Symmetry, Integrability and Geometry: Methods and Applications The Algebra of a q-Analogue of Multiple Harmonic Series ⋆

We introduce an algebra which describes the multiplication structure of a family of q-series containing a q-analogue of multiple zeta values. The double shuffle relations are formulated in our framework. They contain a q-analogue of Hoffman's identity for multiple zeta values. We also discuss the dimension of the space spanned by the linear relations realized in our algebra.


Introduction
In this article we introduce an algebra to formalize the multiplication structure of a q-analogue of multiple zeta values.
The vector space spanned by MZVs over Q is closed under multiplication. There are two ways to calculate the product of MZVs. One way is to calculate the product directly from the above definition of MZVs shuffling the indices n i . Another way is to use an iterated integral representation, called the Drinfel'd integral [2,10]. By calculating the product of MZVs in two ways above, we obtain different expressions. As a result we get linear relations among MZVs, which are called the double shuffle relations.
In [4] Hoffman gives an algebraic formulation to describe the multiplication structure of MZVs. The two ways to calculate the product are realized as two different operations of multiplication on a non-commutative polynomial ring, which we call in this paper the harmonic product and the integral shuffle product, respectively. Using the algebraic setup, an extension of the double shuffle relations is given in [6], and it is conjectured that it contains all linear relations among MZVs.
The harmonic product of qMZVs can be defined naturally, and we also have an iterated integral representation of qMZV [11]. However the vector space spanned by qMZVs over Q is not presumably closed under the multiplication arising from the integral representation. To overcome the difficulty we consider a larger class of q-series allowing the factor q n /[n] in the sum (1). Such extension is proposed also in [8]. Then the enlarged vector space of q-series is closed under the harmonic product and the integral shuffle product. The main result of this paper is to formulize the multiplication structure by extending Hoffman's algebra. Thus we can consider the double shuffle relations for qMZVs.
There are many linear relations over Q among qMZVs. An important feature in the qanalogue case is that there are inhomogeneous linear relations in the following sense. For an admissible index k = (k 1 , . . . , k r ), the modified qMZV ζ q (k) is defined bȳ where |k| is the weight of k defined by |k| := r i=1 k i . In [9] it is observed that there are linear relations among the modified qMZVs with different weight. Taking the limit q → 1 in such relations, the highest weight terms only survive and we obtain linear relations for MZVs. It suggests that we should consider the vector space spanned by the modified qMZVs rather than the original qMZVs.
Our double shuffle relations contain linear relations for the modified qMZVs. However they do not suffice to get all linear relations. In this article we also give some relations among qseries containing the factor q n /[n], which we call the resummation duality, as a supply of linear relations (see Theorem 4 below). By computer experiment it is checked that the double shuffle relations and the resummation duality give all linear relations among the modified qMZVs up to weight 7.
The paper is organized as follows. In Section 2 we give the algebraic setup to formalize the multiplication structure of qMZVs. To define the integral shuffle product we make use of an extended version of a q-analogue of multiple polylogarithms (of one variable). In Section 3 we discuss the double shuffle relations. As an example we prove Hoffman's identity for qMZV (see Proposition 7 below) in our algebraic framework. Note that it is derived from Ohno's relation and the duality for qMZV [1]. At last we prove the resummation duality and show some computer experiment about the dimension of the Q-linear space spanned by the relations among the modified qMZVs obtained in this paper.
Define the C-submodule H 0 of H 1 by We denote by H 0 the C-submodule of H 0 generated by 1 and the words z k 1 . . . z kr with k 1 ≥ 2 and k 2 , . . . , k r ≥ 1.
Hereafter we fix a complex parameter q such that 0 < |q| < 1. We endow C with C-module structure such that acts as multiplication by 1−q. Denote by z the C-submodule of H generated by A. For a positive integer n we define the C-linear map I · (n) : z → C by Note that Now we define the C-linear map Z q : H 0 → C by Z q (1) = 1 and where r ≥ 1 and u i ∈ A. The infinite sum in the right hand side absolutely converges because there exists a positive constant M such that |1/[n]| ≤ M for all n ≥ 1. If k = (k 1 , . . . , k r ) is an admissible index, the value Z q (z k 1 . . . z kr ) is equal to the qMZV (1).

Harmonic product
We define the harmonic product on H 1 generalizing the algebraic formulation given in [6].
Consider the commutative product • on z by setting for k, l ≥ 1 and extending by C-linearity. Define the C-bilinear product * on H 1 inductively by setting for w, w ∈ H 1 and u 1 , u 2 ∈ A. It is commutative and associative because the product • is commutative and associative. Let us call * the harmonic product on H 1 . Then the C-submodule H 0 is a subalgebra of H 1 with respect to the harmonic product.
Proof . For a positive integer N we define the C-linear map F · (N ) : H 1 → C by F 1 (N ) = 1 and where u i ∈ A. Note that F uw (N ) = N >m>0 I u (m)F w (m) for u ∈ A and w ∈ H 1 . We for any w ∈ H 0 , and hence it suffices to prove that F w * w (N ) = F w (N )F w (N ) for words w, w in A starting with ξ or z k (k ≥ 2). Let us prove it by induction on the sum of the degrees of w and w . Note that if w = 1 or w = 1 the equality is trivial. Let w, w ∈ H 0 be words and u 1 , u 2 ∈ A. Then Now the desired equality follows from the induction hypothesis and for k, l ≥ 2 and m ≥ 1.

Integral shuf f le product
Let us define the C-bilinear product x on H inductively as follows. We set 1 x w = w x 1 = w for any w ∈ H. For u, v ∈ {x, y, ρ} and w, w ∈ H, we set for u ∈ {x, y, ρ}. Then the product x is commutative because of the symmetry of α.
Proof . From the property (5), we see that for u ∈ {x, y, ρ} and w, w ∈ H. Using this formula repeatedly we find that ρw The product x is associative.
3) by induction on the sum of the degrees of w 1 , w 2 and w 3 . If w i = 1 for some i, it is trivial. Suppose that w i = u i w i (i = 1, 2, 3) for u i ∈ {x, y, ρ} and w i ∈ H. Lemma 1 implies that if u i = ρ for some i, the desired equality follows from the induction hypothesis. Thus we should check the associativity in the case where any u i is x or y. Here let us consider the case where (u 1 , u 2 , u 3 ) = (x, y, y). We have Now apply the induction hypothesis and use the equality which follows from Lemma 1. Then we obtain It is equal to xw 1 x (yw 2 x yw 3 ). The proof for the other cases is similar.
Thus an associative commutative product x is defined on H. We call it the integral shuffle product.
Proof . First let us prove that for k, l ≥ 1 and w, w ∈ H 1 . If k = l = 1 it follows from α(y, y) = −yρ. If k = 1 and l ≥ 2, we find the above property by induction on l using It remains to prove that ξw x z k w and ξw x ξw belong to H 0 for w, w ∈ H 1 and k ≥ 2. It follows from the property (6) and In the rest of this section we prove the following theorem.
Thus we define the two operations of multiplication, the harmonic product and the integral shuffle product, on H 0 . They describe the multiplication structure of a family of q-series Z q (w) containing qMZVs. Note that we can formally restore Hoffman's algebra for MZVs [4] by setting = 0 and ρ = 0.

A q-analogue of multiple polylogarithms
To prove Theorem 2 we introduce an extended version of a q-analogue of multiple polylogarithms (of one variable). Denote by F the ring of holomorphic functions on the unit disk |t| < 1. We consider F as a C-module by ( f )(t) : Consider the q-difference operator D q defined by To describe the function D q L w (w ∈ H 0 ) we introduce the two maps ∆ j (j = 0, 1) as follows. Set For w ∈ h ≥1 and r ≥ 0 it holds that Proof . The equality (8) follows from D q (t n ) = [n]t n−1 for n ≥ 0. Let us prove (9).
because of (3). Therefore Here we used the equality for any non-negative integer k. Now the equality (9) follows from for k ≥ 1.

Multiplication structure of the q-analogue of multiple polylogarithms
Let us prove that the set of functions {L w } w∈ H 0 is closed under multiplication. Define the C-linear map I 0 : and the C-linear map I 1 : H 0 → h ≥1 by for w ∈ H 1 and k ≥ 2. We have the following property.   (2) Let w ∈ H 0 and r ≥ 0. Suppose that f ∈ F satisfies f (0) = 0 and Then f = L ξρ r I 1 (w) .
Proof . Note that the initial value problem D q f = g and f (0) = 0 for a given g ∈ F has a unique solution in F if it exists. Therefore it suffices to check that the function f given above is a solution to the initial value problems in (1) or (2). We have f (0) = 0 because the image of I 0 or ξρ r I 1 is contained in H 0 . Proposition 3 and Lemma 2 imply that the function f is a solution.
To write down the structure of multiplication of the functions L w (w ∈ H 0 ), let us define the commutative C-bilinear product on H 0 . Set 1 w = w 1 = w for w ∈ H 0 . In general we define the product inductively as follows. For w, w ∈ h ≥2 we set For w ∈ h ≥2 , w ∈ h ≥1 and r ≥ 0, set For w, w ∈ h ≥1 and r, s ≥ 0, ξρ r w ξρ s w = ξρ r I 1 (∆ 1 (w) ξρ s w ) + ξρ s I 1 ξρ r w ∆ 1 (w ) − ξρ r+s+1 I 1 ∆ 1 (w) ∆ 1 (w ) .
Since the image of I 0 is contained in H 0 , the product is well-defined.
Proposition 5. For any w, w ∈ H 0 we have L w w = L w L w .
Proof . It suffices to prove in the case where w and w are homogeneous. Let us prove it by induction on the sum of the degrees of w and w . Note that the desired equality is trivial if w or w belongs to C. Otherwise the function L w L w has a zero at t = 0. Now calculate D q (L w L w ) by using the formula for f, g ∈ F. The q-difference of L w and L w is written in terms of the maps ∆ 0 and ∆ 1 as described in Proposition 3. Here note that if w is homogeneous the degree of ∆ 0 (w) is less than that of w. Now the induction hypothesis implies that D q (L w L w ) is given in terms of the product . Use Proposition 4 to restore the original function L w L w , and we get the desired equality from the definition of the product .

Proof of Theorem 2
Let us prove Theorem 2. First we describe a relation between the qMZV and the function L w . Proof . It follows from q n /[n] = I ξ (n) and q n /[n] k = I e(z k ) (n) for k ≥ 2 and n ≥ 1.
Note that the map e given in Lemma 3 is an isomorphism on the C-module H 0 . Its inverse is given by e −1 (1) = 1, e −1 (ξw) = ξw and for w ∈ H 1 and k ≥ 2. Now Theorem 2 is reduced to the following proposition because of Proposition 5 and Lemma 3. In the proof of Proposition 6 we use the properties below.
The proof is straightforward.
Proof . We can assume without loss of generality that w and w are homogeneous. If w = 1 or w = 1, it is trivial since I 1 ∆ 1 is the identity on h ≥1 (Lemma 2). Let us prove the desired equality by induction on the sum of the degrees of w and w . First consider the case where w = z 1 ρ r w 1 and w = z 1 ρ s w 2 for r, s ≥ 0 and w 1 , w 2 ∈ H 1 . From the definition of ∆ 1 and we have Apply the induction hypothesis and we get It is equal to z 1 ρ r w 1 x z 1 ρ s w 2 because of Lemma 1.
Use Lemma 4 (6) to calculate the image of the first term by I 1 . Now we can apply the induction hypothesis and see that Therefore It is equal to z 1 ρ r w 1 x z k w 2 .
Finally suppose that w = z k w 1 and w = z l w 2 for k, l ≥ 2 and w 1 , w 2 ∈ H 1 . From Lemma 4 (3) we get Using the induction hypothesis we have Since e −1 (z k w 1 ) and e −1 (z l w 2 ) belong to h ≥2 , we see that I 1 (e −1 (z k w 1 ) e −1 (z l w 2 )) is equal to using Lemma 4 (3), (4) and (6). Now apply the induction hypothesis again. As a result we find that I 1 (∆ 1 (w) ∆ 1 (w )) is equal to This completes the proof.
Proof of Proposition 6. It suffices to prove that e(w w ) = e(w) x e(w ) for homogeneous elements w, w ∈ H 0 . If w = 1 or w = 1, then it is trivial. Now we divide into four cases: (i) w = ξρ r w 1 and w = ξρ s w 2 for r, s ≥ 0 and w 1 , w 2 ∈ h ≥1 , (ii) w = z k w 1 and w = ξρ r w 2 for k ≥ 2, w 1 ∈ H 1 , r ≥ 0 and w 2 ∈ h ≥1 , (iii) w = z 2 w 1 and w = z l w 2 for l ≥ 2 and w 1 , w 2 ∈ H 1 , (iv) w = z k w 1 and w = z l w 2 for k, l ≥ 3 and w 1 , w 2 ∈ H 1 .
Case (ii) We proceed by induction on k. Let k = 2. Using the result in the Case (i) and Lemma 5 we see that e(w w ) = eI 0 e −1 (e(ξw 1 ) x e(ξρ r w 2 )) + ξρ r (z 2 w 1 x w 2 ).
This completes the proof for the case k = 2.
Note that e(z k w 1 ) ∈ h ≥1 . From the induction hypothesis and Lemma 5, we get Because of Lemma 5 (1), the second term in the right hand side is equal to Let us calculate the first term. Set Then e(z k−1 w 1 ) = xθ k−1 w 1 . Hence Note that the first term in the right hand side belongs to a≥2 z a H 1 . Using Lemma 5 (5) we find that Thus we obtain Case (iii) We proceed by induction on l. Let l = 2. From the result in the Case (ii) we have e(w w ) = eI 0 e −1 (e(ξw 1 ) x e(z 2 w 2 ) + e(z 2 w 1 ) x e(ξw 2 ) − e(ξw 1 ) x e(ξw 2 )) .
Using Lemma 5 (1) we have It belongs to a≥2 z a H 1 . Hence Lemma 5 (5) implies that This completes the proof.
3 Linear relations among the modif ied qMZVs

Double shuf f le relation
We regard H 0 as a graded Q-module by setting the degree of x, y, ρ and to be one, and call the degree the weight on H 0 . Denote the homogeneous component of weight d by H 0 d . Now we define the Q-linear mapZ q : H 0 → C byZ q (w) := (1 − q) −d Z q (w) for w ∈ H 0 d . If k = (k 1 , . . . , k r ) is an admissible index,Z q (z k 1 . . . z kr ) is equal to the modified qMZVζ q (k) defined by (2). Set From the definition of the harmonic product * and the integral shuffle product x, we see that H 0 = ⊕ d≥0 H 0 d is a commutative graded Q-algebra with respect to either * or x. Now we obtain the following theorem from Theorems 1 and 2. Thus we obtain linear relations among the modified qMZVs as the image of S d ∩ H 0 . Let us call such relations the double shuffle relations.
Proof . The proof is similar to that for MZVs given in [5]. From the definition of the harmonic product we have For α ≥ 1 and w ∈ H 1 , it holds that y j ξy α−j w + y α (y x w).
Using these formulas we obtain Hence we get the desired equality from Theorem 3.
By computer experiment we can find a lower bound of the dimension of Z ≤d [9], and calculate the dimension of N ≤d . The result up to weight 7 is given as follows: The second line above gives the number of admissible indices whose weight is less than or equal to d. We see that the sum of the values in the third line and the fourth one is equal to the number of admissible indices. Therefore the third line gives the dimension of Z ≤d exactly and the space N ≤d describes all linear relations among the modified qMZVs up to weight 7.