper is devoted to the combinatorics ofϕ-deformed shuffle algebras and to the ef- fective constructions of pairs of dual bases. Its organisation is as follows:

• Section 2 is a short reminder of well-known facts about the combinatorics of the q- stuffle product [4], which encompasses the shuffle [41] and the quasi-shuffle products [26, 27].

• In Section 3, we thoroughly investigate algebraic and combinatorial aspects of the ϕ-deformed shuffle products and explain how to use bases in duality to get a local system of coordinates on the (infinite-dimensional) Lie group of group-like series.

Throughout the paper, we have a particular concern for Lie series and their correspondence with the Hausdorff group.

2. ASURVEY OF SHUFFLE PRODUCTS

For standard definitions and facts pertaining to the (algebraic) combinatorics on words, we refer the reader to the classical books by Lothaire [32] and Reutenauer [41].

Throughout the paper,Kstands for a (unital, associative and commutative)Q-algebra con- taining a parameterq. In this section, we review the known combinatorics of bases in duality and local coordinates on the infinite-dimensional Lie group of group-like series (Hausdorff group). The parameter q allows for a unified treatment between shuffle (q = 0) and quasi- shuffle (q= 1) products.

LetY = {y_i}i≥1 be an alphabet, totally ordered byy₁ > y₂ > · · ·. The free monoid and the set of Lyndon words overY are denoted byY^∗ andLyn Y, respectively. The unit ofY^∗ is denoted by1Y^∗. We also writeY⁺ =Y^∗\ {1Y^∗}.

Theq-stuffle [4], which interpolates between the shuffle [40], quasi-shuffle [33] (or stuffle) and minus-stuffle products [10], forq= 0,1, and−1, respectively, is defined as follows:

u _q1_Y^∗ = 1_Y^∗ _qu=u, (1)

y_su _qy_tv =y_s(u _qy_tv) +y_t(y_su _qv) +q y_s+t(u _qv), (2) or its dual co-product, as follows, for anyy_s, y_t∈Y andu, v ∈Y^∗,

∆ _q(1_Y^∗) = 1_Y^∗⊗1_Y^∗, (3)

∆ _q(y_s) =y_s⊗1_Y^∗+ 1_Y^∗⊗y_s+q X

s1+s2=s

y_s1 ⊗y_s2. (4)

We now turn to the study of the combinatorial q-stuffle Hopf algebra, which we do by stressing the importance of the Lie elements⁷studied by Ree [40], and show how Schützen- berger’s factorization extends to this new structure.

Theq-stuffle is commutative, associative and unital. With the co-unit defined by(P) = hP |1_Y^∗i, forP ∈KhYi, we get

H _q = (KhYi,conc,1_Y^∗,∆ _q, ) and

H^∨ _q = (KhYi, _q,1_Y^∗,∆_conc, )

which are mutually dual bialgebras and, in fact, Hopf algebras because they areN-graded by the weight.

LetD_Y be the diagonal series overH _q, i.e., D_Y = X

w∈Y^∗

w⊗w. (5)

Then⁸

logD_Y = X

w∈Y⁺

w⊗π₁(w), (6)

whereπ₁ is the extended Eulerian projector⁹overH _q, defined by (see [4]) π₁(w) =w+X

k≥2

(−1)^k−1 k

X

u1,...,uk∈Y⁺

hw|u_{1 q}· · · _qu_kiu₁. . . u_k. (7)

Let{Π_l}_{l∈Lyn Y} be defined by (Π_y =π₁(y), fory∈Y,

Π_l = [Π_s,Π_r], for the standard factorization(s, r)ofl∈ Lyn Y −Y. (8) Then it forms a basis of the Lie algebra of primitive elements ofH _q (see [4]).

Let{Π_w}w∈Y^∗ be defined, for anyw ∈Y^∗ such thatw =lⁱ₁¹. . . l_k^ik withl₁ > . . . > l_kand l₁. . . , l_k ∈ Lyn Y, by

Π_w = Πⁱ_l¹

1. . .Πⁱ_l^k

k. (9)

Then, by the PBW theorem, the set{Πw}w∈Y^∗ is a basis ofKhYi(see [4]).

7Following Ree [40], the Lie elements contain the non-commutative power series which are Lie series (as the Chen non-commutative generating series of iterated integrals), i.e., they are group-like for the co-product of the shuffle.

8The diagonal series lives inKhhY^∗⊗Y^∗ii '(KhYi ⊗KhYi)^∗.

9In fact,π1 is a projector which mapsH _q onto the space of its primitive elementsP rim(H _q), see Lemma 7.

Let {Σ_w}w∈Y^∗ be the family dual¹⁰ to {Π_w}w∈Y^∗ in the quasi-shuffle algebra. Then {Σw}w∈Y^∗freely generates the quasi-shuffle algebra, and the subset{Σl}l∈Lyn Y forms a tran- scendence basis of(KhYi, q,1Y^∗). TheΣw can be obtained as follows (see [4]):















Σ_y =y, fory ∈Y,

Σ_l =X

(!)

qⁱ⁻¹ i! y_sk

1+···+s_kiΣ_l1···ln, forl =y_s1. . . y_sk ∈ Lyn Y, Σ_w = Σ_l ^qi1

1 q· · · _qΣ_l ^qik

k

i₁!· · ·i_k! , forw=l₁ⁱ¹. . . lⁱ_k^k,

(10)

andl_{1 lex} · · · _lex l_k ∈ Lyn Y. In the second expression of (10), the sum(!)is taken over all subsequences{k₁, . . . , k_i} ⊂ {1, . . . , k}and all Lyndon wordsl_{1 lex} · · · _lex l_n such that (y_s1, . . . , y_sk) ⇐^∗ (y_sk

1, . . . , y_ski, l₁, . . . , l_n), where ⇐^∗ denotes the transitive closure of the relation on standard sequences, denoted by⇐(see [4]).

In this case, since {Πw}w∈Y^∗ and {Σw}w∈Y^∗ are multiplicative, we get the q-extended Schützenberger’s factorization as follows (see [4]):

D_Y = X

w∈Y^∗

Σ_w⊗Π_w =

&

Y

l∈Lyn Y

exp(Σ_l⊗Π_l). (11) This series, in the factorized form, encompasses a large part of the combinatorics of Dyson’s functional expansions in quantum field theory [18, 34]. It is the infinite-dimensional analogue of the theorem of Wei and Norman [2, 43, 44].

3. ALGEBRAIC ASPECTS OFϕ-SHUFFLE BIALGEBRAS

From now on, we will work with an alphabetY ={y_i}i∈I withI an arbitrary index set¹¹, which needs not be totally ordered unless we write it explicitly.

3.1. First properties. Let us consider the following recursion in order to construct a map

Y^∗×Y^∗ −→KhYi. (12)

i) For anyw∈Y^∗,

(Init) 1_Y^∗ _ϕw=w _ϕ1_Y^∗ =w. (13) ii) For anya, b∈Y andu, v ∈Y^∗,

(Rec) au _ϕbv=a(u _ϕbv) +b(au _ϕv) +ϕ(a, b)(u _ϕv), (14) whereϕis an arbitrary mapping defined by its structure coefficients

ϕ:Y ×Y −→ KY, (15)

(yi, yj) 7−→ X

k∈I

γ_y^y_i^k_,yj yk. (16) The following proposition guarantees the existence of a unique bilinear law onKhYisatisfy- ing(Init)and(Rec).

10The duality pairing is given byhu|vi=δ_u,v, foru, v∈Y^∗.

11The indexing is one-to-one, i.e., there is no repetition.

Proposition 1 ([14]). The recursion (Rec) together with the initialization (Init) defines a unique mapping

ϕ :Y^∗ ×Y^∗ −→ KhYi, which can, at once, be extended by linearity as a law

ϕ :KhYi ⊗KhYi −→ KhYi.

The spaceKhYiendowed with the law _ϕ is an algebra (with unit1_KhYi by definition).

It will be called theϕ-shuffle algebra. In full generality, this algebra need not be associative or commutative ifϕis not so. In the next example, we give a table of well known laws which can be defined according to this pattern (in whichϕis reasonable).

Example1. Below, a summary table ofϕ-deformed cases found in the literature is given. The last case (infiltration product) comes from computer science (see [37, 38, 15])

Name (recursion) Formula ϕ

Shuffle au^ttbv =a(u^ttbv) +b(au^ttv) ϕ≡0 Quasi-shuffle x_iu x_jv =x_i(u x_jv) +x_j(x_iu v) ϕ(x_i, x_j) =x_i+j

Stuffle +x_i+j(u v)

Min-shuffle x_iu x_jv =x_i(u x_jv) +x_j(x_iu v) ϕ(x_i, x_j) =−x_i+j

−x_i+j(u v)

Muffle x_iu ^a x_jv =x_i(u ^a x_jv) +x_j(x_iu ^a v) ϕ(x_i, x_j) =x_i×j +xi×j(u ^a v)

q-stuffle x_iu _qx_jv =x_i(u _qx_jv) +x_j(x_iu _qv) ϕ(x_i, x_j) =qx_i+j +qx_i+j(u^ttv)

q-stuffle x_iu^tt_qx_jv =x_i(u^tt_qx_jv) +x_j(x_iu^tt_qv) ϕ(x_i, x_j) = q^i×jx_i+j (character) +q^i×jx_i+j(u^ttv)

LDIAG(1, q_s)

non-crossed, au∗bv =a(u∗bv) +b(au∗v) ϕ(a, b) = q^|a||b|s (a.b) non-shifted +qs^|a||b|(a.b)(u∗v) (a.b)assoc.

B-shuffle au^tt_Bbv=a(u^tt_Bbv) +b(au^tt_Bv) ϕ(a, b) = ha, bi +ha, bi(u^tt_Bv) =hb, ai Semigroup- x_tu^tt_⊥x_sv =x_t(u^tt_⊥x_sv) +x_s(x_tu^tt_⊥v) ϕ(x_t, x_s) =x_t⊥s

-shuffle +x_t⊥s(u^tt_⊥v)

q-Infiltration au↑bv =a(u↑bv) +b(au↑v) ϕ(a, b) =qδ_a,ba +qδ_a,ba(u↑v)

Now we set forth the first properties of _ϕ (see [21]): associativity, commutativity and dualizability.

Definition 1([14]). A lawµ:KhYi ⊗KhYi →KhYiis said to bedualizableif there exists a (linear) mapping

∆_µ:KhYi −→KhYi ⊗KhYi (necessarily unique) such that the dual mapping

KhYi ⊗KhYi∗

−→KhhYii

restricts¹²toµ. Or, equivalently,

for allu, v, w∈Y^∗ : hµ(u⊗v)|wi=hu⊗v |∆_µ(w)i^⊗2. Theorem 1([14]). We have:

(1) The law _ϕ is associative (respectively commutative) if and only if the extension ϕ:KY ⊗KY −→KY is so.

(2) Letγ_x,y^z :=hϕ(x, y)|zibe the structure constants ofϕ. Then _ϕis dualizable if and only ifϕis also dualizable, that is to say, there exists a mapδ : KY −→ KY ⊗KY such that for allx, y, z∈X we have

hϕ(x, y)|zi=hx⊗y|δ(z)i. This mapδis given by¹³

δ(z) = X

x,y∈Y

γ_x,y^z x⊗y .

For the proof of the theorem we need the following auxiliary result.

Lemma 1([14]). Let∆be the morphismKhYi −→AhhY^∗⊗Y^∗iidefined on the letters by¹⁴

∆(y_s) =y_s⊗1 + 1⊗y_s+ X

n,m∈I

γ_y^y_n^s_,ymy_n⊗y_m. Then

(1) for allu, v, w ∈Y^∗,hu _ϕv |wi=hu⊗v |∆(w)i^⊗2. (2) for allw∈Y⁺,∆(w) = w⊗1 + 1⊗w+ X

u,v∈Y⁺

h∆(w)|u⊗viu⊗v.

Proof of Theorem 1 (sketch). The theorem follows by application of items (1) and (2) in Lem-

ma 1.

If ϕis associative (which is fulfilled in all cases of Table 1), we extend ϕto Y⁺ by the universal property of the free semigroupY⁺,

(ϕ(x) =x, forx∈Y,

ϕ(xw) =ϕ(x, ϕ(w)), forx∈Y andw∈Y⁺, (17) and we extend the definition of the structure constants accordingly: forx1. . . xl ∈Y⁺,

γ_x^y_1...x

l =hy |ϕ(x₁. . . x_l)i= X

t1,...,tl−2∈Y

γ_x^y_1,t1γ_x^t1_2,t2. . . γ_x^tl−2_l−1,x

l. (18)

Note that the fact thatϕis dualizable can be rephrased as:

for ally∈Y : {w∈Y²|γ_w^y 6= 0}is finite. (19)

12through the pairingsh− | −i.

13Note that all these conditions are equivalent to the fact that(γ_x,y^z )_x,y,z∈Y satisfies:

for allz∈Y : #{(x, y)∈Y²|γ_x,y^z 6= 0}<+∞.

14Ifϕis dualizable, this expression can be written

∆(ys) =ys⊗1 + 1⊗ys+δ(ys).

In this case, it can be checked immediately that, for an arbitrarily fixedN ≥1,

for ally∈Y : {w∈Y^N|γ_w^y 6= 0}is finite, (20) but, by no means, we have in general that

for ally∈Y : {w∈Y⁺|γ_w^y 6= 0}is finite. (21) Remark1. i) Condition (21) is strictly stronger than (19) as the example of any group law on Y, with|Y| ≥2and finite, shows.

ii) Non-dualizable laws occur with the alphabet Y_Z = {y_j}_j∈Z and the stuffle on it (ϕ(y_i, y_j) = y_i+j). This alphabet naturally appears in the theory of polylogarithms at nega- tive integers in [17] where another non-dualizable law (called>) arises. See also Example 2 below.

Definition 2. An associative lawϕonKY will be said to bemoderateif and only if it fulfils condition(21).

Let us now state the structure theorem from [14].

Theorem 2 ([14]). Let us suppose that ϕis dualizable and associative. We still denote its dual co-multiplication by

∆ _ϕ :KhYi −→KhYi ⊗KhYi.

ThenB_ϕ = (KhYi,conc,1_Y^∗,∆ _ϕ, ε)is a bialgebra. If, moreover, ϕis commutative, the following conditions are equivalent:

(1) Bϕ is an enveloping bialgebra.

(2) Bϕ is isomorphic to(KhYi,conc,1Y^∗,∆tt, )as a bialgebra.

(3) For ally∈Y, the following series is a polynomial (P) y+X

l≥2

(−1)^l−1 l

X

x1,...,x_l∈Y

hy|ϕ(x₁. . . x_l)ix₁. . . x_l. (4) ϕis moderate.

Proof (sketch). 4=⇒3) Obvious.

3=⇒2) One first constructs an endomorphism of (KhYi,conc,1_Y^∗)sending each letter y ∈ Y to the polynomial form (P) and then proves that it is an automorphism of AAU¹⁵ which sends(KhYi,conc,1_Y^∗,∆ _ϕ, ε)to(KhYi,conc,1_Y^∗,∆tt, ).

2=⇒1) Because(KhYi,conc,1_Y^∗,∆tt, )is an enveloping bialgebra.

1=⇒4) Observe that, for each lettery∈Y, we have

h∆⁽ⁿ⁻¹⁾_ϕ (y)|x₁⊗x₂⊗ · · · ⊗x_ni=γ_x^y_1...x

l .

Example2. (1) The muffle product (see Table 1), which determines the product of Hur- witz polyzetas with rational centers and correspond toϕ(x_i, x_j) =x_i.j fori, j ∈Q^∗+, is not dualizable (γ_n,1/n¹ = 1for alln ≥1).

15Abbreviation for associative algebra with unit.

(2) The q-infiltration bialgebra (see again Table 1) has its origin in computer science [37, 38] and appears as a generic solution in [15]. It provides a bialgebra

Hq−inflitr = (KhYi,conc,1_X^∗,∆↑_q, )

(q ∈ K) based on aϕwhich is an associative, commutative and dualizable law, but, if Y 6= ∅ this law is moderate only if and only ifq is nilpotent in the Q-algebraK. Indeed, for allx∈Y,(1 +qx)is group-like and it has an inverse inKhXiif and only ifqis nilpotent. In this case the antipode is the involutive antiautomorphism defined on the letters by

S(x) = −x 1 +qx.

3.2. Structural properties. Here, we only assume thatϕis associative.

The bialgebra

H^∨ _ϕ = (KhYi, _ϕ,1_Y^∗,∆_conc, ) (22) is a Hopf algebra because it is co-nilpotent¹⁶. Its antipode can be computed by a(1_Y^∗) = 1 and, forw∈Y⁺,

a _ϕ(w) =X

k≥1

(−1)^−k X

u1,...,uk∈Y⁺ u1...uk=w

u_{1 ϕ}· · · _ϕu_k. (23)

Due to the finite number of decompositions of any wordu₁. . . u_k = w ∈ Y⁺ into factors u₁, . . . , u_k ∈Y⁺, we can, at this very early stage, define an endomorphismΦ(S)ofKhYias follows:

Φ(S)[w] =X

k≥1

ak

X

u1,...,uk∈Y⁺ u1...uk=w

u1 ϕ· · · ϕuk, (24)

associated to any univariate formal power seriesS =a₁X+a₂X²+a₃X³+· · ·. The case of

log(1 +X) = X

k≥1

(−1)^k−1

k X^k (25)

will be of particular importance. It reads here in the style of formula (23), ˇ

π₁(w) =X

k≥1

(−1)^k−1 k

X

u1,...,uk∈Y⁺ u1...uk=w

u_{1 ϕ}· · · _ϕu_k. (26)

16The law∆_conc, dual to the concatenation is, of course, defined by

∆_conc(w) = X

uv=w

u⊗v.

The correspondingn-fold∆⁺_conc(∆⁺= ∆minus the primitive part) reads

∆⁺⁽ⁿ⁻¹⁾_conc (w) = X

u₁u₂···u_n=w u_i∈Y⁺

u1⊗u2⊗ · · · ⊗un,

from which it is clear that∆⁺⁽ⁿ⁻¹⁾_conc (w) = 0forn >|w|.

Thisπˇ₁ ∈End(KhYi)has an adjointπ₁ ∈End(KhhYii)which reads π1(S) = X

w∈Y^∗

hS |πˇ1(w)iw (27)

=X

k≥1

(−1)^k−1 k

X

u1,...,uk∈Y⁺

hS |u_{1 ϕ}· · · _ϕu_kiu₁. . . u_k. (28) It is an easy exercise to check that the family in the sums of (27) is summable¹⁷. It is easy to check that the dominant term of all terms in a _ϕ product is the corresponding^tt product.

This explains why we still have the theorem of Radford.

Theorem 3 (RADFORD’S THEOREM). When ϕ is commutative, the associative and com- mutative algebra with unit (KhYi, _ϕ,1_Y^∗) is a polynomial algebra. More precisely, the morphism β : K[Lyn Y] → (KhYi, _ϕ,1_Y^∗) defined by β(l) = l for l ∈ Lyn Y is an isomorphism. In other words, the family

l₁ ^ϕi1 ϕ· · · ϕl_k ^ϕik

k≥0,{l1,l2,...,lk}⊂Lyn Y (i1,i2,...,ik)∈(N+)^k

is a linear basis ofKhYi.

Proof. One checks that

l₁ ^ϕi1 _ϕ· · · _ϕl_k ^ϕik =l^tt₁ ⁱ¹tt. . .ttl^tt_k ^ik + X

|v|<P

1≤j≤kij|lj|

c_vv.

The result follows.

The theorem of Radford is important in the classical cases because it is the left-hand side of Schützenberger’s factorization in which we have the move

PBW→Radford;

see [12] for a discussion of the converse.

Lemma 2(ϕ-EXTENDED FRIEDRICHS’CRITERION). We denote¹⁸by

∆ _ϕ :KhhYii →KhhY^∗ ⊗Y^∗ii the dual of _ϕapplied to series, i.e., defined by

∆ _ϕ(S) = X

u,v∈Y^∗

hS |u _ϕviu⊗v.

Let nowS ∈KhhYii. Then we have:

(1) IfhS | 1_Y^∗i = 0thenS is primitive (i.e., ∆ _ϕ(S) = S⊗1_Y^∗ + 1_Y^∗ ⊗S)¹⁹if and only if we havehS |u _ϕvi= 0for anyuandv ∈Y⁺.

(2) If hS | 1_Y^∗i = 1, then S is group-like (i.e., ∆ _ϕ(S) = S⊗S)²⁰if and only if we havehS|u _ϕvi=hS |uihS |vifor anyuandv ∈Y^∗.

17A family of (simple, double, etc.) series is summable if it is locally finite (see [14] for a complete development).

18As in the classical case,∆ _ϕis aconc-morphism as can be seen by transposition of the fact that∆_conc is a ϕ-morphism.

19Tensor products of linear forms.

20idem

Proof. The expected equivalences are due to the following facts:

∆ _ϕ(S) = S⊗1Y^∗ + 1Y^∗⊗S− hS |1Y^∗i1Y^∗⊗1Y^∗+ X

u,v∈Y⁺

hS |u ϕviu⊗v,

∆ _ϕ(S) = X

u,v∈Y^∗

hS |u _ϕviu⊗v and S⊗S = X

u,v∈Y^∗

hS|uihS |viu⊗v.

Lemma 3. LetS∈KhhYiibe such thathS|1_Y^∗i= 1. ThenS is group-like if and only if²¹ log(S)is primitive.

Proof. Since∆ _ϕ and the maps T 7→ T ⊗1Y^∗, T 7→ 1Y^∗ ⊗T are continuous homomor- phisms, then, iflogSis primitive, we have (see Lemma 2(1))

∆ _ϕ(logS) = logS⊗1Y^∗+ 1Y^∗ ⊗logS, and, sincelogS⊗1_Y^∗and1_Y^∗⊗logS commute, we get successively

∆ _ϕ(S) = ∆ _ϕ(exp(logS))

= exp(∆ _ϕ(logS))

= exp(logS⊗1_Y^∗) exp(1_Y^∗⊗logS)

= (exp(logS)⊗1_Y^∗)(1_Y^∗⊗exp(logS))

=S⊗S.

This means, together withhS | 1_Y^∗i, that S is group-like. The converse can be obtained in

the same way.

Remark2. i) In fact, Lemma 3 establishes a nice log-exp correspondence for the Lie group of group-like series.

ii) Through the canonical pairing h− | −i : KhhYii ⊗KhYi → K, we haveKhhYii ' (KhYi)^∗. Group-like (respectively primitive) series are in bijection with characters (respec- tively infinitesimal characters) of the algebra(KhYi, ϕ,1Y^∗).

Lemma 4. (1) Group-like series form a group (for concatenation).

(2) The space Prim(KhhYii)is a Lie algebra (for the bracket derived from concatena- tion).

Proof. As in the classical case.

We extend the transposition process in the same way as in Lemma 2 and note, forn ≥ 1, that

∆⁽ⁿ⁻¹⁾

ϕ :KhhYii →Khh(Y^∗)^⊗nii, (29)

the dual of ⁽ⁿ⁻¹⁾ϕ applied to series, i.e., defined by

∆⁽ⁿ⁻¹⁾_ϕ (S) = X

u1,u2,...un∈Y^∗

hS|u1 ϕ· · · ϕuniu1⊗ · · · ⊗un. (30)

21For anyh∈KhhYii, ifhh|1_Y^∗i= 0, we define

log(1Y^∗+h) =X

n≥1

(−1)ⁿ⁻¹

n hⁿ and exp(h) =X

n≥0

hⁿ n!, and we have the usual formulaslog(exp(h)) =handexp(log(1Y^∗+h)) = 1Y^∗+h.

We will use the following lemma several times, which gives the combinatorics of products of primitive series (and the polynomials).

Lemma 5 (HIGHER ORDER CO-MULTIPLICATIONS OF PRODUCTS). Let us consider the languageMover the alphabetA={a₁, a₂, . . . , , a_m},

M={w∈ A^∗ |w=a_j1. . . a_j|w|, j₁ <· · ·< j|w|,1≤ |w| ≤m}, and the morphism

µ:KhAi −→ KhhYii, a_i 7−→ S_i, whereS₁, . . . , S_mare primitive series inKhhYii. Then

∆⁽ⁿ⁻¹⁾_ϕ (S₁. . . S_m) = X

w1,...,wn∈M

|w₁|+···+|w_n|=m a1···am∈supp(w₁tt...ttwn)

µ(w₁)⊗ · · · ⊗µ(w_n).

Proof (sketch). Let S = (S1, . . . , Sm) be this family of primitive series and, for I = {i1, . . . , ik} ⊂[1· · ·m]in increasing order, let us writeS[I]for the productSi1· · ·Si_k. Then we have

∆⁽ⁿ⁻¹⁾_ϕ (S₁. . . S_m) = X

I1+···+I_n=[1···m]

S[I₁]⊗ · · · ⊗ S[I_n].

Settingw_i = (a₁a₂. . . a_m)[I], one gets the expected result.

Lemma 6(PAIRING OF PRODUCTS). LetS₁, . . . , S_m be primitive series inKhhYii, and let P₁, . . . , P_nbeproper²²polynomials inKhYi. Then one has in general

hP_{1 ϕ}· · · _ϕP_n|S₁. . . S_mi= X

w1,...,wn∈M

|w1|+···+|wn|=m a1···am∈supp(w1tt...ttwn)

n

Y

i=1

hP_i |µ(w_i)i.

In particular, we have the following exhaustive list of circumstances:

(1) Ifn > m, thenhP_{1 ϕ}· · · _ϕP_n|S₁. . . S_mi= 0.

(2) Ifn=m, then

hP_{1 ϕ}· · · _ϕP_n|S₁. . . S_ni= X

σ∈Sn

n

Y

i=1

hP_i |S_σ(i)i.

(3) Ifn < m, then one has the general form in which every product in the sum contains at least a factorhP_i |µ(w_i)iwith|w_i| ≥2.

Proof. It is a consequence of Lemma 5 through the equality

hP_{1 ϕ}· · · _ϕP_n |S₁. . . S_ni=hP₁⊗ · · · ⊗P_n|∆⁽ⁿ⁻¹⁾

ϕ (S₁. . . S_n)i.

22i.e., polynomials without constant term; see [1].

In the sequel, we assume thatϕis associative and dualizable.

Now, we have the following two structures:

H _ϕ = (KhYi,conc,1_Y^∗,∆ _ϕ, ), (31) H^∨ _ϕ = (KhYi, _ϕ,1_Y^∗,∆_conc, ), (32) which are mutually dual²³bialgebras. The bialgebraH _ϕ need not be a Hopf algebra,even if∆ _ϕ is cocommutative(see Example 2.2).

Now, let us consider

I := span_K{u _ϕv}u,v∈Y⁺, (33)

K+hYi:={P ∈KhYi | hP |1_Y^∗i= 0}, (34) P := Prim(H _ϕ) = {P ∈KhYi |∆⁺

ϕ(P) = 0}, (35) where

∆⁺ _ϕ(P) = ∆ _ϕ(P)−(P ⊗1_Y^∗+ 1_Y^∗⊗P) +hP |1_Y^∗i1_Y^∗⊗1_Y^∗ . (36) Remark3. At this stage (ϕnot necessarily moderate), it can happen thatPrim(H _ϕ) ={0}.

This is for example the case with theq-infiltration bialgebra on one letter atq= 1, H _ϕ = (K[x],conc,1_x^∗,∆↑₁, ),

and, more generally, whenqis not nilpotent²⁴.

We can also endowEnd(KhYi), theK-module of endomorphisms ofKhYi, with thecon- volutionproduct defined by

for allf, g∈End(KhYi), f ? g=conc◦(f⊗g)◦∆ _ϕ, (37) i.e.,for allP ∈KhYi, (f ? g)(P) = X

u,v∈Y^∗

hP |u _ϕvif(u)g(v). (38) Then End(KhYi) becomes a K-associative algebra with unity (AAU), its unit being e = 1_KhYi◦.

It is convenient to represent every f ∈ End(KhYi) by its graph, a double series which reads

Γ(f) = X

w∈Y^∗

w⊗f(w). (39)

This representation is faithful and, by direct computation, one gets

Γ(f)Γ(g) = Γ(f ? g), (40)

where the multiplication of double series is performed by the stuffle on the left and the con- catenation on the right.

Definition 3. Heretis a real parameter. Let us define define D_Y := Γ(Id_KhYi) = X

w∈Y^∗

w⊗w, Haus_Y := logD_Y, σ_Y(t) := exp(tHaus_Y).

From now on, we assume thatϕis associative, commutative and dualizable.

23This duality is separating; see [5].

24Recall thatqis an element of the ringK(see example 2.2).

Lemma 7 (π₁ IS A PROJECTOR ON THE PRIMITIVE SERIES). The endomorphism π₁ is a projector, the image of which is exactly the space of primitive series,Prim(KhhYii).

Proof (sketch). The proof follows the lines of [41] with the difference thatπ₁(w)might not be a polynomial and the operator defined in Lemma 2 is not a genuine co-product. The diagonal seriesD_Y (when considered as a series in(KhhYii)hhYii, the coefficient ring,KhhYii, being endowed with the _ϕ product) is group-like in the sense of Lemma 2. Then, using

log(D_Y) = X

w∈Y^∗

w⊗π₁(w)

(which can be established by summability of the family(w⊗π₁(w))w∈Y^∗; but remember that theπ1(w)are, in general, series²⁵), one gets that π1(w)is a primitive series for allw. Now, from

π₁(S) = X

w∈Y^∗

hS |wiπ₁(w),

one hasπ1(S)∈Prim(KhhYii). Conversely, from Friedrichs’ criterion, one getsπ1(S) =S

ifS ∈Prim(KhhYii).

In the remainder of the paper, we suppose thatϕis moderate (and still dualizable, associa- tive and commutative).

Definition 4(PROJECTORS, [41]). LetI₊ : KhYi −→ KhYibe the linear mapping defined by

I₊(1_Y^∗) = 0, and for allw∈Y⁺, I₊(w) =w.

One defines²⁶

π₁ := log(e+ I₊) = X

n≥1

(−1)ⁿ⁻¹

n I₊^?n, where I₊^?n :=concn−1◦I₊^⊗n◦∆⁽ⁿ⁻¹⁾_ϕ . It follows immediately that

exp(π₁) = e+X

n≥1

1

n!π₁^?n =X

n≥0

π_n, (41)

where e = 1_KhYi ◦ is the orthogonal complement of I₊ and neutral for the convolution product. Theπ_nso obtained is called then-th Eulerian projector.

Lemma 8. The endomorphismπˇ₁defined in(26)is the adjoint ofπ₁. One has ˇ

π₁ =X

n≥1

(−1)ⁿ⁻¹ n

(n−1)

ϕ ◦I₊^⊗n◦∆⁽ⁿ⁻¹⁾_conc .

Proof. Immediate.

25In greater detail, this equality amounts to checking the summability of the family (−1)^k−1

k w⊗ hw|u1 ϕ· · · ϕukiu1· · ·uk

w∈Y^∗, k≥1 u₁,...,u_k∈Y⁺

(which is immediate) and rearranging the sums.

26The series below are summable because the family(I+?n

)n≥0is locally nilpotent (see [14] for complete proofs). Note that this definition gives the same result as the computation of the adjoint ofπˇ1given in (27).

Proposition 2. (GRAPH OF π₁, VALUES AND ITS EXPONENTIAL AS RESOLUTION OF UNITY).

(1) For allY andϕ(moderate, associative, commutative and dualizable), one has logD_Y = X

w∈Y⁺

w⊗π₁(w) = X

w∈Y⁺

ˇ

π₁(w)⊗w.

(2) LetP ∈KhYibe a primitive polynomial, for∆ _ϕ. Then π₁(P) = P, for allk, n∈N+, π_n(P^k) = δ_k,nP^k. (3) One has

Id_KhYi =e+I₊ =X

n≥0

π_n. (42)

Equation(42)is a resolution of identity with mutually orthogonal summands.

(4) We have

K+hYi=P⊕I^⊥ =P ⊕ M

n≥2

π_n(KhYi) .

Proof. The only statement which cannot be proved through an isomorphism with the shuf- fle algebra is the first equality of the point (4). The fact that P ∩ I = {0} comes from Friedrichs’ criterion, andP +I = K+hYiis a consequence of the fact (seen again through any isomorphism with the shuffle algebra) that

(H _ϕ)+ = span_K([

n≥1

(P1 ϕ· · · ϕPn)_P

i∈Prim(H _ϕ)).

Remark4. (1) The first equality of Proposition 2.(4), i.e., K+hYi=P⊕I,^⊥ is known as the theorem of Ree [40].

(2) The projector on P parallel to I is not in general in the descent algebra (see [19]).

This proves that, although they are isomorphic, the spaces I and L

n≥2π_n(KhYi) are, in general, not identical.

Proposition 2.(1) leads to the following corollary.

Corollary 1. We haveπ₁(1_Y^∗) = ˇπ₁(1_Y^∗) = 0and, for allw∈Y⁺, π₁(w) = w+X

k≥2

(−1)^k−1 k

X

u1,...,uk∈Y⁺

hw |u_{1 ϕ}· · · _ϕu_kiu₁. . . u_k,

ˇ

π₁(w) = w+X

k≥2

(−1)^k−1 k

X

u1,...,u_k∈Y⁺

hw |u₁. . . u_kiu_{1 ϕ}· · · _ϕu_k. In particular,π1(1Y^∗) = ˇπ1(1Y^∗) = 0, for anyy∈Y,πˇ1(y) =y, and

π₁(y) = y+X

l≥2

(−1)^l−1 l

X

x1...xl∈Y^∗

γ_x^y_1,...,x

l x₁. . . x_l.

Remark5. We already knew that, as soon asϕis associative,πˇ₁(w)is a polynomial. Here, becauseϕis moderate, dualizable, and associative,π₁(w)is also a polynomial, and because ϕis commutative, it is primitive.

Proposition 3. We have:

(1) The expression ofσ_Y(t)is given by σY(t) =X

n≥0

tⁿ X

w∈Y^∗

w⊗πn(w) =X

n≥0

tⁿ X

w∈Y^∗

ˇ

πn(w)⊗w,

whereπˇ_nis the adjoint ofπ_n. These are given byπ_n(1_Y^∗) = ˇπ_n(1_Y^∗) =δ_0,nand, for allw∈Y⁺,

π_n(w) = 1 n!

X

u1,...,un∈Y⁺

hw|πˇ₁(u₁) _ϕ· · · _ϕπˇ₁(u_n)iπ₁(u₁). . . π₁(u_n), ˇ

πn(w) = 1 n!

X

u1,...,un∈Y⁺

hw|π1(u1). . . π1(un)iˇπ1(u1) ϕ· · · ϕπˇ1(un)).

(2) For anyw∈Y^∗, we have w=X

k≥0

1 k!

X

u1,...,uk∈Y⁺

hw|u_{1 ϕ}· · · _ϕu_kiπ₁(u₁). . . π₁(u_k)

=X

k≥0

1 k!

X

u1,...,u_k∈Y⁺

hw|u₁. . . u_kiˇπ₁(u₁) _ϕ· · · _ϕπˇ₁(u_k).

In particular, for anyy_s ∈Y, we havey_s = ˇπ₁(y_s)and y_s=X

k≥1

1 k!

X

ys1,...,y_sk∈Y

γ_y^y_s^s

1,...,y_skπ₁(y_s1). . . π₁(y_sk).

Proof. Direct computation.

Applying the tensor product²⁷of isomorphisms of algebras²⁸α⊗Id_Y to the diagonal series D_Y, we obtain a group-like element, and then computing the logarithm of this element (or equivalently, applyingα⊗Id_Y toHaus_Y) we obtainS which is, by Lemma 3, primitive:

S = X

w∈Y^∗

α(w)π₁(w) = X

w∈Y^∗

α◦πˇ₁(w)w. (43)

Lemma 9. For anyw∈Y⁺, one hasπ₁(w)∈Prim(KhYi).

Proof. Immediate from Lemma 7.

A primitive projector,π :KhYi −→ KhYi, is defined in the same way as a Lie projector by the three following conditions:

π◦π=π, π(1_Y^∗) = 0, π(KhYi) = Prim(KhYi) =P. (44) For example,π₁defined in Definition 4 (see also Proposition 2) is a primitive projector which will be used to construct bases of P and its enveloping algebra (see Theorem 5 below).

Another example of a primitive projector is the orthogonal projector on P attached to the decomposition in Remark 4.

27Extended to series.

28In order to clarify the ideas at this point, the reader can also take the alphabet duplication isomorphism for ally¯∈Y ,¯ y¯=α(y),

and use{w}_w∈Y¯^∗as a basis forKhY¯i.