Bo-Hae Im

KHAC Hojin Kim

Nhuan Le

Tuan Ngo Dac
email: tuan.ngodac@unicaen.fr

Lan Huong Pham

HOPF ALGEBRAS AND ALTERNATING MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC

Keywords: Mathematics Subject Classification. Primary 11M32; Secondary 11M38, 16S10, 16T30, 11R58 multiple zeta values, zeta and L-functions in characteristic p, Hopf algebras, shuffle algebras, horizontal maps, arithmetic of function fields

 and Racinet [42] in the classical setting. The proof is based on two new ingredients: the first one is a direct and explicit construction of the shuffle Hopf algebra structure, and the second one is the notion of horizontal maps.

Let N = {1, 2, . . . } be the set of positive integers. The multiple zeta values (MZV's for short) are infinite sum defined by

ζ(n 1 , . . . , n r ) = 0<k1<•••<kr 1 k n1 1 k n2 2 . . . k nr r ,
for n i ∈ N with n r ≥ 2 for convergence. Here r and w := n 1 + • • • + n r are said to be the depth and weight of the presentation ζ(n 1 , . . . , n r ), respectively. In the 18th century, Euler [START_REF] Euler | Meditationes circa singulare serierum genus[END_REF] studied MZV's for depth 1 and 2 cases. After two centuries, Zagier [START_REF] Zagier | Values of zeta functions and their applications[END_REF] studied the general MZV's, initiating extensive works connected to various fields in mathematics and physics, including number theory, K-theory, knot theory, and higher energy physics (see for example [START_REF] Brown | Mixed Tate motives over Z[END_REF][START_REF] Deligne | Groupes fondamentaux motiviques de Tate mixte[END_REF][START_REF] Drinfeld | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF][START_REF] Hoffman | The algebra of multiple harmonic series[END_REF][START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF][START_REF] Ihara | Derivation and double shuffle relations for multiple zeta values[END_REF][START_REF] Le | Kontsevich's integral for the Kauffman polynomial[END_REF][START_REF] Terasoma | Mixed Tate motives and multiple zeta values[END_REF]). There exist more general objects called colored multiple zeta values. Letting N ∈ N be a positive integer and Γ N be the group of N -th roots of unity, we define colored multiple zeta values by ζ 1 . . . for i ∈ Γ N and n i ∈ N with (n r , r ) = (1, 1) for convergence. These objects were first systematically studied by Arakawa-Kaneko, Deligne, Goncharov and Racinet (see [START_REF] Arakawa | Multiple zeta values, poly-Bernoulli numbers, and related zeta functions[END_REF][START_REF] Arakawa | On multiple L-values[END_REF][START_REF] Deligne | Le groupe fondamental unipotent motivique de Gm -µ N , pour N = 2, 3, 4, 6 ou 8[END_REF][START_REF] Deligne | Groupes fondamentaux motiviques de Tate mixte[END_REF][START_REF] Goncharov | Multiple polylogarithms and mixed tate motives[END_REF][START_REF] Goncharov | Galois symmetries of fundamental groupoids and noncommutative geometry[END_REF][START_REF] Racinet | Doubles mélanges des polylogarithmes multiples aux racines de l'unité[END_REF]). Since then they have been also studied by a lot of mathematicians and physicists, including Broadhurst [START_REF] Broadhurst | On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory[END_REF][START_REF] Broadhurst | Conjectured Enumeration of irreducible Multiple Zeta Values, from Knots and Feynman Diagrams[END_REF], Hoffman [START_REF] Hoffman | An odd variant of multiple zeta values[END_REF], Kaneko-Tsumura [START_REF] Kaneko | Double zeta values, double Eisenstein series, and modular forms of level[END_REF], and Zhao [START_REF] Zhao | Multiple zeta functions, multiple polylogarithms and their special values[END_REF] in various fields of mathematics and physics. For N = 1 we recover MZV's. For N = 2 the colored multiple zeta values are also known as alternating multiple zeta values (AMZV's for short) or Euler sums.

The main goal of the theory of colored multiple zeta values is to find all linear and algebraic relations among colored multiple zeta values over a suitable coefficient field. It led to several important conjectures formulated by Ihara-Kaneko-Zagier [START_REF] Ihara | Derivation and double shuffle relations for multiple zeta values[END_REF], Zagier [START_REF] Zagier | Values of zeta functions and their applications[END_REF] and Hoffman [START_REF] Hoffman | The algebra of multiple harmonic series[END_REF]. We will explain below Ihara-Kaneko-Zagier's conjecture and refer the reader to the excellent book [START_REF] Gil | Multiple zeta values: from numbers to motives[END_REF] for more details about Zagier and Hoffman's conjectures.

For MZV's (the case N = 1) Ihara-Kaneko-Zagier [START_REF] Ihara | Derivation and double shuffle relations for multiple zeta values[END_REF] introduced the double shuffle relations and regularization. Roughly speaking, they used Hoffman's algebra h (see [START_REF] Hoffman | Quasi-shuffle products[END_REF]) and subalgebras h 0 ⊂ h 1 ⊂ h which are related to the MZV's, and presented two different algebra structures (h 1 , * ) and (h, ¡). These structures can be extended to Hopf algebra structures (see for example the work of Hoffman [START_REF] Hoffman | Quasi-shuffle products[END_REF]). By regularization one gets the double shuffle relations by 'comparing' two products in the level of MZV's. Double shuffle relations and regularization produce linear relations among MZV's, and Ihara-Kaneko-Zagier conjectured that these relations are enough to understand all of the linear relations among MZV's (see [START_REF] Ihara | Derivation and double shuffle relations for multiple zeta values[END_REF]Conjecture 1]).

For colored multiple zeta values the previous works of Ihara-Kanako-Zagier and Hoffman were extended by Racinet in [START_REF] Racinet | Doubles mélanges des polylogarithmes multiples aux racines de l'unité[END_REF]. However, we warn the reader that as mentioned in loc. cit., for general N the double shuffle relations and regularization fail to exhaust all the linear relations among colored multiple zeta values. normalized as v ∞ (θ) = -1, | • | ∞ = q -v∞(•) the corresponding absolute value on K.

Carlitz [START_REF] Carlitz | On certain functions connected with polynomials in Galois field[END_REF] defined the Carlitz zeta values ζ A (n) for n ∈ N as

ζ A (n) := a∈A+ 1 a n ∈ K ∞ .
Note that this definition is an analogue of the classical zeta values ζ(n) = k∈N 1 k n . Extending this, Thakur [START_REF] Thakur | Function field arithmetic[END_REF] defined the multiple zeta value in positive characteristic for tuples s = (s 1 , . . . , s r ) ∈ N r as

ζ A (s) = 1 a s1 1 . . . a sr r .
Here the sum is over all tuples of monic polynomials with decreasing degree, i.e., (a 1 , . . . , a r ) ∈ A r + with deg(a 1 ) > • • • > deg(a r ). One can see the analogy between the classical MZV's and MZV's in positive characteristic. We define the depth and weight of ζ A (s) by r and w(s) := s 1 + • • • + s r respectively. Note that Carlitz zeta values are MZV's of depth one. Thakur showed some of their properties including non-vanishing [START_REF] Thakur | Power sums with applications to multizeta and zeta zero distribution for Fq[t][END_REF], sum-shuffle relations [START_REF] Thakur | Shuffle relations for function field multizeta values[END_REF], etc. For further references on the MZV's in positive characteristic, see [START_REF] Anderson | Tensor powers of the Carlitz module and zeta values[END_REF][START_REF] Anderson | Multizeta values for Fq[t], their period interpretation, and relations between them[END_REF][START_REF] Gezmis | Trivial multiple zeta values in Tate algebras[END_REF][START_REF] Goss | Basic Structures of function field arithmetic[END_REF][START_REF] Lara Rodriguez | Zeta-like multizeta values for Fq[t][END_REF][START_REF] Lara Rodriguez | Zeta-like multizeta values for higher genus curves[END_REF][START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF][START_REF] Thakur | Function field arithmetic[END_REF][START_REF] Thakur | Relations between multizeta values for Fq[t][END_REF][START_REF] Thakur | Shuffle relations for function field multizeta values[END_REF][START_REF] Thakur | Multizeta values for function fields: a survey[END_REF][START_REF] Thakur | t-motives: Hodge structures, transcendence and other motivic aspects[END_REF][START_REF] Yu | Transcendence and special zeta values in characteristic p[END_REF].

In [START_REF] Harada | Alternating multizeta values in positive characteristic[END_REF] Harada introduced the notion of alternating multiple zeta values in positive characteristic. Let s = (s 1 , . . . , s r ) ∈ N r and ε = (ε 1 , . . . , ε r ) ∈ (F * q ) r , we define Here, the sum is over all tuples of monic polynomials with decreasing degree as MZV's, i.e., (a 1 , . . . , a r ) ∈ A r + with deg(a 1 ) > • • • > deg(a r ). We define the depth and weight of ζ A ε s in a similar manner, that is, r is the depth and w(s) = s 1 + • • • + s r is the weight. In loc. cit., Harada also showed some fundamental properties of AMZV's which hold for MZV's including non-vanishing, sum-shuffle relations, period interpretation and linear independence.

Zagier-Hoffman's conjectures for AMZV's in positive characteristic.

The primary objective of the theory of MZV's and AMZV's in positive characteristic is to comprehend all of their K-linear relations as in the classical setting. The positive characteristic version of Zagier-Hoffman's conjectures for MZV's is suggested by Todd [START_REF] Todd | A conjectural characterization for Fq(t)-linear relations between multizeta values[END_REF] and Thakur [START_REF] Thakur | Multizeta values for function fields: a survey[END_REF]. The algebraic part of these conjectures was solved by the fourth author [START_REF] Dac | On Zagier-Hoffman's conjectures in positive characteristic[END_REF] using ingredients of Chen [START_REF] Chen | On shuffle of double zeta values over Fq[t][END_REF], Thakur [START_REF] Thakur | Shuffle relations for function field multizeta values[END_REF][START_REF] Thakur | Multizeta values for function fields: a survey[END_REF] and Todd [START_REF] Todd | A conjectural characterization for Fq(t)-linear relations between multizeta values[END_REF]. The transcendental part was solved for the small weight cases by the fourth author in [START_REF] Dac | On Zagier-Hoffman's conjectures in positive characteristic[END_REF] by applying Anderson's theory of motives [START_REF] Anderson | t-motives[END_REF][START_REF] Brownawell | A rapid introduction to Drinfeld modules, t-modules and t-motives[END_REF][START_REF] Hartl | Pink's theory of Hodge structures and the Hodge conjectures over function fields[END_REF], as well as Anderson-Brownawell-Papanikolas criterion [START_REF] Anderson | Determination of the algebraic relations among special Γ-values in positive characteristic[END_REF] as a tool for transcendence.

In [START_REF] Im | Zagier-Hoffman's conjectures in positive characteristic[END_REF], by discovering a deep link among AMZV's and alternating Carlitz multiple polylogarithms, we solved both the MZV's and AMZV's version of the Zagier-Hoffman's conjectures in positive characteristic. We mention that the latter is much harder than the former (see [31, §0.3] for a detailed discussion).

More precisely, we show that

dim K AZ w = s(w)
where AZ w is the vector space spanned by the AMZV's in the positive characteristic of weight w over K, and s(w) is a Fibonacci-like sequence defined by

s(w) =          1 if w = 0, (q -1)q w-1 if 1 ≤ w ≤ q -1, (q -1)(q w-1 -1) if w = q, (q -1) q-1 i=1 s(w -i) + s(w -q) if w > q.
Further, we give an explicit basis over K of AZ w . Unlike the MZV's case, this basis consists of linear combinations of AMZV's, not AMZV's. This crucial point has been observed in the classical setting by Charlton [START_REF] Charlton | On motivic multiple t values, Saha's basis conjecture, and generators of alternating MZV[END_REF] and Deligne [START_REF] Deligne | Le groupe fondamental unipotent motivique de Gm -µ N , pour N = 2, 3, 4, 6 ou 8[END_REF].

Algebraic structures of MZV's in positive characteristic.

In our recent paper [START_REF] Im | Hopf algebras and multiple zeta values in positive characteristic[END_REF] we addressed a question raised by a referee of a previous paper [START_REF] Dac | On Zagier-Hoffman's conjectures in positive characteristic[END_REF] by the fourth author investigating the algebraic structure of MZV's in positive characteristic. It turns out 1 that in 2016, Shuji Yamamoto [START_REF] Thakur | Multizeta values for function fields: a survey[END_REF] gave a construction of the word algebra for the shuffle product, and Shi [START_REF] Shi | Multiple zeta values over Fq[END_REF] equipped it with a conjectural Hopf algebra structure, but left some axioms for the Hopf algebra open (see [START_REF] Shi | Multiple zeta values over Fq[END_REF]Conjecture 3.2.2 and 3.2.11]). Following this, Thakur wrote to Deligne, and he [START_REF] Deligne | Letter to Dinesh Thakur[END_REF] responded with a helpful letter in 2017. In it, Deligne stated that he had never seen such a construction before and advised on how to proceed with proving and investigating it.

Conjecture 1.1. The set of MZV's in positive characteristic can be equipped with a shuffle Hopf algebra structure.

In [START_REF] Im | Hopf algebras and multiple zeta values in positive characteristic[END_REF] we proved Conjecture 1.1:

Theorem 1.2 ([32, Theorem B]). Conjecture 1.1 is true.
Further, based on the previous link among AMZV's and alternating Carlitz multiple polylogarithms introduced in [START_REF] Im | Zagier-Hoffman's conjectures in positive characteristic[END_REF], we deduced a Hopf algebra of MZV's for the stuffle product (see [32, §9]).

1.3. Main results and key ingredients.

Main results.

In this paper we present a systematic study of algebraic structures of AMZV's in positive characteristic, and hence extend our previous work for MZV's. It could be considered as an analogue of Hoffman's and Racinet's results mentioned above in the classical setting.

Let us stress that the extension from the setting of MZV's to that of AMZV's is highly nontrivial. This fact is somewhat expected as in [START_REF] Im | Zagier-Hoffman's conjectures in positive characteristic[END_REF] we have encountered a similar problem while extending the proof of Zagier-Hoffman's conjectures for MZV's to that for AMZV's.

Roughly speaking, in the case of MZV's, if we consider the corresponding word algebra denoted by C (see §3.2), then there is only one letter of fixed weight n which is denoted by x n . Consequently, it follows that there is also a unique and canonical way to define the coproducts ∆(x n ) so that the compatibility for x 1 and x n is satisfied for all n ∈ N (see §3.3). Unfortunately, when we work with AMZV's, the situation is much more complicated. Although it is not difficult to define the word algebra D of AMZV's and to show that it is associative, the definition of a coproduct is far from evident. The main issue is that in D there are q -1 letters of fixed weight, and there are no canonical ways to define coproducts for letters by using the compatibility for letters of weight 1 and other letters. To move forward we would need some new ideas.

Key ingredients.

The solution that we give here differs from that for the MZV's case at some crucial points and requires two new ingredients.

The first new ingredient is giving direct and explicit formulas for the coproduct ∆ (see §6.1). We do not define it by the recursive induction on weight as it is the case for MZV's.

The second new ingredient is introducing so-called horizontal maps which allow us to deal with letters of the same weight (see §5.2). In particular, if some property is verified for a word beginning with a specific letter, says x, then by using horizontal maps we are able to deduce this property for all words beginning with a letter of the same weight as x.

With these tools we prove that the coproduct verifies all the desired properties so that the word algebra D of AMZV's is indeed a connected graded Hopf algebra.

Generalization for colored MZV's in positive characteristic.

We now give some comments about our results. First, as mentioned above we have discovered a connection among AMZV's and the counterpart of Carlitz multiple polylogarithms in our previous work [31, §1]. Using this connection we could define another Hopf algebra structure on D called the stuffle Hopf algebra of AMZV's which extends [32, §9]. As the extension can be done without difficulty, we leave it to the reader as an exercise.

Second, our construction of the Hopf algebra structure for the word algebra D of AMZV's can be extended in a straightforward manner to the word algebra D N of colored MZV's for all N ∈ N. More precisely, the letters of D N consist of x n,ε where n ∈ N and ε is an N -th root of unity in F q . However, we warn the reader that in general, colored MZV's do not belong to K ∞ but only to a finite extension of K ∞ .

Plan of the manuscript.

The structure of this paper is as follows.

• In §2 we review some notions on the Hopf algebra. Our presentation is identical to that given in [START_REF] Im | Hopf algebras and multiple zeta values in positive characteristic[END_REF]. • In §3, we recall the word algebra C of MZV's (called the composition space in [START_REF] Im | Hopf algebras and multiple zeta values in positive characteristic[END_REF]) and the shuffle Hopf algebra structure of C. Then we review main results of loc. cit and state all necessary results that we will need in this paper so that the reader could read it easily without referring to [START_REF] Im | Hopf algebras and multiple zeta values in positive characteristic[END_REF]. • In §4 we review the definitions and basic properties of the AMZV's in positive characteristic introduced by Harada [START_REF] Harada | Alternating multizeta values in positive characteristic[END_REF].

• In §5 we introduce the word algebra or composition space D of AMZV's and equip it with the shuffle algebra structure. We then define horizontal maps in §5.2. Finally, using these maps we prove that D is an algebra (i.e., the associativity) and that there exists a shuffle map for AMZV's (see Theorem 5.9 and Theorem 5.11). • In §6 we define the coproduct on D. This definition differs from that on C as it is direct, explicit and does not use any induction on weight. Using horizontal maps we prove the compatibility and coassociativity results (see Theorem 6.5 and Theorem 6.8, respectively), establishing the main theorem of this paper (see Theorem 6.9 for a precise statement):

Theorem 1.3. The word algebra D of AMZV's equipped with the shuffle product and the above coproduct is a connected graded Hopf algebra of finite type over F q .

Review on Hopf algebra

We give a brief review on basic notions of Hopf algebras, referring to our previous paper [31, §2] on the Hopf algebra structure of MZV's in positive characteristic.

Hopf algebra.

Let k be a field. From now on, tensor products are taken over k. Let H be a k-vector space, and ι : H ⊗ H → H ⊗ H be the transposition map, defined as ι(x ⊗ y) = y ⊗ x and to be linear. Definition 2.1. An algebra over k is a triple (H, m, u) consisting of a k-vector space H together with k-linear maps m : H ⊗ H → H called the multiplication and u : k → H called the unit such that the following diagrams are commutative:

(1) associativity

H ⊗ H ⊗ H H ⊗ H H ⊗ H H m⊗id id⊗m m m (2) unitary H ⊗ k H ⊗ H k ⊗ H H id⊗u m u⊗id
where the diagonal arrows are canonical isomorphisms.

The algebra is said to be commutative if the following diagram is commutative:

H ⊗ H H ⊗ H H m ι m
Coalgebra over k is dual to the algebra over k, obtained by reversing the arrows.

Definition 2.2. A coalgebra over k is a triple (H, ∆, ) consisting of a k-vector space H together with k-linear maps ∆ : H → H ⊗ H called the coproduct and : H → k called the counit such that the following diagrams are commutative:

(1) coassociativity

H ⊗ H ⊗ H ⊗H H ⊗ H H ∆⊗id id⊗∆ ∆ ∆ (2) counitary H ⊗ k H ⊗ H k ⊗ H H id⊗ ⊗id ∆
where the diagonal arrows are canonical isomorphisms.

The coalgebra is said to be cocommutative if the following diagram is commutative:

H ⊗ H H ⊗ H H ι ∆ ∆
Definition 2.3. A bialgebra over k is a 5-tuple (H, m, u, ∆, ) consisting of an algebra (H, m, u) over k and a coalgebra (H, ∆, ) over k which are compatible, i.e., the following diagrams are commutative:

(1) product and coproduct

H ⊗ H H H ⊗ H H ⊗ H ⊗ H ⊗ H H ⊗ H ⊗ H ⊗ H ∆⊗∆ m ∆ id⊗ι⊗id m⊗m
(2) unit and coproduct

H H ⊗ H k k ⊗ k ∆ u canonical u⊗u
(3) counit and product

H H ⊗ H k k ⊗ k ⊗ m canonical (
H ⊗ H H ⊗ H H k H H ⊗ H H ⊗ H S⊗id m ∆ ∆ u id⊗S m
We note that there exist a bialgebra which does not admit an antipode (see [START_REF] Gil | Multiple zeta values: from numbers to motives[END_REF]Exercise 3.83] for an example).

Graded Hopf algebras.

We recall the notions of connected graded bialgebras.

Definition 2.5.

(1) A bialgebra (H, m, u, ∆, ) over k is said to be graded if one can write H as a direct sum of k-subspaces

H = ∞ n=0 H n ,
such that for all integers r, s ≥ 0, we have

m(H r ⊗ H s ) ⊆ H r+s and ∆(H r ) ⊆ i+j=r H i ⊗ H j .
A graded bialgebra is said to be connected if H 0 = k. (2) A graded Hopf algebra is a Hopf algebra H whose underlying bialgebra is graded and the antipode S satisfies S(H n ) ⊆ H n . (3) A graded Hopf algebra is said to be connected if H 0 = k. (4) A graded Hopf algebra is said to be of finite type if H n is a k-vector space of finite dimension.

The following proposition shows that a connected graded bialgebra automatically admits an antipode, thus it is always a Hopf algebra. Proposition 2.6. Let (H, m, u, ∆, ) be a connected graded bialgebra over k.

(1) For each element x ∈ H n with n ≥ 1, we have

∆(x) = 1 ⊗ x + x ⊗ 1 + x (1) ⊗ x (2) , where x (1) ⊗ x (2) ∈ i,j>0 i+j=n H i ⊗ H j .
Moreover, the counit vanishes on H n for all n ≥ 1.

(2) We continue the notation as in (1) and define recursively a k-linear map S : H → H given by

S(x) = x if x ∈ H 0 , -x -m(S(x (1) ) ⊗ x (2) ) if x ∈ H n with n ≥ 1.
Then H is a graded Hopf algebra whose antipode is S.

Proof. See [19, Lemma 2.1].

Review of shuffle Hopf algebra of MZV's

We briefly review some results of our previous work [START_REF] Im | Hopf algebras and multiple zeta values in positive characteristic[END_REF] on algebra structures of MZV's in positive characteristic.

Alphabet.

Let I be a countable set and A = {x n } n∈I be a set of variables indexed by I, equipped with weights w(x n ) ∈ N. The set A will be called an alphabet and its elements will be called letters. A word over the alphabet A is a finite string of letters. In particular, the empty word will be denoted by 1. The depth depth(a) of a word a is the number of letters in the string of a, so that depth(1) = 0. The weight of a word is the sum of the weights of its letter and we put w(1) = 0. Let A denote the set of all words over A. We endow A with the concatenation product defined by the following formula:

x i1 . . . x in • x j1 . . . x jm = x i1 . . . x in x j1 . . . x jm .
Let F q A be the free F q -vector space with basis A . The concatenation product extends to F q A by linearity. For a letter x a ∈ A and an element a ∈ F q A , we write simply x a a instead of x a • a. For each nonempty word a ∈ A , we can write a = x a a -where x a is the first letter of a and a -is the word obtained from a by removing x a .

Alphabet and shuffle algebra associated to MZV's.

We recall the alphabet associated to MZV's as given in [32, §4.2]. The set I will be N and we denote by Σ = {x n } n∈N with weight w(x n ) = n the alphabet attached to MZV's. We put C = F q Σ , i.e., the free F q -vector space with basis Σ .

For positive integers r, s and for all positive integers i, j such that i + j = r + s, we put

∆ i r,s = (-1) r-1 i-1 r-1 + (-1) s-1 i-1 s-1 if (q -1) | i, 0 otherwise. (3.1)
Then we define recursively two products on C as F q -bilinear maps

: C × C -→ C and ¡: C × C -→ C by setting 1 a = a 1 = a, 1 ¡ a = a ¡ 1 = a and a b = x a+b (a -¡ b -) + i+j=a+b ∆ j a,b x i (x j ¡ (a -¡ b -)), a ¡ b = x a (a -¡ b) + x b (a ¡ b -) + a b
for any nonempty words a, b ∈ Γ . We call the diamond product and ¡ the shuffle product. The unit u : F q → C is given by sending 1 to the empty word. One of the main results of [START_REF] Im | Hopf algebras and multiple zeta values in positive characteristic[END_REF] reads as follows (see [32, Theorem A]):

Theorem 3.1. The spaces (C, ) and (C, ¡) are commutative F q -algebras.

Shuffle Hopf algebra associated to MZV's.

We briefly recall the coproduct as in [32, §7.1]

∆ : C → C ⊗ C.
We first define it on Σ by induction on weight and extend by F q -linearity to C. We put

∆(1) := 1 ⊗ 1, ∆(x 1 ) := 1 ⊗ x 1 + x 1 ⊗ 1.
Let w ∈ N and we suppose that we have defined ∆(u) for all words u of weight w(u) < w. We now give a formula for ∆(u) for all words u with w(u) = w. For such a word u with depth(u) > 1, we put u = x u v with w(v) < w. Since x u and v are both of weight less than w, we have already defined

∆(x u ) = 1 ⊗ x u + a u ⊗ b u , ∆(v) = a v ⊗ b v .
Then we set

∆(u) := 1 ⊗ u + (a u a v ) ⊗ (b u ¡ b v ).
Our last task is to define ∆(x w ). We know that

x 1 ¡ x w-1 = x w + x 1 x w-1 + x w-1 x 1 + 0<j<w ∆ j 1,w-1 x w-j x j
where all the words x w-j x j have weight w and depth 2 and all ∆ j 1,w-1 defined as in (3.1) belong to F q . Therefore, we set

∆(x w ) := ∆(x 1 ) ¡∆(x w-1 ) -∆(x 1 x w-1 ) -∆(x w-1 x 1 ) - 0<j<w ∆ j 1,w-1 ∆(x w-j x j ).
By induction on weight, we can check that ∆ preserves the grading.

Next, the counit : C → F q is defined by putting (1) = 1 and (u) = 0 otherwise. We proved (see [32, Theorem B]): Theorem 3.2. The connected graded bialgebra (C, ¡,u,∆, ) is a connected graded Hopf algebra over F q .

3.4. Some properties of the coproduct.

In this section we collect some properties of the coproduct that will be useful in the sequel. Lemma 3.3. For all words u ∈ C, we have

∆(u) = 1 ⊗ u + a u ⊗ b u , where a u = 1.
Proof. See [32, Lemma 7.1].

Lemma 3.4. Let u and v be two nonempty words in Σ . Suppose that

∆(u) = 1 ⊗ u + u (1) ⊗ u (2) , and
∆(v) = 1 ⊗ v + v (1) ⊗ v (2) . Then ∆(u v) = 1 ⊗ (u v) + (u (1) v (1) ) ⊗ (u (2) ¡ v (2) ).
Proof. See [START_REF] Im | Hopf algebras and multiple zeta values in positive characteristic[END_REF]Lemma 8.5].

Finally, there is a useful formula for ∆(x n ) for x n ∈ Σ that we briefly explain below.

Lemma 3.5. Let n ∈ N. Then we have (1) for all j < n,

∆ j 1,n = 1 if (q -1) | j 0 otherwise. (2) ∆ n 1,n = 0.
Proof. The result is straightforward from the definition of ∆ j 1,n . See [START_REF] Im | Hopf algebras and multiple zeta values in positive characteristic[END_REF]Lemma 8.1].

Next, we introduce the bracket operator by the following formula: for any a = x i1 . . .

x im ∈ Σ , (3.2) [a] := (-1) m ∆ i1 1,w(a)+1 • • • ∆ im 1,w(a)+1 x i1 ¡ • • • ¡ x im .
As a matter of convention, we also agree that [START_REF] Anderson | t-motives[END_REF] = 1. Then we showed (see [32, Theorem C]):

Proposition 3.6. We keep the above notation. Then for all n ∈ N, we have

∆(x n ) = 1 ⊗ x n + r∈N,a∈ Σ r+w(a)=n r + depth(a) -2 depth(a) x r ⊗ [a].

Alternating multiple zeta values

We recall some notions on MZV's and AMZV's in positive characteristic, referring to [START_REF] Harada | Alternating multizeta values in positive characteristic[END_REF][START_REF] Im | Zagier-Hoffman's conjectures in positive characteristic[END_REF][START_REF] Im | Hopf algebras and multiple zeta values in positive characteristic[END_REF].

Letting

s = (s 1 , . . . , s n ) ∈ N n and ε = (ε 1 , . . . , ε n ) ∈ (F × q ) n , we set s -:= (s 2 , . . . , s n ) and ε -:= (ε 2 , . . . , ε n ). A positive array ε s is an array of the form ε s = ε 1 • • • ε n s 1 • • • s n .
We recall the power sums studied by Thakur [START_REF] Thakur | Shuffle relations for function field multizeta values[END_REF]. For d ∈ Z and for s = (s 1 , . . . , s n ) ∈ N n we introduce We also recall the following analogues of the power sums after Harada [START_REF] Harada | Alternating multizeta values in positive characteristic[END_REF]. For a positive array

S d (s) =
ε s = ε 1 . . . ε n s 1 . . . s n , we introduce S d ε s = a1,...,an∈A+ d=deg a1>•••>deg an≥0 ε deg a1 1 . . . ε deg an n a s1 1 . . . a sn n ∈ K
and

S <d ε s = a1,...,an∈A+ d>deg a1>•••>deg an≥0 ε deg a1 1 . . . ε deg an n a s1 1 . . . a sn n ∈ K.
One verifies easily the following formulas:

S d ε s = ε d S d (s), S d 1 . . . 1 s 1 . . . s n = S d (s 1 , . . . , s n ), S <d 1 . . . 1 s 1 . . . s n = S <d (s 1 , . . . , s n ), S d ε s = S d ε 1 s 1 S <d ε - s - .
In [START_REF] Harada | Alternating multizeta values in positive characteristic[END_REF] Harada introduced the alternating multiple zeta value (AMZV) in positive characteristic by setting

ζ A ε s := d≥0 S d ε s = a1,...,an∈A+ deg a1>•••>deg an≥0 ε deg a1 1 . . . ε deg an n a s1 1 . . . a sn n ∈ K ∞ .
Then he proved several fundamental properties of these values. In particular, it is shown that the product of two AMZV's can be expressed as a linear combination with coefficients in F q of AMZV's. To do so, we first recall Chen's formula in [START_REF] Chen | On shuffle of double zeta values over Fq[t][END_REF].

For integers a, b with b ≥ 0, we recall the binomial number defined by

a b := a(a -1) . . . (a -b + 1) b! .
It should be remarked that a b = 0 if b > a ≥ 0. Then refining the work of Thakur [START_REF] Thakur | Shuffle relations for function field multizeta values[END_REF], Chen [14, Theorem 3.1] showed that for positive integers r, s and for all d ∈ N,

S d (r)S d (s) = S d (r + s) + i+j=r+s ∆ j r,s S d (i, j),
where the coefficients ∆ j r,s are defined as in (3.1). As a direct consequence of Chen's formula, we obtain Lemma 4.1. Let α, β be two elements in F * q , and let r, s be two positive integers. Then for all d ∈ N, we have

S d α r S d β s = S d αβ r + s + i+j=r+s ∆ j r,s S d αβ 1 i j ,
where the indices i, j are positive integers. Proof. See for example [25, Lemma 2.3].

Shuffle algebra of AMZV's

In the following sections §5 and §6 we construct the Hopf shuffle algebra of AMZV's and note that the extension from the setting of MZV's to that of AMZV's is not straightforward as we have explained in the introduction. The key ingredient of our construction is the notion of so-called horizontal maps that we introduce in this section. These maps will allow us to deal with letters of the same weight (see §5.2). In the following section, we will give direct and explicit formulas for the coproduct and then prove that such formulas verify all desired properties. 5.1. Alphabet and shuffle algebra associated to AMZV's.

We put

I = {(n, ε) : n ∈ N, ε ∈ F * q }. Let Γ = {x n,ε } n∈N,ε∈F *
q be the alphabet associated to AMZV's indexed by I (see §3.1 for notation). We define the weights by w(x n,ε ) = n. We define D := F q Γ to be the free F q -vector space with basis Γ .

Inspired by §3 we define recursively two products on D as F q -bilinear maps

: D × D -→ D and ¡: D × D -→ D by setting 1 a = a 1 = a, 1 ¡ a = a ¡ 1 = a and a b = x a+b,αβ (a -¡ b -) + i+j=a+b ∆ j a,b x i,αβ (x j,1 ¡ (a -¡ b -)), a ¡ b = x a,α (a -¡ b) + x b,β (a ¡ b -) + a b
for any nonempty words a, b ∈ Γ with a = x a,α a -, b = x b,β b -. We call the diamond product and ¡ the shuffle product. We proceed the proof by induction on depth(a) + depth(b). If one of a or b is the empty word, then (5.1) holds trivially. We assume that (5.1) holds when depth(a) + depth(b) < n with n ∈ N and n ≥ 2. We need to show that (5.1) holds when depth(a) + depth(b) = n.

Indeed, assume that a = x a,α a -, b = x b,β b -, we have

a b = x a+b,αβ (a -¡ b -) + i+j=a+b ∆ j a,b x i,αβ (x j,1 ¡ (a -¡ b -)), b a = x b+a,βα (b -¡ a -) + i+j=b+a ∆ j b,a x i,βα (x j,1 ¡ (b -¡ a -)).
It follows from the induction hypothesis that a -¡b -= b -¡a -, hence a b = b a.

On the other hand, we have

a ¡ b = x a,α (a -¡ b) + x b,β (a ¡ b -) + a b, b ¡ a = x b,β (b -¡ a) + x a,α (b ¡ a -) + b a.
It follows from the induction hypothesis and the above arguments that a

-¡ b = b ¡ a -, a ¡ b -= b -¡ a and a b = b a, hence a ¡ b = b ¡ a.
This proves the proposition.

Horizontal maps.

In this section, we introduce horizontal maps which are crucial in the sequel. For each element α ∈ F * q , we consider the F q -linear map

ϕ α : C -→ D,
which maps the empty word 1 ∈ Σ to the empty word 1 ∈ Γ and maps a nonempty word x i1 x i2 . . . x in to x i1,α x i2,1 . . . x in,1 . For all nonempty words a ∈ Σ with a = x a a -, we get

ϕ α (a) = ϕ α (x a )ϕ 1 (a -) = x a,α ϕ 1 (a -).
As w(ϕ α (a)) = w(a) for all α ∈ F * q and a ∈ C, we call these maps horizontal maps. We note that they are all injective. Lemma 5.2. Let a, b be two words in Σ . Then we have

ϕ 1 (a b) = ϕ 1 (a) ϕ 1 (b) and ϕ 1 (a ¡ b) = ϕ 1 (a) ¡ ϕ 1 (b).
Proof. The proof can be done by induction on depth(a) + depth(b). We omit the details.

As a direct consequence, ϕ 1 is a homomorphism of algebras from (C, ) to (D, ) (resp. from (C, ¡) to (D, ¡)).

Notation 5.3. Since ϕ 1 is injective, from now on for a ∈ C we write a ∈ D instead of ϕ 1 (a) ∈ D.

Further, we extend the horizontal map as follows:

ϕ α : D → D; 1 → 1, u → x u,αε u -
for nonempty words u ∈ Γ with u = x u,ε u -. This is a natural extension under the identification of a ∈ C and ϕ 1 (a) ∈ D. We note that if u ∈ Γ is a nonempty word written as u = x u,ε u -, then for all α ∈ F * q , ϕ α (u) = ϕ α (x u,ε )u -.

Under this extension, the above lemma holds for a, b ∈ Γ . Proof. Consider the horizontal maps defined on Σ . We write a = x a a -, b = x b b - with x a , x b ∈ Σ and a -, b -∈ Σ . From Lemma 5.2, we have

ϕ αβ (a b) = ϕ αβ x a+b (a -¡ b -) + i+j=a+b ∆ j a,b x i (x j ¡ (a -¡ b -)) = x a+b,αβ (a -¡ b -) + i+j=a+b ∆ j a,b x i,αβ (x j ¡ (a -¡ b -)) = x a,α a -x b,β b - = ϕ α (a) ϕ β (b).
The proof for the horizontal maps on Γ is parallel. Let a = x a,ε1 a -, b = x b,ε2 b - with x a,ε1 , x b,ε2 ∈ Γ and a -, b -∈ Γ . From Lemma 5.2 and its following remark, we have

ϕ αβ (a b) = ϕ αβ x a+b,ε1ε2 (a -¡ b -) + i+j=a+b ∆ j a,b x i,ε1ε2 (x j ¡ (a -¡ b -)) = x a+b,αβε1ε2 (a -¡ b -) + i+j=a+b ∆ j a,b x i,αβε1ε2 (x j ¡ (a -¡ b -)) = x a,αε1 a -x b,βε2 b - = ϕ α (a) ϕ β (b).
This proves the lemma.

We end this section by noting that horizontal maps are associative. More precisely, we prove Lemma 5.5. Let α, β ∈ F * q . For any word a ∈ Γ , we have ϕ αβ (a) = ϕ α (ϕ β (a)).

Proof. The proof is straightforward from the definition of horizontal maps.

Associativity.

In this subsection, we first show that the diamond product is associative over Γ (see Proposition 5.6), and then prove the associativity of the diamond product and the shuffle product on D (see Proposition 5.8).

Proposition 5.6. Let x a,α , x b,β , x c,γ be letters in Γ. Then we have

(x a,α x b,β ) x c,γ = x a,α (x b,β x c,γ ).
Proof. From Lemma 5.4, we have

(x a,α x b,β ) x c,γ = (ϕ α (x a ) ϕ β (x b )) ϕ γ (x c ) = ϕ αβ (x a x b ) ϕ γ (x c ) = ϕ αβγ ((x a x b ) x c ), x a,α (x b,β x c,γ ) = ϕ α (x a ) (ϕ β (x b ) ϕ γ (x c )) = ϕ α (x a ) ϕ βγ (x b x c ) = ϕ αβγ (x a (x b x c )).
From the associativity of the diamond product on C (see Theorem 3.1), we have (x a x b ) x c = x a (x b x c ). This proves the proposition.

Next, we define recursively a product on D as a F q -bilinear map

: D × D -→ D by setting 1 a = a 1 = a and a b = x a,α (a -¡ b)
for any nonempty words a, b ∈ Γ with a = x a,α a -. We call the triangle product. The triangle product is neither commutative nor associative, as one verifies at once. Lemma 5.7. For all words a, b ∈ Γ with a = x a,α a -, b = x b,β b -, we have

(1) a b = (x a,α x b,β ) (a -¡ b -). (2) a ¡ b = a b + b a + a b.
Proof. We have

a b = x a+b,αβ (a -¡ b -) + i+j=a+b ∆ j a,b x i,αβ (x j ¡ (a -¡ b -)) = x a+b,αβ (a -¡ b -) + i+j=a+b ∆ j a,b (x i,αβ x j ) (a -¡ b -) =   x a+b,αβ + i+j=a+b ∆ j a,b x i,αβ x j   (a -¡ b -) = (x a,α x b,β ) (a -¡ b -).
This proves part [START_REF] Anderson | t-motives[END_REF]. Part (2) is straightforward from the commutativity of the shuffle product (see Proposition 5.1). This completes the proof.

We now ready to prove the main result of this subsection.

=   x a+b,αβ (a -¡ b -) + i+j=a+b ∆ j a,b x i,αβ (x j ¡ (a -¡ b -))   c = x a+b,αβ (a -¡ b -) c + i+j=a+b ∆ j a,b x i,αβ (x j ¡ (a -¡ b -)) c = (x a+b,αβ x c,γ ) ((a -¡ b -) ¡ c -) + i+j=a+b ∆ j a,b (x i,αβ x c,γ ) ((x j ¡ (a -¡ b -)) ¡ c -).
For all i, j ∈ N with i + j = a + b, it follows from the induction hypothesis that

(x i,αβ x c,γ ) ((x j ¡ (a -¡ b -)) ¡ c -) = (x i,αβ x c,γ ) (x j ¡ ((a -¡ b -) ¡ c -)) =   x i+c,αβγ + i1+j1=i+c ∆ j1 i,c x i1,αβγ x j1   (x j ¡ ((a -¡ b -) ¡ c -)) = x i+c,αβγ (x j ¡ ((a -¡ b -) ¡ c -)) + i1+j1=i+c ∆ j1 i,c x i1,αβγ x j1 (x j ¡ ((a -¡ b -) ¡ c -)) = x i+c,αβγ (x j ¡ ((a -¡ b -) ¡ c -)) + i1+j1=i+c ∆ j1 i,c x i1,αβγ ((x j1 ¡ x j ) ¡ ((a -¡ b -) ¡ c -)) = (x i+c,αβγ x j ) ((a -¡ b -) ¡ c -) + i1+j1=i+c ∆ j1 i,c x i1,αβγ (x j1 ¡ x j ) ((a -¡ b -) ¡ c -) =   x i+c,αβγ x j + i1+j1=i+c ∆ j1 i,c x i1,αβγ (x j1 ¡ x j )   ((a -¡ b -) ¡ c -) = ((x i,αβ x j ) x c,γ ) ((a -¡ b -) ¡ c -). Thus (a b) c = (x a+b,αβ x c,γ ) ((a -¡ b -) ¡ c -) + i+j=a+b ∆ j a,b ((x i,αβ x j ) x c,γ ) ((a -¡ b -) ¡ c -) =   (x a+b,αβ x c,γ ) + i+j=a+b ∆ j a,b (x i,αβ x j x c,γ )   ((a -¡ b -) ¡ c -) = ((x a,α x b,β ) x c,γ ) ((a -¡ b -) ¡ c -).
On the other hand, from Proposition 5.1 and the above arguments, we deduce that

a (b c) = (b c) a = ((x b,β x c,γ ) x a,α ) ((b -¡ c -) ¡ a -) = (x a,α (x b,β x c,γ )) (a -¡ (b -¡ c -)).
It follows from Proposition 5.6 that (x a,α x b,β ) x c,γ = x a,α (x b,β x c,γ ). Moreover, it follows from the induction hypothesis that (a We have

-¡ b -) ¡ c -= a -¡ (b -¡ c -).
(a b) c = x a,α (a -¡ b) c = (x a,α x c,γ ) ((a -¡ b) ¡ c -), a (c b) = a x c,γ (c -¡ b) = (x a,α x c,γ ) (a -¡ (c -¡ b)).
From the induction hypothesis and commutativity of shuffle product, one deduces that (a b) c = a (c b).

We have

(b a) c = x b,β (b -¡ a) c = x b,β ((b -¡ a) ¡ c) and (b c) a = x b,β (b -¡ c) a = x b,β ((b -¡ c) ¡ a).
From the induction hypothesis and commutativity of shuffle product, one deduces that (b a) c = (b c) a.

It follows from the induction hypothesis and that From the above arguments, we conclude that (a ¡ b) ¡ c = a ¡ (b ¡ c). This completes the proof.

(a b) c =   x a+b,αβ (a -¡ b -) + i+j=a+b ∆ j a,b x i,αβ (x j ¡ (a -¡ b -))   c = x a+b,αβ (a -¡ b -) c + i+j=a+b ∆ j a,b x i,αβ (x j ¡ (a -¡ b -)) c = x a+b,αβ ((a -¡ b -) ¡ c) + i+j=a+b ∆ j a,b x i,αβ ((x j ¡ (a -¡ b -)) ¡ c) = x a+b,αβ ((a -¡ b -) ¡ c) + i+j=a+b ∆ j a,b x i,αβ (x j ¡ ((a -¡ b -) ¡ c)) = x a+b,αβ ((a -¡ b -) ¡ c) + i+j=a+b ∆ j a,b x i,αβ x j ((a -¡ b -) ¡ c) =   x a+b,αβ + i+j=a+b ∆ j a,b x i,αβ x j   ((a -¡ b -) ¡ c) = (x a,α x b,β ) ((a -¡ b -) ¡ c) and a (b c) = a x b,β (b -¡ c) = (x a,α x b,β ) (a -¡ (b -¡ c)).
As for C, we define the unit u : F q → D, sending 1 ∈ F q to an empty word 1 ∈ D. Then, as a direct consequence of Proposition 5.1 and Proposition 5.8, we obtain the following result.

Theorem 5.9. The spaces (D, ) and (D, ¡) are commutative F q -algebras.

The following proposition summarizes several properties of different products , and ¡ that are shown above and will be useful in the sequel.

Shuffle map for AMZV's in positive characteristic.

For each d ∈ Z, we define the following two F q -linear maps

S <d : D → K ∞ and ζ A : D → K ∞ ,
both of which map the empty word 1 ∈ D to 1 ∈ K ∞ , and the word x s1,ε1 . . . x sr,εr

to S <d ε 1 • • • ε n s 1 • • • s n and ζ A ε 1 • • • ε n s 1 • • • s n respectively, identifying presenta- tions S <d (x s1,ε1 . . . x sr,εr ) = S <d ε 1 • • • ε n s 1 • • • s n , ζ A (x s1,ε1 . . . x sr,εr ) = ζ A ε 1 • • • ε n s 1 • • • s n .
Theorem 5.11. For a, b ∈ D and d ∈ Z we have

S <d (a ¡ b) = S <d (a)S <d (b). ζ A (a ¡ b) = ζ A (a)ζ A (b).
Proof. The ¡ product is defined to satisfy this equation. We can check that the ¡ follows the inductive steps for product of power sums and ζ A as follows.

When any of a and b is the empty word, the theorem holds trivially. Let a = x a,α a -and b = x b,β b -be nonempty words. We proceed the proof by induction on depth(a) + depth(b). Following the proof of [25, Lemma 2.5] and applying Lemma 4.1,

S <d (a)S <d (b) = m<d S m (x a,α )S <m (a -)S <m (b) + m<d S m (x b,β )S <m (a)S <m (b -) + m<d S m (x a,α )S m (x b,β )S <m (a -)S <m (b -) = m<d S m (x a,α )S <m (a -)S <m (b) + m<d S m (x b,β )S <m (a)S <m (b -) + m<d S m (x a+b,αβ )S <m (a -)S <m (b -) + m<d ∆ j a,b i+j=a+b S m (x i,αβ x j )S <m (a -)S <m (b -).
This is parallel to the recursive definition of ¡. Proof is the same for ζ A .

Shuffle Hopf algebra of AMZV's

In §5 we have defined the shuffle algebra structure of AMZV's. This section is devoted to the construction of the Hopf algebra structure of AMZV's. In §6.1 we present the coproduct for the word algebra D. In §6.2 we prove the compatibility of the coproduct in Theorem 6.5. Unlike the MZV's case, the proof of this theorem uses the induction on depth instead of that on weight. Then in §6.3 we prove the coassociativity of the coproduct in Theorem 6.8. Putting all together we get the main result of this paper (see Theorem 6.9).

Coproduct.

This section is inspired by our previous work [32, §8]. We will introduce the coproduct

∆ : D → D ⊗ D.
First, we define the coproduct of depth one. We put

∆(1) := 1 ⊗ 1, ∆(x 1,ε ) := 1 ⊗ x 1,ε + x 1,ε ⊗ 1
for all ε ∈ F * q . Further, we define the coproduct of depth one as follows. Definition 6.1. For all n ∈ N and ε ∈ F * q , we set

∆(x n,ε ) = 1 ⊗ x n,ε + r∈N,a∈ Σ r+w(a)=n r + depth(a) -2 depth(a) x r,ε ⊗ [a].
Here we recall that [a] is given as in (3.2) and we identify a ∈ Σ with ϕ 1 (a) ∈ Γ.

Next, we define the coproduct for words of depth greater than one. Let n ∈ N with n > 1 and we suppose that we have defined ∆(u) for all words u of depth(u) < n. We now give a formula for ∆(u) for all words u with depth(u) = n. For such a word u, we put u = x u,ε v with depth(v) = n -1 < n. Since x u,ε and v are both of depth strictly less than n, we have already defined

∆(x u,ε ) = 1 ⊗ x u,ε + a u,ε ⊗ b u,ε , and ∆(v) = a v ⊗ b v , with a u,ε ∈ Γ and b u,ε , a v , b v ∈ Γ . Then we set ∆(u) := 1 ⊗ u + (a u,ε a v ) ⊗ (b u,ε ¡ b v ) = 1 ⊗ u + (a u,ε a v ) ⊗ (b u,ε ¡ b v ).
Here the last equality holds as a u,ε ∈ Γ. We end this section by proving the following result: Proposition 6.2. For any nonempty word u ∈ Γ , we have

∆ (ϕ ε (u)) = (ϕ ε ⊗ Id) (∆(u)) + (Id ⊗ϕ ε -ϕ ε ⊗ Id)(1 ⊗ u). Proof. Note that for a = x a a -∈ Γ , b ∈ Γ and ε ∈ F * q , ϕ ε (a) b = x a,ε (a -¡ b) = ϕ ε (a b).
We mention that this does not hold when a = 1. We first consider the case where depth(u) = 1, i.e., u belongs to Γ. We put u = x n,α with n ∈ N and α ∈ F * q . Thus ϕ ε (u) = x n,εα = ϕ εα (x n ). Recall that (see Proposition 3.6)

∆(x n ) = 1 ⊗ x n + r∈N,a∈ Σ r+w(a)=n r + depth(a) -2 depth(a) x r ⊗ [a].
Therefore, Definition 6.1 can be written as

∆ (u) = ∆ (ϕ α (x n )) = (ϕ α ⊗ Id) (∆(x n ) -1 ⊗ x n ) + (Id ⊗ϕ α ) (1 ⊗ x n ) = (ϕ α ⊗ Id) (∆(x n )) + (Id ⊗ϕ α -ϕ α ⊗ Id) (1 ⊗ x n ) .
Similarly,

∆ (ϕ ε (u)) = ∆ (ϕ εα (x n )) = (ϕ εα ⊗ Id) (∆(x n ) -1 ⊗ x n ) + (Id ⊗ϕ εα ) (1 ⊗ x n ) = (ϕ εα ⊗ Id) (∆(x n )) + (Id ⊗ϕ εα -ϕ εα ⊗ Id) (1 ⊗ x n ) .
By Lemma 5.5, we conclude that for u = x n,α ,

∆ (ϕ ε (u)) = (ϕ ε ⊗ Id) (∆(u)) + (Id ⊗ϕ ε -ϕ ε ⊗ Id)(1 ⊗ u). Next, suppose that u = x u,α v ∈ Γ with depth(u) ≥ 2. Let ∆(x u ) = 1 ⊗ x u + a u ⊗ b u , and ∆(v) = a v ⊗ b v .
We have a u ∈ Σ, b u ∈ Σ with a u = 1 by Lemma 3.3, and a v , b v ∈ Γ . We have seen that

∆(x u,α ) = 1 ⊗ x u,α + a u,α ⊗ b u .
By definition,

∆(u) = 1 ⊗ u + (a u,α a v ) ⊗ (b u ¡ b v ) = 1 ⊗ u + (a u,α a v ) ⊗ (b u ¡ b v ).
The last equality follows from the fact that depth(a u,α ) = 1.

Since

ϕ ε (u) = x u,εα v and ∆ (ϕ ε (x u,α )) = 1 ⊗ ϕ ε (x u,α ) + ϕ ε (a u,α ) ⊗ b u , ∆ (ϕ ε (u)) = ∆ (ϕ ε (x u,α )v) = 1 ⊗ ϕ ε (u) + (ϕ ε (a u,α )a v ) ⊗ (b u ¡ b v ) = 1 ⊗ ϕ ε (u) + ϕ ε (a u,α a v ) ⊗ (b u ¡ b v ).
Thus we get

∆ (ϕ ε (u)) = (ϕ ε ⊗ Id) (∆(u)) + (Id ⊗ϕ ε -ϕ ε ⊗ Id)(1 ⊗ u).
6.2. Compatibility.

Compatibility for letters.

We want to prove the following result (for MZV's, see [32, Proposition 7.5]):

Proposition 6.3. Let u, v ∈ N and α, β ∈ F * q . Then ∆(x u,α ¡ x v,β ) = ∆(x u,α ) ¡ ∆(x v,β ). Proof. Let ∆(x u ) = 1 ⊗ x u + a u ⊗ b u , ∆(x v ) = 1 ⊗ x v + a v ⊗ b v .
By Proposition 6.2, we have

∆(x u,α ) = 1 ⊗ x u,α + a u,α ⊗ b u , ∆(x v,β ) = 1 ⊗ x v,β + a v,β ⊗ b v ,
where a u,α = ϕ α (a u ) and a v,β = ϕ β (a v ). Here we recall that a u and a v are of depth 1.

Since

x u,α ¡ x v,β = x u,α x v,β + x v,β x u,α + ϕ αβ (x u x v ) (by Lemma 5.4), ∆ (x u,α ¡ x v,β ) =1 ⊗ x u,α x v,β + a u,α ⊗ (b u ¡ x v,β ) + (a u,α a v,β ) ⊗ (b u ¡ b v ) + 1 ⊗ x v,β x u,α + a v,β ⊗ (b v ¡ x u,α ) + (a v,β a u,α ) ⊗ (b v ¡ b u ) + ∆(ϕ αβ (x u x v )).
On the other hand,

∆(x u,α ) ¡ ∆(x v,β ) =1 ⊗ (x u,α ¡ x v,β ) + a u,α ⊗ (b u ¡ x v,β ) + a v,β ⊗ (b v ¡ x u,α ) + (a u,α ¡ a v,β ) ⊗ (b u ¡ b v ).
By expanding ¡ in each tensorand and canceling, the proposition holds when

∆(ϕ αβ (x u x v )) = 1 ⊗ ϕ αβ (x u x v ) + (a u,α a v,β ) ⊗ (b u ¡ b v ) = 1 ⊗ ϕ αβ (x u x v ) + (ϕ αβ ⊗ Id) (a u a v ) ⊗ (b u ¡ b v ) .
By Proposition 6.2, the proposition is reduced into

∆(x u x v ) = 1 ⊗ (x u x v ) + (a u a v ) ⊗ (b u ¡ b v ),
which is a special case of Lemma 3.4. 6.2.2. Compatibility for words of arbitrary depth. Lemma 6.4. Let u, v ∈ Γ . Then u ¡ v can be presented as the sum of words whose depths are all less than or equal to depth(u) + depth(v).

Proof. We proceed with the induction on the total depth depth(u) + depth(v).

The lemma is trivial when u or v is trivial. If u = x u,α and v = x v,β , then the lemma is also immediate from the expression

u ¡ v = x u+v,αβ + x u,α x v,β + x v,β x u,α + i+j=u+v ∆ j u,v x i,αβ x j .
So we assume that depth(u), depth(v) ≥ 1 with at least one of u and v has depth > 1. Now assume the lemma holds for all a, b ∈ Γ such that depth(a

) + depth(b) < depth(u) + depth(v). Let u = x u,α u -, v = x v,β v -. Then x u,α u -¡ x v,β v -= x u+v,αβ (u -¡ v -) + x u,α (u -¡ v) + x v,β (u ¡ v -) + i+j=u+v ∆ j u,v x i,αβ (x j ¡ u -¡ v -)
From the induction hypothesis, the first three terms can be presented as a sum of words with depths less than or equal to depth(u) + depth(v). For each j, x j ¡ u -= u -¡ x j can be presented as a sum of words whose depths are less than or equal to depth(u) from the induction hypothesis. Therefore, x i (x j ¡ u -¡ v -) can be presented as sum of words with depth depth(u) + depth(v). This proves the lemma.

The proof of the compatibility of ∆ and ¡ for Γ is parallel to that for Σ . Theorem 6.5. Let u, v ∈ Γ . Then ∆ and ¡ are compatible for u, v, that means

∆(u ¡ v) = ∆(u) ¡ ∆(v).
Note that the theorem is immediate when u or v is trivial. Also, depth(u) = depth(v) = 1 case is shown in Proposition 6.3. Before proving this theorem, we need some preparatory results. Lemma 6.6. Let u = x u,α u -, v = x v,β v -be two words in Γ , whose depths are ≥ 1. Assume that ∆ and ¡ are compatible for all words a, b ∈ Γ with depth(a) + depth(b) < depth(u) + depth(v). Further, we suppose that ∆(

x u u -¡ x v v -) = ∆(x u u -) ¡ ∆(x v v -).
Then we have

∆(u ¡ v) = ∆(u) ¡ ∆(v). Proof. Let ∆(x u ) = 1 ⊗ x u + a u ⊗ b u , ∆(x v ) = 1 ⊗ x v + a v ⊗ b v , ∆(u -) = a u-⊗ b u-, ∆(v -) = a v-⊗ b v-, with x u , x v , a u , a v ∈ Σ, b u , b v ∈ Σ and a u-, b u-, a v-, b v-∈ Γ . We have ∆(x u,α ) = 1 ⊗ x u,α + a u,α ⊗ b u , ∆(x v,β ) = 1 ⊗ x v,β + a v,β ⊗ b v , where a u,α = ϕ α (a u ), a v,β = ϕ β (a v ). Note that ∆(x u,α u -) = 1 ⊗ x u,α u -+ a u,α a u-⊗ (b u ¡ b u-), ∆(x v,β v -) = 1 ⊗ x v,β v -+ a v,β a v-⊗ (b v ¡ b v-).
Therefore, we have

∆(u) ¡ ∆(v) = ∆(x u,α u -) ¡ ∆(x v,β v -) (6.1) = 1 ⊗ x u,α u -+ a u,α a u-⊗ (b u ¡ b u-) ¡ 1 ⊗ x v,β v -+ a v,β a v-⊗ (b v ¡ b v-) =1 ⊗ (x u,α u -¡ x v,β v -) + a v,β a v-⊗ (x u,α u -¡ b v ¡ b v-) + a u,α a u-⊗ (x v,β v -¡ b u ¡ b u-) + (a u,α a u-¡ a v,β a v-) ⊗ (b u ¡ b v ¡ b u-¡ b v-).
On the other hand, from Lemma 5.4 and Lemma 5.7

u ¡ v = u v + v u + ϕ αβ (x u u -x v v -), thus we have ∆(u ¡ v) = S 1 + S 2 + S 3 (6.2)
where

S 1 = ∆(u v) = ∆(x u,α (u -¡ v)), S 2 = ∆(v u) = ∆(x v,β (v -¡ u)), S 3 = ∆(ϕ αβ (x u u -x v v -)) = (ϕ αβ ⊗ Id)∆(x u u -x v v -) + (Id ⊗ϕ αβ -ϕ αβ ⊗ Id)(1 ⊗ (x u u -x v v -)).
We calculate S 1 and S 2 . Since depth(u -) + depth(v) < depth(u) + depth(v), the induction hypothesis yields

∆(u -¡ v) = ∆(u -) ¡ ∆(x v,β v -) = a u-⊗ b u-¡ 1 ⊗ x v,β v -+ a v,β a v-⊗ (b v ¡ b v-) = a u-⊗ (b u-¡ x v,β v -) + (a u-¡ a v,β a v-) ⊗ (b v ¡ b u-¡ b v-), so S 1 = 1 ⊗ (x u,α (u -¡ v)) + a u,α a u-⊗ (b u ¡ b u-¡ x v,β v -) + a u,α (a u-¡ a v,β a v-) ⊗ (b u ¡ b v ¡ b u-¡ b v-).
A similar calculation yields

S 2 = 1 ⊗ (x v,β (u ¡ v -)) + a v,β a v-⊗ (b v ¡ b v-¡ x u,α u -) + a v,β (a v-¡ a u,α a u-) ⊗ (b u ¡ b v ¡ b u-¡ b v-).
What we want to show is that two equations (6.1) and (6.2) coincide, that is,

X := ∆(u ¡ v) -∆(u) ¡ ∆(v) = 0.
First, we collect the terms of the form 1⊗ . The equation for the right tensorands of such terms in X is

x u,α (u -¡ v) + x v,β (u ¡ v -) + ϕ αβ (x u u -x v v -) -(x u u -x v v -) -u ¡ v = x u,α (u -¡ v) + x v,β (u ¡ v -) + ϕ αβ (x u u -x v v -) -(x u u -x v v -) -u v -v u -u v = -(x u u -x v v -)
by Lemma 5.4, that is, the terms of the form 1 ⊗ in X are collected as

-1 ⊗ (x u u -x v v -).
Next we collect the terms of the form

⊗ (b u ¡ b v ¡ b u-¡ b v-).
For fixed indices, the equation for the left tensorands of such terms in X is

a u,α (a u-¡ a v,β a v-) + a v,β (a v-¡ a u,α a u-) -(a u,α a u-¡ a v,β a v-) = -ϕ αβ (a u a u-a v a v-),
by expanding the shuffle product of the third term and reducing it. That is, the terms of such a form in X are collected as

-(ϕ αβ ⊗ Id) (a u a u-a v a v-) ⊗ (b u ¡ b v ¡ b u-¡ b v-).
Some of the remaining terms cancel each other. One can check that

X =(ϕ αβ ⊗ Id)∆(x u u -x v v -) -1 ⊗ (x u u -x v v -) -(ϕ αβ ⊗ Id) (a u a u-a v a v-) ⊗ (b u ¡ b v ¡ b u-¡ b v-), which vanishes when (6.3) ∆(x u u -x v v -) = 1 ⊗ (x u u -x v v -) + (a u a u-a v a v-) ⊗ (b u ¡b v ¡b u-¡b v-).
Recall the assumption of the lemma, ∆(

x u u -¡ x v v -) = ∆(x u u -) ¡ ∆(x v v -).
Expanding this by using definitions of ¡, and calculating, we can conclude the desired Equation (6.3). Thus the lemma holds. Proposition 6.7. Let x u u, x v v be two words in Γ , whose first letters are with trivial character 1 ∈ F * q . If ∆ and ¡ are compatible for all words a, b ∈ Γ with

depth(a) + depth(b) < depth(x u u) + depth(x v v), then ∆(x u u ¡ x v v) = ∆(x u u) ¡ ∆(x v v).
The proof closely follows that of [START_REF] Im | Hopf algebras and multiple zeta values in positive characteristic[END_REF]Proposition 7.4], with the only difference being that the induction step is now based on depth rather than weight induction. For the sake of clarity, we include a complete proof.

Note that the proof is valid for u = 1 or v = 1 cases. In those cases, some sum presentation is vacuous, i.e., if u = 1, the indexed set {(a u , b u )} is the empty set in the sum presentation of ∆(u) = 1 ⊗ u + a u ⊗ b u .

Proof of Proposition 6.7. By the definition of the ¡ and Lemma 5.7, we have

x u u ¡ x v v (6.4) = (x u u) (x v v) + (x v v) (x u u) + (x u u) (x v v) = x v (x u u ¡ v) + x u (u ¡ (x v v)) + (x u x v ) (u ¡ v) = x v (x u u ¡ v) + x u (u ¡ (x v v)) + x u+v (u ¡ v) + i+j=u+v ∆ j u,v x i (x j ¡ u ¡ v).
Therefore, we get

∆(x u u ¡ x v v) -∆(x u u) ¡ ∆(x v v) (6.5) =∆(x v (x u u ¡ v)) + ∆(x u (u ¡ x v v)) + ∆(x u+v (u ¡ v)) + i+j=u+v ∆ j u,v ∆(x i (x j ¡ u ¡ v)) -∆(x u u) ¡ ∆(x v v).
We calculate each term on the right hand side to conclude that this vanishes. We put

∆(u) = 1 ⊗ u + a u ⊗ b u , and ∆(v) = 1 ⊗ v + a v ⊗ b v .
Also for all j ∈ N, we put

∆(x j ) = 1 ⊗ x j + a j ⊗ b j .
In particular,

∆(x u ) = 1 ⊗ x u + a u ⊗ b u , and ∆(x v ) = 1 ⊗ x v + a v ⊗ b v . The first term ∆(x v (x u u ¡ v)).
From the definition of ∆,

∆(x u u) = 1 ⊗ x u u + a u ⊗ (b u ¡ u) + (a u a u ) ⊗ (b u ¡ b u ). (6.6)
As depth(x u u) + depth(v) < depth(x u u) + depth(x v v), by the induction hypothesis we obtain

∆(x u u ¡ v) =∆(x u u) ¡ ∆(v) =1 ⊗ ((x u u) ¡ v) + a u ⊗ (b u ¡ u ¡ v) + a v ⊗ ((x u u) ¡ b v ) + (a u a u ) ⊗ (b u ¡ b u ¡ v) + (a u ¡ a v ) ⊗ (b u ¡ u ¡ b v ) + ((a u a u ) ¡ a v ) ⊗ (b u ¡ b u ¡ b v ). Thus ∆(x v (x u u ¡ v)) =1 ⊗ x v (x u u ¡ x v ) + a v ⊗ (b v ¡ (x u u) ¡ v) (6.7) + (a v a u ) ⊗ (b u ¡ b v ¡ u ¡ v) + (a v (a u a u )) ⊗ (b u ¡ b v ¡ b u ¡ v) + (a v a v ) ⊗ (b v ¡ (x u u) ¡ b v ) + (a v (a u ¡ a v )) ⊗ (b u ¡ b v ¡ u ¡ b v ) + (a v ((a u a u ) ¡ a v )) ⊗ (b u ¡ b v ¡ b u ¡ b v ). The second term ∆(x u (u ¡ (x v v)).
Similarly, we get

∆(x u (u ¡ (x v v)) =1 ⊗ x u (u ¡ x v v) + a u ⊗ (b u ¡ u ¡ (x v v)) (6.8) + ((a u a v ) ⊗ (b u ¡ b v ¡ u ¡ v) + (a u (a v a v )) ⊗ (b u ¡ b v ¡ u ¡ b v ) + (a u a u ) ⊗ (b u ¡ b u ¡ (x v v)) + (a u (a u ¡ a v )) ⊗ (b u ¡ b u ¡ b v ¡ v) + (a u (a u ¡ (a v a v )) ⊗ (b u ¡ b v ¡ b u ¡ b v ). The third term ∆(x u+v (u ¡ v)).
We put

∆(u ¡ v) = 1 ⊗ (u ¡ v) + a u¡v ⊗ b u¡v .
As depth(u)+depth(v) < depth(x u u)+depth(x v v), the induction hypothesis implies that

∆(u ¡ v) = ∆(u) ¡ ∆(v) = 1 ⊗ u + a u ⊗ b u ¡ 1 ⊗ v + a v ⊗ b v . Thus 1 ⊗ (u ¡ v) + a u¡v ⊗ b u¡v = 1 ⊗ u + a u ⊗ b u ¡ 1 ⊗ v + a v ⊗ b v =1 ⊗ (u ¡ v) + a u ⊗ (b u ¡ v) + a v ⊗ (u ¡ b v ) + (a u ¡ a v ) ⊗ (b u ¡ b v ),
which implies (6.9)

a u¡v ⊗b u¡v = a u ⊗(b u ¡v)+ a v ⊗(u¡b v )+ (a u ¡a v )⊗(b u ¡b v ).
Thus we have

∆(x u+v (u ¡ v)) =1 ⊗ (x u+v (u ¡ v)) + a u+v ⊗ (b u+v ¡ u ¡ v) (6.10)
+ (a u+v a u¡v ) ⊗ (b u+v ¡ b u¡v ). The fourth terms ∆(x i (x j ¡u¡v)) for all 0 ≤ i, j ≤ u + v with i + j = u + v.

As depth(u) + depth(x j ) < depth(x u u) + depth(x v v), we have ∆(x j ¡ u) = ∆(u ¡x j ) = ∆(u) ¡∆(x j ) by induction hypothesis. Further by Lemma 6.4, x j ¡u can be represented as the sum of words with depth less than or equal to depth(x j u).

Thus ∆(x j ¡u¡v) = ∆(x j ¡u)¡∆(v) = ∆(x j )¡∆(u)¡∆(v) = ∆(x j )¡∆(u¡v)
by the induction hypothesis again. Therefore,

∆(x j ¡ u ¡ v) =∆(x j ) ¡ ∆(u ¡ v) = 1 ⊗ x j + a j ⊗ b j ¡ 1 ⊗ (u ¡ v) + a u¡v ⊗ b u¡v =1 ⊗ (x j ¡ u ¡ v) + a j ⊗ (b j ¡ u ¡ v) + a u¡v ⊗ (x j ¡ b u¡v ) + (a j ¡ a u¡v ) ⊗ (b j ¡ b u¡v ). This leads to ∆(x i (x j ¡ u ¡ v)) =1 ⊗ (x i (x j ¡ u ¡ v)) + a i ⊗ (b i ¡ x j ¡ u ¡ v) (6.11) + (a i a j ) ⊗ (b i ¡ b j ¡ u ¡ v) + (a i a u¡v ) ⊗ (b i ¡ x j ¡ b u¡v ) + (a i (a j ¡ a u¡v )) ⊗ (b i ¡ b j ¡ b u¡v ). The last term ∆(x u u) ¡ ∆(x v v).
Note that ∆(x u u) is given by (6.6). A parallel calculation gives

∆(x v v) = 1 ⊗ x v v + a v ⊗ (b v ¡ v) + (a v a v ) ⊗ (b v ¡ b v ). Thus ∆(x u u) ¡ ∆(x v v) (6.12) = 1 ⊗ x u u + a u ⊗ (b u ¡ u) + (a u a u ) ⊗ (b u ¡ b u ) ¡ 1 ⊗ x v v + a v ⊗ (b v ¡ v) + (a v a v ) ⊗ (b v ¡ b v ) =1 ⊗ (x u u ¡ x v v) + a v ⊗ ((x u u) ¡ b v ¡ v) + (a v a v ) ⊗ ((x u u) ¡ b v ¡ b v ) + a u ⊗ (b u ¡ u ¡ (x v v)) + (a u ¡ a v ) ⊗ (b u ¡ b v ¡ u ¡ v) + (a u ¡ (a v a v )) ⊗ (b u ¡ u ¡ b v ¡ b v ) + (a u a u ) ⊗ (b u ¡ b u ¡ (x v v)) + ((a u a u ) ¡ a v ) ⊗ (b u ¡ b u ¡ b v ¡ v) + ((a u a u ) ¡ (a v a v )) ⊗ (b u ¡ b u ¡ b v ¡ b v ).
Plugging the equations (6.7), (6.8), (6.10), (6.11), (6.12) into (6.5) yields

∆(x u u ¡ x v v) -∆(x u u) ¡ ∆(x v v) = S 0 -S 1 -S 2 -S u -S v + S 3 + S 4 .
Here the sums S 0 , S 1 , S 2 , S 3 , S 4 , S u and S v are given as follows:

S 0 =1 ⊗ x v (x u u ¡ v) + 1 ⊗ x u (u ¡ (x v v)) + 1 ⊗ x u+v (u ¡ v) + i+j=u+v ∆ j u,v 1 ⊗ x i (x j ¡ u ¡ v) -1 ⊗ ((x u u) ¡ (x v v)). S 1 = (a u ¡ a v ) ⊗ (b u ¡ b v ¡ u ¡ v) -(a u a v ) ⊗ (b u ¡ b v ¡ u ¡ v) - (a v a u ) ⊗ (b u ¡ b v ¡ u ¡ v). S 2 = ((a u a u ) ¡ (a v a v )) ⊗ (b u ¡ b u ¡ b v ¡ b v ) - (a u (a u ¡ (a v a v ))) ⊗ (b u ¡ b u ¡ b v ¡ b v ) - (a v ((a u a u ) ¡ a v ) ⊗ (b u ¡ b u ¡ b v ¡ b v ). S 3 = a u+v ⊗ (b u+v ¡ u ¡ v) + i+j=u+v ∆ j u,v a i ⊗ (b i ¡ x j ¡ u ¡ v) + i+j=u+v ∆ j u,v (a i a j ) ⊗ (b i ¡ b j ¡ u ¡ v). S 4 = (a u+v a u¡v ) ⊗ (b u+v ¡ b u¡v ) + i+j=u+v ∆ j u,v (a i a u¡v ) ⊗ (b i ¡ x j ¡ b u¡v ) + i+j=u+v ∆ j u,v (a i (a j ¡ a u¡v )) ⊗ (b i ¡ b j ¡ b u¡v ),
and

S u = (a u ¡ (a v a v )) ⊗ (b u ¡ b v ¡ u ¡ b v ) -(a u (a v a v )) ⊗ (b u ¡ b v ¡ u ¡ b v ) - (a v (a u ¡ a v )) ⊗ (b u ¡ b v ¡ u ¡ b v ), S v = ((a u a u ) ¡ a v ) ⊗ (b u ¡ b v ¡ b u ¡ v) -(a u (a u ¡ a v )) ⊗ (b u ¡ b v ¡ b u ¡ v) - (a v (a u a u )) ⊗ (b u ¡ b v ¡ b u ¡ v). We claim that (1) S 0 = 0. (2) S 1 -S 3 = 0. (3) S 2 + S u + S v -S 4 = 0.
If these claims are true, the proposition is also true.

We prove three claims below. The claim (1), S 0 = 0, is immediate from (6.4). We show the claim (2). We will show that

S 1 = S 3 = (a u a v ) ⊗ (b u ¡ b v ¡ u ¡ v).
With Lemma 5.7, we have

S 1 = (a u ¡ a v ) ⊗ (b u ¡ b v ¡ u ¡ v) -(a u a v ) ⊗ (b u ¡ b v ¡ u ¡ v) - (a v a u ) ⊗ (b u ¡ b v ¡ u ¡ v) = (a u ¡ a v -a u a v -a v a u ) ⊗ (b u ¡ b v ¡ u ¡ v) = (a u a v ) ⊗ (b u ¡ b v ¡ u ¡ v). Now we calculate S 3 . Recall x u x v = x u+v + i+j=u+v ∆ j u,v x i x j . Lemma 3.4 gives that ∆(x u x v ) = 1 ⊗ (x u x v ) + (a u a v ) ⊗ (b u ¡ b v ), so 1 ⊗ x u+v + a u+v ⊗ b u+v + i+j=u+v ∆ j u,v 1 ⊗ x i x j + a i ⊗ (b i ¡ x j ) + (a i a j ) ⊗ (b i ¡ b j ) =1 ⊗ (x u x v ) + (a u a v ) ⊗ (b u ¡ b v ), therefore a u+v ⊗ b u+v + i+j=u+v ∆ j u,v a i ⊗ (b i ¡ x j ) + i+j=u+v ∆ j u,v (a i a j ) ⊗ (b i ¡ b j ) (6.13)
We now consider the term S 4 . We have

S 4 = (a u+v a u¡v ) ⊗ (b u+v ¡ b u¡v ) + i+j=u+v ∆ j u,v (a i a u¡v ) ⊗ (b i ¡ x j ¡ b u¡v ) + i+j=u+v ∆ j u,v (a i (a j ¡ a u¡v )) ⊗ (b i ¡ b j ¡ b u¡v ) = (a u+v a u¡v ) ⊗ (b u+v ¡ b u¡v ) + i+j=u+v ∆ j u,v (a i a u¡v ) ⊗ (b i ¡ x j ¡ b u¡v ) + i+j=u+v ∆ j u,v ((a i a j ) a u¡v ) ⊗ (b i ¡ b j ¡ b u¡v ) = ((a u a v ) a u¡v ) ⊗ (b u ¡ b v ¡ b u¡v ).
Here the second equality follows from Proposition 5.10. The third one is a direct consequence of (6.13). This proves S 2 + S u + S v = S 4 as claimed. Now Proposition 6.7 is proven with the claims (1), (2), and (3), as

∆(x u u ¡ x v v) -∆(x u u) ¡ ∆(x v v) = S 0 -S 1 -S 2 -S u -S v + S 3 + S 4 = S 0 -(S 1 -S 3 ) -(S 2 + S u + S v -S 4 ) = 0.
Proof of Theorem 6.5. By combining Lemma 6.6 and Proposition 6.7 we conclude that, if Theorem 6.5 holds for all words (a, b) ∈ Γ × Γ with depth(a)+depth(b) < n for some n ∈ N then the theorem holds for all words (x u u, x v v) such that depth(x u u) + depth(x v v) = n, and thus for (x u,α u, x v,β v) such that depth(x u,α u) + depth(x v,β v) = n, for all u, v ∈ N and α, β ∈ F * q . Since the theorem holds for the initial cases (1, b) and (a, 1) for all a, b ∈ Γ with the arbitrary depths, the theorem holds for all u, v ∈ Γ .

Coassociativity.

Next we want to prove: Theorem 6.8. Let u ∈ Γ . Then we have

(Id ⊗∆)∆(u) = (∆ ⊗ Id)∆(u).
Proof. The proof is formed into two parts as follows:

• Part (1): if the coassociativity holds for u = x u v, then coassociativity holds for u = x u,ε v, for v ∈ Γ with depth(v) ≥ 0, and • Part (2): if the coassociativity holds for all u with depth(u) ≤ n, then the coassociativity holds for u = x u v for u ∈ N and depth v = n.

Note that the initial case (i.e., when u = x u,ε ) is established by the coassociativity in C and Part [START_REF] Anderson | t-motives[END_REF].

For Part (1), we assume that (Id ⊗∆)∆(

x u v) = (∆ ⊗ Id)∆(x u v). Let ∆(x u ) = 1 ⊗ x u + a u ⊗ b u , ∆(v) = a v ⊗ b v .
Here a u ∈ Σ, b u ∈ Σ , and a v , b v ∈ Γ . Thus

∆(x u v) = 1 ⊗ x u v + a u a v ⊗ (b u ¡ b v ),
and

(Id ⊗∆)∆(x u v) = 1 ⊗ 1 ⊗ x u v + 1 ⊗ a u a v ⊗ (b u ¡ b v ) + a u a v ⊗ ∆(b u ¡ b v ), (∆ ⊗ Id)∆(x u v) = 1 ⊗ 1 ⊗ x u v + ∆(a u a v ) ⊗ (b u ¡ b v ).
Then the induction hypothesis that is the coassociativity for x u v is equivalent to where a u,ε = ϕ ε (a u ). By definition, we have

1 ⊗ a u a v ⊗ (b u ¡ b v ) + a u a v ⊗ ∆(b u ¡ b v ) = ∆(a u a v ) ⊗ (b u ¡ b v ).
∆(u) = 1 ⊗ u + a u,ε a v ⊗ (b u ¡ b v ).
One can calculate

(Id ⊗∆)∆(u) =1 ⊗ ∆(u) + a u,ε a v ⊗ ∆(b u ¡ b v ) = 1 ⊗ 1 ⊗ u + 1 ⊗ a u,ε a v ⊗ (b u ¡ b v ) + a u,ε a v ⊗ ∆(b u ¡ b v ), (∆ ⊗ Id)∆(u) =1 ⊗ 1 ⊗ u + ∆(a u,ε a v ) ⊗ (b u ¡ b v ) =1 ⊗ 1 ⊗ u + (ϕ ε ⊗ Id)∆(a u a v ) ⊗ (b u ¡ b v ) + 1 ⊗ a u,ε a v ⊗ (b u ¡ b v ) -1 ⊗ a u a v ⊗ (b u ¡ b v ).
Thus the coassociativity holds for x u,ε v if

(ϕ ε ⊗ Id)∆(a u a v ) ⊗ (b u ¡ b v ) = a u,ε a v ⊗ ∆(b u ¡ b v ) + 1 ⊗ a u a v ⊗ (b u ¡ b v ),
which is obtained by applying ϕ ε ⊗ Id ⊗ Id on Equation (6.14). This proves that the coassociativity for x u v implies the coassociativity for x u,ε v. We now prove Part (2). Let n ≥ 1. We assume that the coassociativity holds for all words with depth ≤ n. Let u = x t,ε t with depth ≤ n. We write ∆(x t,ε ) = 1 ⊗ x t,ε + a t,ε ⊗ b t , and ∆(t) = a t ⊗ b t .

where a t,ε ∈ Γ and b t,ε , a t , b t ∈ Γ . Then ∆(x t,ε t) = 1 ⊗ x t,ε + a t,ε a t ⊗ (b t ¡ b t ), and the coassociativity for x t,ε t is reduced into (∆(a t,ε a t ) -1 ⊗ a t,ε a t ) ⊗ (b t ¡ b t ) = a t,ε a t ⊗ (∆(b t ) ¡ ∆(b t )). It follows that

∆(x u x v,β v) = 1 ⊗ x u x v,β v + a u ⊗ (b u ¡ x v,β v) + a u a v,β a v ⊗ (b u ¡ b v ¡ b v ), thus (Id ⊗∆)∆(x u x v,β v) =1 ⊗ 1 ⊗ x u x v,β v + 1 ⊗ a u ⊗ (b u ¡ x v,β v) + 1 ⊗ a u a v,β a v ⊗ (b u ¡ b v ¡ b v ) + a u ⊗ ∆(b u ¡ x v,β v) + a u a v,β a v ⊗ ∆(b u ¡ b v ¡ b v ), (∆ ⊗ Id)∆(x u x v,β v) =1 ⊗ 1 ⊗ x u x v,β v + ∆(a u ) ⊗ (b u ¡ x v,β v) + ∆(a u a v,β a v ) ⊗ (b u ¡ b v ¡ b v ),
so what we want to show is that

1 ⊗ a u ⊗ (b u ¡ x v,β v) + 1 ⊗ a u a v,β a v ⊗ (b u ¡ b v ¡ b v ) (6.16) + a u ⊗ ∆(b u ¡ x v,β v) + a u a v,β a v ⊗ ∆(b u ¡ b v ¡ b v ), = ∆(a u ) ⊗ (b u ¡ x v,β v) + ∆(a u a v,β a v ) ⊗ (b u ¡ b v ¡ b v ).
Note that, by Equation (6.15) we have

1 ⊗ a u ⊗ (b u ¡ x v,β v) + a u ⊗ ∆(b u ¡ x v,β v) -∆(a u ) ⊗ (b u ¡ x v,β v) = a u ⊗ ∆(b u ¡ x v,β v) -(∆(a u ) -1 ⊗ a u ) ⊗ b u ¡ (1 ⊗ 1 ⊗ x v,β v) = a u ⊗ ∆(b u ¡ x v,β v) -a u ⊗ ∆(b u ) ¡ (1 ⊗ 1 ⊗ x v,β v) = a u ⊗ (∆(b u ) ¡ ∆(x v,β v)) -a u ⊗ (∆(b u ) ¡ (1 ⊗ x v,β v)) = a u ⊗ (∆(b u ) ¡ (∆(x v,β v) -1 ⊗ x v,β v)) = a u ⊗ (∆(b u ) ¡ (a v,β a v ⊗ (b v ¡ b v )))
so Equation (6.16) is equivalent to

∆(a u a v,β a v ) ⊗ (b u ¡ b v ¡ b v ) (6.17) = 1 ⊗ a u a v,β a v ⊗ (b u ¡ b v ¡ b v ) + a u a v,β a v ⊗ ∆(b u ¡ b v ¡ b v ) + a u ⊗ (∆(b u ) ¡ (a v,β a v ⊗ (b v ¡ b v ))) .
From now on, tracing the indices is important. When needed, we use the notation u which stands for the sum over the pairs (a u , b u ), and similarly (v,β) , v , etc. We now fix a u , b u . The last two terms of (6.17) become

a u a v,β a v ⊗ (∆(b u ) ¡ ∆(b v ) ¡ ∆(b v )) + a u ⊗ (∆(b u ) ¡ (a v,β a v ⊗ (b v ¡ b v ))) = (1 ⊗ ∆(b u )) ¡ a u a v,β a v ⊗ (∆(b v ) ¡ ∆(b v )) + a u ⊗ a v,β a v ⊗ (b v ¡ b v ) = (a u ⊗ 1 ⊗ 1) • (1 ⊗ ∆(b u )) ¡ a v,β a v ⊗ (∆(b v ) ¡ ∆(b v )) + 1 ⊗ a v,β a v ⊗ (b v ¡ b v )
From the coassociativity of C, we have the coassociativity for u ∈ Σ, and then for u ∈ Γ by Part [START_REF] Anderson | t-motives[END_REF]. By applying Parts (2) and (1) repeatedly, we can cover arbitrary depth of u ∈ Γ . Hence this completes the proof. 6.4. Hopf algebra structure on D.

Let the counit : D → F q defined as (1) = 1; (u) = 0 otherwise.

We already checked that (D, ¡,u,∆, ) is a bialgebra by Theorem 5.9, Theorem 6.5, and Theorem 6.8. By induction it is easily verified that ¡ and ∆ preserve the weight, and the weight zero subset is isomorphic to F q . Therefore, (D, ¡,u,∆, ) is a connected graded bialgebra. By Theorem 2.6, we have the following theorem. Theorem 6.9. (D, ¡,u,∆, ) is a connected graded Hopf algebra of finite type over F q .

r n 1 .

 1

  a1,...,an∈A+ d=deg a1>•••>deg an≥0 1 a s1 1 . . . a sn n ∈ K and S <d (s) = a1,...,an∈A+ d>deg a1>•••>deg an≥0 1 a s1 1 . . . a sn n ∈ K.

Proposition 5 . 1 .

 51 The diamond product and the shuffle product on D are commutative. Proof. Let a, b ∈ Γ be two arbitrary words. It suffices to show that a b = b a and a ¡ b = b ¡ a.

Lemma 5 . 4 .

 54 Let α, β be elements of F * q , and let a, b be nonempty words in Σ . We have ϕ α (a) ϕ β (b) = ϕ αβ (a b). Further, for nonempty words a, b ∈ Γ , ϕ α (a) ϕ β (b) = ϕ αβ (a b).

Proposition 5 . 8 .

 58 The diamond product and the shuffle product on D are associative. Proof. Let a, b, c ∈ Γ be arbitrary words. It suffices to show that (a b) c = a (b c) and (a ¡ b) ¡ c = a ¡ (b ¡ c). We proceed the proof by induction on depth(a) + depth(b) + depth(c). If one of a, b or c is the empty word, then (5.3) holds trivially. We assume that (5.3) holds when depth(a) + depth(b) + depth(c) < n with n ∈ N and n ≥ 3. We need to show that (5.3) holds when depth(a) + depth(b) + depth(c) = n. We first show that (a b) c = a (b c). Assume that a = x a,α a -, b = x b,β b -= c = x c,γ c -. From Lemma 5.7, we have (a b) c

  We thus conclude that (a b) c = a (b c). Next, we show that (a ¡ b) ¡ c = a ¡ (b ¡ c). From Lemma 5.7, we have (a ¡ b) ¡ c = (a b + b a + a b) ¡ c = (a b) ¡ c + (b a) ¡ c + (a b) ¡ c = ((a b) c + c (a b) + (a b) c) + ((b a) c + c (b a) + (b a) c) + ((a b) c + c (a b) + (a b) c) = (a b) c + (a b) c + (b a) c + (b a) c + (a b) c + (c (a b) + c (b a) + c (a b)) + (a b) c = (a b) c + (a b) c + (b a) c + (b a) c + (a b) c + c (a ¡ b) + (a b) c and a ¡ (b ¡ c) = a ¡ (b c + c b + b c) = a ¡ (b c) + a ¡ (c b) + a ¡ (b c) = (a (b c) + (b c) a + a (b c)) + (a (c b) + (c b) a + a (c b))+ (a (b c) + (b c) a + a (b c)) = (b c) a + a (b c) + (c b) a + a (c b) + (b c) a + (a (b c) + a (c b) + a (b c)) + a (b c) = (b c) a + a (b c) + (c b) a + a (c b) + (b c) a + a (b ¡ c) + a (b c).We now compare the above expansions. We have showed that (a b) c = a (b c). On the other hand, we have c (a¡b) = x c,γ (c -¡(a¡b)) and (c b) a = x c,γ (c -¡b) a = x c,γ ((c -¡b)¡a).

From

  the induction hypothesis and commutativity of shuffle product, one deduces that c (a ¡ b) = (c b) a. Similarly, one deduces that (a b) c = a (b ¡ c).

From

  the induction hypothesis, one deduces that (a b) c = a (b c). Similarly, one deduces that (b a) c = (b c) a.

Proposition 5 . 10 .

 510 For all nonempty words a, b, c ∈ Γ , we have(1) (a b) c = a (b ¡ c),(2)(a b) c = a (c b) = (a c) b.

(6. 14 )

 14 Now let u = x u,ε v. We have ∆(x u,ε ) = 1 ⊗ x u,ε + a u,ε ⊗ b u

(6. 15 )

 15 As a special case, let t = 1. Then ∆(t) = 1 ⊗ 1, and(∆(a t,ε ) -1 ⊗ a t,ε ) ⊗ b t = a t,ε ⊗ ∆(b t ). Now assume u = x u x v,β v with depth n + 1. Say ∆(x u ) = 1 ⊗ x u + a u ⊗ b u , ∆(x v ) = 1 ⊗ x v + a v ⊗ b v , so ∆(x v,β ) = 1 ⊗ x v,β + a u,β ⊗ b v , ∆(v) = a v ⊗ b v , where a u , a v ∈ Σ, b u , b v ∈ Σ , a v , b v ∈ Γ , and a u,β = ϕ β (a u ). Then, ∆(x v,β v) = 1 ⊗ x v,β v + a v,β a v ⊗ (b v ¡ b v ).

We are thankful to Thakur for his assistance with event orders.

Acknowledgments. We would like to thank Dinesh Thakur for his interest and guidance with literature and for helpful discussions. Bo-Hae Im was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by grant funded by the Korea government (MSIT) (NRF-2023R1A2C1002385). KN. Le and T. Ngo Dac were partially supported by the Excellence Research Chair "L-functions in positive characteristic and applications" financed by the Normandy Region. T. Ngo Dac and LH. Pham were partially supported by the Vietnam Academy of Science and Technology (VAST) under grant no. CTTH00.02/23-24 "Arithmetic and Geometry of schemes over function fields and applications".

We conclude that

and thus S 1 = S 3 as desired.

We now show the claim [START_REF] Anderson | Tensor powers of the Carlitz module and zeta values[END_REF]. Precisely, we show that

We have

Here the last equality is from Lemma 5.7 and Proposition 5.10 as

By Lemma 5.7 and Proposition 5.10 again,

and

With the equations for S 2 , S u , S v and (6.9), again we can apply Lemma 5.7 and Proposition 5.10 to have

Recall that • is the concatenation. The last equality is from the induction hypothesis, Equation (6.15). Equation (6.17) becomes

Next we fix a u , a v,β , a v . So the corresponding b's are also fixed. Let

Then, we have 2) 2)

Plugging this and the sum representation of ∆(a v,β a v ) into the equation (6.18) yields u,(v,β),v   (1),( 2)

Finally, we fix a u , a v,β , a v , c (2) . The corresponding summands are

By direct calculations, we get

Now we can apply Equation (6.15) for the sum of the summand over a u to have

that is, Equation (6.18) is true. Thus the induction step is proved. Email address: hojinkim@kaist.ac.kr