A Three-Parameter Hopf Deformation of the Algebra of Feynman-like Diagrams

G H E Duchamp^{a}, P Blasiak^{b}, A Horzela^{b}, K A Penson^{c}, A I Solomon^{c,d},

a Institut Galilée, LIPN, CNRS UMR 7030 99 Av. J.-B. Clement, F-93430 Villetaneuse, France

b H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences ul. Eliasza-Radzikowskiego 152, PL 31342 Kraków, Poland

c Laboratoire de Physique Théorique de la Matière Condensée Université Pierre et Marie Curie, CNRS UMR 7600 Tour 24 - 2ième ét., 4 pl. Jussieu, F 75252 Paris Cedex 05, France

d The Open University, Physics and Astronomy Department Milton Keynes MK7 6AA, United Kingdom

E-mail:

ghed@lipn-univ.paris13.fr, a.i.solomon@open.ac.uk
pawel.blasiak@ifj.edu.pl, andrzej.horzela@ifj.edu.pl,
penson@lptl.jussieu.fr,

Abstract. We construct a three-parameter deformation of the Hopf algebra LDIAG. This is the algebra that appears in an expansion in terms of Feynman-like diagrams of the product formula in a simplified version of Quantum Field Theory. This new algebra is a true Hopf deformation which reduces to LDIAG for some parameter values and to the algebra of Matrix Quasi-Symmetric Fuctions (MQSym) for others, and thus relates LDIAG to other Hopf algebras of contemporary physics. Moreover, there is an onto linear mapping preserving products from our algebra to the algebra of Euler-Zagier sums.

Contents

1 Introduction 2

2 How and why these Feynman-like Diagrams arise 2

3 Non-commutative lifting (classical case) 5 3.1 Free monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Labeling the nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.3 Coding ldiag with “lists of monomials” . . . . . . . . . . . . . . . . . . . 7 4 The Hopf algebra LDIAG (non-deformed case) 7

4.1 The monoid (𝔐𝔒𝔑^{+}(X))^{∗} and the submonoid of codes of diagrams … 8 4.2 The Hopf algebras C_{ldiag} and LDIAG . . . . . . . . . . . . . . . . . . . . 9 4.3 Subalgebras of LDIAG . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3.1 Graphic primitive elements . . . . . . . . . . . . . . . . . . . . . . 12 4.3.2 Level subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3.3 BELL and LBELL . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 The algebra LDIAG(q_{c}, q_{s}, q_{t}) (deformed case) 14 5.1 Counting crossings (q_{c}) and superpositions (q_{s}) . . . . . . . . . . . . . . . 14 5.2 Modified laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Coproducts 18

7 More on LDIAG(q_{c}, q_{s}, q_{t}) : structure, images and the link with Euler-Zagier sums 19

8 Concluding remarks 20 1. Introduction

We briefly describe the passage from the product formula, as described by by Bender et al. [3], and the related Feynman-like diagrams, to the description of Hopf algebra structures [15] on the diagrams themselves compatible with their evaluations. First, C. M. Bender, D. C. Brody, and B. K. Meister [3] introduced a special field theory which proved to be particularly rich in combinatorial links and by-products. Second, the Feynman-like diagrams produced by this theory label monomials; these monomials combine in a manner compatible with the monomial multiplication and coaddition‡. This is the Hopf algebra DIAG. Third, the natural noncommutative pull-back of this algebra, LDIAG, has a basis (the labeled diagrams) which is in one-to-one correspondence with that of the Matrix Quasi-Symmetric Functions (the packed matrices of MQSym), but their algebra and co-algebra structures are completely different. In particular, in this basis, the multiplication of MQSym implies a sort of shifted shuffle with overlappings reminiscent of Hoffmann’s shuffle used in the theory of of polyzeta functions[11]. The superpositions and overlappings involved there are not present in the (non-deformed) LDIAG and, moreover, the coproduct of LDIAG is co-commutative while that of MQSym is not.

The aim of this paper is to introduce a “parametric algebra” which mediates between the two Hopf algebras LDIAG and MQSym. The striking result is that when we introduce parameters which count the crossings and overlappings of the shifted shuffle, one notes that the resulting law is associative (graded with unit). We also show how to interpolate with a coproduct which makes, at each stage, our algebra a Hopf algebra. The result is thus a three-parameter Hopf algebra deformation which reduces to LDIAG at (0, 0, 0) and to MQSym at (1, 1, 1). Moreover it appears that, for one set of parameters, the multiplication rule of LDIAG recovers that of Euler-Zagier sums.

Acknowkedgements : The authors are pleased to acknowledge the hospitality of institutions in Paris, Cracow and New York, as well as that of Catherine Borgen, owner of “The Firs”(Exeter, UK). Also, we would like to thank L. Foissy and J. Stasheff for their thorough reading.

2. How and why these Feynman-like Diagrams arise

The beginning of the story was fully explained in [36, 37, 38, 28, 5, 6], and the Hopf algebra structure was made precise in [15, 40]. In this note we shall emphasize the latter part of the analysis, where the algebraic structure constructed on the diagrams themselves arise.

Our starting point is the formula (product formula) of Bender and al. [3], which can be considered as an expression of the Hadamard product for an exponential generating

‡ i.e. the commultiplication obtained by replacing each variable by the sum of two (independent) copies of it.

series. That is, using

F (z) = ∑_{n≥0} a_{n} \frac{z^{n}}{n!} , G(z) = ∑_{n≥0} b_{n} \frac{z^{n}}{n!} , H(F, G) := ∑_{n≥0} a_{n}b_{n} \frac{z^{n}}{n!}  (1)

one can check that

H(F, G) = F (z \frac{d}{dx}) G(x)|_{x=0} .  (2)

When F (0) and G(0) are not zero one can renormalize the functions in this bilinear product so that F (0) = G(0) = 1. We wish to obtain compact and generic formulas. If we write the functions as

F (z) = exp (∑_{n=1}^{∞} L_{n} \frac{z^{n}}{n!} ) ,  G(z) = exp (∑_{n=1}^{∞} V_{n} \frac{z^{n}}{n!} ) .  (3)

that is, as free exponentials, then by using Bell polynomials in the sets of variables 𝕃, 𝕍 (see [15] for details), we obtain

H(F, G) = ∑_{n≥0} \frac{z^{n}}{n!} ∑_{P_{1},P_{2}∈UP_{n}} 𝕃^{T ype(P_{1})}𝕍^{T ype(P_{2})}  (4)

where UP_{n} is the set of unordered partitions of [1 ⋯ n]. An unordered partition P of a set X is a subset of P ⊂ 𝔓(X) − {∅}§ (that is an unordered collection of blocks, i. e. non-empty subsets of X) such that

• the union ⋃_{Y ∈P} Y = X (P is a covering)

• P consists of disjoint subsets, i. e.

Y_{1}, Y_{2} ∈ P and Y_{1} ∩ Y_{2} ≠ ∅ ⟹ Y_{1} = Y_{2}.

The type of P ∈ UP_{n} (denoted above by T ype(P)) is the multiindex (α_{i})_{i∈ℕ^{+}} such that α_{k} is the number of k-blocks, that is the number of members of P with cardinality k.

At this point the formula entangles and the diagrams of the theory arise. Note particularly that

• the monomial 𝕃^{T ype(P_{1})}𝕍^{T ype(P_{2})} needs much less information than that which is contained in the individual partitions P_{1}, P_{2} (for example, one can relabel the elements without changing the monomial),

• two partitions have an incidence matrix from which it is still possible to recover the types of the partitions.

The construction now proceeds as follows.

§ The set of subsets of X is denoted by 𝔓(X) (this notation [9] is that of the former German school).

(i) Take two unordered partitions of [1 ⋯ n], say P_{1}, P_{2} (ii) Write down their incidence matrix (card(Y ∩ Z))_{(Y,Z)∈P_{1}×P_{2}} (iii) Construct the diagram representing the multiplicities of the incidence matrix : for

each block of P_{1} draw a black spot (resp. for each block of P_{2} draw a white spot) (iv) Draw lines between the black spot Y ∈ P_{1} and the white spot Z ∈ P_{2}; there are

card(Y ∩ Z) such.

(v) Remove the information of the blocks Y, Z, · · ·.

In so doing, one obtains a bipartite graph with p (= card(P_{1})) black spots, q (= card(P_{2})) white spots, no isolated vertex and integer multiplicities. We denote the set of such diagrams by diag.

[figure: {1}; {2, 3, 4}; {5, 6, 7, 8, 9}; {10, 11}; {2, 3, 5}; {1, 4, 6, 7, 8}; {9, 10, 11}]

Fig 1. — Diagram from P_{1}, P_{2} (set partitions of [1 ⋯ 11]). P_{1} = {{2, 3, 5}, {1, 4, 6, 7, 8}, {9, 10, 11}} and P_{2} = {{1}, {2, 3, 4}, {5, 6, 7, 8, 9}, {10, 11}} (respectively black spots for P_{1} and white spots for P_{2}). The incidence matrix corresponding to the diagram (as drawn) or these partitions is ( \begin{matrix}0 & 2 & 1 & 0 \\ 1 & 1 & 3 & 0 \\ 0 & 0 & 1 & 2 \end{matrix} ). But, due to the fact that the defining partitions are unordered, one can permute the spots (black and white, between themselves) and, so, the lines and columns of this matrix can be permuted. the diagram could be represented by the matrix ( \begin{matrix}0 & 0 & 1 & 2 \\ 0 & 2 & 1 & 0 \\ 1 & 0 & 3 & 1 \end{matrix} ) as well.

The product formula now reads

H(F, G) = ∑_{n≥0} \frac{z^{n}}{n!} ∑_{{d∈diag \atop |d|=n}} mult(d)𝕃^{α(d)}𝕍^{β(d)}  (5)

where α(d) (resp. β(d)) is the “white spots type” (resp. the “black spots type”) i.e. the multiindex (α_{i})_{i∈ℕ^{+}} (resp. (β_{i})_{i∈ℕ^{+}}) such that α_{i} (resp. β_{i}) is the number of white spots (resp. black spots) of degree i (i lines connected to the spot) and mult(d) is the number of pairs of unordered partitions of [1 ⋯ |d|] (here |d| = |α(d)| = |β(d)| is the number of lines of d) with associated diagram d.

Now one may naturally ask Q1) “Is there a (graphically) natural multiplicative structure on diag such that the arrow

d ↦ 𝕃^{α(d)}𝕍^{β(d)} (6) be a morphism ?”

The answer is “yes”. The desired product just consists in concatenating the diagrams (the result, i.e. the diagram obtained in placing d_{2} at the right of d_{1}, will be denoted by [d_{1}|d_{2}]_{D}). One must check that this product is compatible with the equivalence of the permutation of white and black spots among themselves, which is rather straightforward (see [15]). We have

Proposition 2.1 Let diag be the set of diagrams (including the empty one).

i) The law (d_{1}, d_{2}) ↦ [d_{1}|d_{2}]_{D} endows diag with the structure of a commutative monoid with the empty diagram as neutral element(this diagram will, therefore, be denoted by 1_{diag}). ii) The arrow d ↦ 𝕃^{α(d)}𝕍^{β(d)} is a morphism of monoids, the codomain of this arrow being the monoid of (commutative) monomials in the alphabet 𝕃 ∪ 𝕍 i.e.

𝔐𝔒𝔑(𝕃 ∪ 𝕍) = {𝕃^{α}𝕍^{β}}_{α,β∈(ℕ^{+})^{(ℕ)}} = ⋃_{n,m≥1} {L_{1}^{α_{1}}L_{2}^{α_{2}} ⋯ L_{n}^{α_{n}}V_{1}^{β_{1}}V_{2}^{β_{2}} ⋯ V_{m}^{β_{m}}}_{α_{i},β_{j}∈ℕ}.

iii) The monoid (diag, [−|−]_{D}, 1_{diag}) is a free commutative monoid. Its letters are the connected (non-empty) diagrams.

Remark 2.2 The reader who is not familiar with the algebraic structure of 𝔐𝔒𝔑(𝕏) can find rigorous definitions in paragraph (3.1) where this structure is needed for the proofs relating to deformations.

3. Non-commutative lifting (classical case)

The “classical” construction of the Hopf algebra LDIAG was given in [15]. We give the proofs below, using a coding through “lists of monomials” needed for the deformed (quantum) case. The entries of a list can be considered as “coordinate functions” for the diagrams (see introduction of section (4)).

3.1. Free monoids

We recall here the construction of the free and free-commutative monoids generated by a given set of variables (i.e. an alphabet). Let 𝕏, be a set. We denote by 𝕏^{∗} the set of lists of elements of 𝕏, including the empty one. In many works, and in the sequel, the list [x_{1}, x_{2}, ⋯, x_{n}] will be considered as a word x_{1}x_{2} ⋯ x_{n} so that the concatanation of two lists [x_{1}, x_{2}, ⋯, x_{n}], [y_{1}, y_{2}, ⋯, y_{m}] is just the word x_{1}x_{2} ⋯ x_{n}y_{1}y_{2} ⋯ y_{m}. For this (associative) law, the empty list [ ] is the neutral element and will therefore be denoted by 1_{𝕏^{∗}}

Similarly, we denote by ℕ^{(𝕏)} [7] the set of multisubsets of 𝕏 (i.e. the set of - multiplicity - mappings with finite support 𝕏 ↦ ℕ). Every element α of ℕ^{(X)} can be written multiplicatively, following the classical multiindex notation

𝕏^{α} = ∏_{x∈𝕏}x^{α(x)} (7)

and the set 𝔐𝔒𝔑(X) = {𝕏^{α}}_{α∈ℕ^{(X)}} is exactly the set of (commutative) monomials with variables in 𝕏. It is a monoid, indeed a (multiplicative) copy of ℕ^{(X)} as 𝕏^{α}𝕏^{β} = 𝕏^{α+β}. The subset of its non-unit elements is a semigroup which will be denoted by 𝔐𝔒𝔑^{+}(X) (= 𝔐𝔒𝔑(X) − {𝕏^{0}}).

3.2. Labeling the nodes

There are (at least) two good reasons to look for non-commutative structures which may serve as a noncommutative pullback for diag.

• Rows and Columns of matrices are usually (linearly) ordered and we have seen that a diagram is not represented by a matrix but by a class of matrices

• The complexity of diag and its algebra is not sufficient to relate it to other (noncommutative or non-cocommutative) algebras relevant to contemporary physics

The solution (of the non-deformed problem [15]) is simple and consists in labeling the nodes from left to right and from “1” to the desired number as follows.

[figure: 1; 2; 3; 4; 1; 2; 3]

Fig 2. — Labelled diagram of format 3 × 4 corresponding to the one of Fig 1.

The set of these graphs (i.e. bipartite graphs on some product [1..p] × [1..q] with no isolated vertex) will be denoted by ldiag. The composition law is, as previously, concatenation in the obvious sense. Explicitly, if d_{i}, i = 1, 2 are two diagrams of dimension [1..p_{i}] × [1..q_{i}], one relabels the black (resp. white) spots of d_{2} from p_{1} + 1 to p_{1} + p_{2} (resp. from q_{1} + 1 to q_{1} + q_{2}) the result will be noted [d_{1}|d_{2}]_{L}. One has

Proposition 3.1 Let ldiag be the set of labeled diagrams (including the empty one).

i) The law (d_{1}, d_{2}) ↦ [d_{1}|d_{2}]_{L} endows ldiag with the structure of a noncommutative monoid with the empty diagram (p = q = 0) as neutral element(which will, therefore, be denoted by 1_{ldiag}). ii) The arrow from ldiag to diag, which implies “forgetting the labels of the vertices” is a morphism of monoids. iii) The monoid (ldiag, [−|−]_{L}, 1_{ldiag}) is a free (noncommutative) monoid. Its letters are the irreducible diagrams (denoted from now on by irr(ldiag)).

Remark 3.2 i) In a general monoid (M, ⋆, 1_{M}), the irreducible elements are the elements x ≠ 1_{M} such that x = y ⋆ z ⟹ 1_{M} ∈ {y, z}. ii) It can happen that an irreducible of ldiag has an image in diag which splits, as shown by the simple example of the cross defined by the incidence matrix ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ).

3.3. Coding ldiag with “lists of monomials”

One can code every labelled diagram by a “list of (commutative) monomials” in the following way.

• Let 𝕏 = {x_{i}}_{i≥1} be an infinite set of indeterminates and d ∈ ldiag_{p×q} a diagram (ldiag_{p×q} is the set of diagrams with p black spots and q white spots).

• Associate with d the multiplicity function [1..p] × [1..q] → ℕ such that d(i, j) is the number of lines from the black spot i to the white spot j.

• The code associated with d is φ_{lm}(d) = [m_{1}, m_{2}, ⋯, m_{p}] such that m_{i} = ∏_{j=1}^{q} x_{j}^{m(i,j)}

Fig 3. — Coding the diagram of fig 2 by a word of monomials. The code here is [x_{2}^{2}x_{3} , x_{1}x_{2}x_{3}^{3} , x_{3}x_{4}^{2}]

As a data structure, the lists of monomials are elements of (𝔐𝔒𝔑^{+}(X))^{∗}, the free monoid whose letters are 𝔐𝔒𝔑^{+}(X) = 𝔐𝔒𝔑(X) − {𝕏^{0}}, the semigroup of non-unit monomials over 𝕏.

It is not difficult to see that, through this coding, concatenation is reflected in the following formula

φ_{lm}([d_{1}|d_{2}]_{L}) = φ_{lm}(d_{1}) ∗ T_{max(IndAlph(φ_{lm}(l_{1})))}(φ_{lm}(d_{2})) (8)

where T_{p} is the translation operator which changes the variables according to T_{p}(x_{i}) = x_{i+p} (which corresponds to the relabelling of the white spots) and p_{1} is the number of black spots of d_{1}. For example, one has

T_{2}([x_{2}^{2}x_{3}, x_{1}x_{2}x_{3}^{3}, x_{3}x_{4}^{2}]) = [x_{4}^{2}x_{5}, x_{3}x_{4}x_{5}^{3}, x_{5}x_{6}^{2}] ; T_{6}([x_{1}, x_{2}^{2}]) = [x_{7}, x_{8}^{2}] (9)

4. The Hopf algebra LDIAG (non-deformed case)

In [15], we defined a Hopf algebra structure on the space of diagrams LDIAG. The aim of this section is to give complete proofs and details for this construction through the use of the special space of coordinates constructed above (the complete vector of coordinates of a diagram being its code).

4.1. The monoid (𝔐𝔒𝔑^{+}(X))^{∗} and the submonoid of codes of diagrams

Formula (8) can be written using lists as

l_{1}¯∗l_{2} = l_{1} ∗ T_{max(IndAlph(l_{1}))}(l_{2}) (10)

which defines a monoid structure on (𝔐𝔒𝔑^{+}(X))^{∗} (the set of lists of non-unit monomials) with the empty list as neutral (i.e. [ ] which will, therefore, be denoted by “1_{(𝔐𝔒𝔑^{+}(X))^{∗}}” or simply “1” when the context is clear). We will return to this construction (called shifting [20]) later. The alphabet of a list is the set of variables occurring in the list. Formally

Alph([m_{1}, m_{2}, ⋯ m_{n}]) = ⋃_{1≤i≤k} Alph(m_{i}) (11)

where, classically, for a monomial m = 𝕏^{α}, Alph(m) = {x_{i}}_{α(i)≠0}. Now, we can define the “compacting operator” on k⟨𝔐𝔒𝔑^{+}(X)⟩ by its action on the lists. This operator actually removes the holes in the alphabet of a list by pushing to the left the indices which are at the right of a hole. For example (we denote by cpt the operator)

cpt([x_{2}^{2}x_{10}, x_{3}x_{4}x_{8}^{3}, x_{3}x_{4}^{2}]) = [x_{1}^{2}x_{5}, x_{2}x_{3}x_{4}^{3}, x_{2}x_{3}^{2}] (12)

the alphabet of the list on the LHS is Alph(l) = Alph([x_{2}^{2}x_{10}, x_{3}x_{4}x_{8}^{3}, x_{3}x_{4}^{2}]) = {x_{2}, x_{3}, x_{4}, x_{8}, x_{10}}, its indices are IndAlph(l) = {2, 3, 4, 8, 10} and the re-indexing function is the unique strictly increasing mapping from {2, 3, 4, 8, 10} to [[5]]. Here the compacting operator is just the substitution

x_{1} ← x_{2}; x_{2} ← x_{3}; x_{3} ← x_{4}; x_{4} ← x_{8}; x_{5} ← x_{10}

The formal definitions are the following

• IndAlph(l) = {i | x_{i} ∈ Alph(l)}

• l being given, let ϕ_{l} be the unique increasing mapping from IndAlph(l) to [[card(IndAlph(l))]] (in fact, card(IndAlph(l)) = card(Alph(l)))

• let s_{l} be the substitution x_{i} ← x_{ϕ_{l}(i)} in the monomials.

• Then, if l = [m_{1}, m_{2}, ⋯ m_{n}], cpt(l) = [s_{l}(m_{1}), s_{l}(m_{2}), ⋯ s_{l}(m_{n})].

Définition 4.1 The compacting operator cpt : k⟨𝔐𝔒𝔑^{+}(X)⟩ ↦ k⟨𝔐𝔒𝔑^{+}(X)⟩ is the extension by linearity of the mapping cpt defined above.

It can be checked easily that, for l ∈ (𝔐𝔒𝔑^{+}(X))^{∗}, the following are equivalent

(i) cpt(l) = l (ii) IndAlph(l) = [[card(IndAlph(l))]] (iii) there is no hole in Alph(l); that is, there exists no i ≥ 1 s.t. x_{i} ∉ Alph(l) and

x_{i+1} ∈ Alph(l)) (iv) l is the code of some (then unique) diagram d.

It follows from the preceding properties that cpt is a projector with range the subspace C_{ldiag} of k⟨𝔐𝔒𝔑^{+}(𝕏)⟩ generated by the codes of the diagrams. Formula (8) proves that C_{ldiag} is closed under the shifted concatenation defined by (10). More precisely

Proposition 4.2 The algebra C_{ldiag} is a free algebra on the set of the codes of irreducible diagrams.

These codes are also the non-empty lists l which are compact (i.e. cpt(l) = l) and cannot be factorized into a product of two non-empty lists i.e. l = l_{1} ∗ l_{2}; l_{i} ≠ [ ] (one can check easily that, if l_{1} ∗ l_{2} is compact, so are l_{1} and l_{2}).

4.2. The Hopf algebras C_{ldiag} and LDIAG

The algebra LDIAG is endowed with the structure of a bi-algebra by the comultiplication

Δ_{BS}(d) = ∑_{I+J=[1..p]} d[I] ⊗ d[J] (13)

where p is the number of black spots and d[I] is the “restriction” of d to the black spots selected by the I ⊂ [1..p]. On the other hand, we have a standard Hopf algebra structure on the free algebra, expressed in terms of concatenation and subwords [26, 34]. Let 𝔸 be an alphabet (a set of letters) and w ∈ 𝔸^{∗} a word, if we write w a a sequence of letters w = a_{1}a_{2} ⋯ a_{n}; a_{i} ∈ 𝔸, the length |w| of w is n and if I = {i_{1}, i_{2}, ⋯ i_{k}} ⊂ [1..n], the subword w[I] is a_{i_{1}}a_{i_{2}} ⋯ a_{i_{k}} (this notation is slightly different from that of [34] where it is w|_{I}). Then, the free algebra k⟨𝔸⟩ is a Hopf algebra with comultiplication [34, 26].

Δ_{LieHopf}(w) = ∑_{I+J=[1..n]} w[I] ⊗ w[J]. (14)

One has the following relation between restrictions of diagrams and subwords

φ_{lm}(d[I]) = cpt(φ_{lm}(d)[I]) (15)

this suggests that the coproduct

Δ_{list}(l) = ∑_{I+J=[1..n]} cpt(l[I]) ⊗ cpt(l[J]) (16)

could be a Hopf algebra comultiplication for the shifted algebra (k⟨𝔐𝔒𝔑^{+}(𝕏)⟩, ¯∗, [ ]). Unfortunately, this fails due to the lack of counit (i and ii of the following Theorem), but the “ground subalgebra” C_{ldiag} is a genuine Hopf algebra (which is exactly what we do need here).

Theorem 4.3 Let A = (k⟨𝔐𝔒𝔑^{+}(𝕏)⟩, ¯∗, [ ]) be the algebra of lists of (non-unit) monomials endowed with the shifted concatenation of formula (10). Then

i) A is a free algebra. ii) The coproduct Δ_{list} (recalled below) is co-associative and a morphism of algebras A ↦ A ⊗ A (i.e. A is a bi-algebra without counit).

Δ_{list}(l) = ∑_{I+J=[1..n]} cpt(l[I]) ⊗ cpt(l[J]) (17)

iii) The algebra C_{ldiag} is a sub-algebra and coalgebra of A which is a Hopf algebra for the following co-unit and antipode.

• Counit

ε(l) = δ_{l,[ ]}  (Kronecker delta)  (18)

• Antipode

S(l) = ∑_{r≥0} ∑_{{I_{1}+I_{2}+...Ir=[1..p] \atop I_{j} ≠∅}}(−1)^{r}cpt(l[I_{1}])cpt(l[I_{2}]) ⋯ cpt(l[I_{r}])  (19)

Proof — i) Throughout the proof, we will denote by ∗ the concatenation between lists and ¯∗ the shifted concatenation defined by the formula (10). We first remark that, if l = l_{1}¯∗l_{2}, then max(IndAlph(l_{1})) < min(IndAlph(l_{2})). This leads us to define, for a (non-shifted) factorization l = l_{1} ∗ l_{2} = l[1..t] ∗ l[t + 1..p] (p = |l|), a gauge of the degree of overlapping of the intervals (of integers) [1..max(IndAlph(l1))] and [min(IndAlph(l2))..∞[, thus the function

ω_{l}(t) = card([1..max(IndAlph(l[1..t])] ∩ [min(IndAlph(l[t + 1..p])..∞[) =

(max(IndAlph(l[1..t])) − min(IndAlph(l[t + 1..p])) + 1)^{+}.  (20)

(We recall that, for a real number x, x^{+} is its positive part x^{+} = max(x, 0) = \frac{1}{2}(|x| + x) [8]). It can be easily checked that the points t where ω_{l}(t) = 0 determine the (unique) factorisation of l in irreducibles. It follows that the monoid ((𝔐𝔒𝔑^{+}(𝕏))^{∗}, ¯∗, [ ]) is free and so is its algebra (k⟨𝔐𝔒𝔑^{+}(𝕏)⟩, ¯∗, [ ]).

ii) If we denote Δ : A ↦ A ⊗ A the standard coproduct given, for a list l of length p, by formula (14), one can remark that

(i) cpt(l_{1})¯∗cpt(l_{2}) = cpt(l_{1}¯∗l_{2}) (ii) Δ_{list} = (cpt ⊗ cpt) ∘ Δ (iii) Δ_{list} ∘ cpt = Δ_{list} (iv) (∀n ∈ ℕ)(cpt(T_{n}(l)) = cpt(l))

(v) (∀n ∈ ℕ)(Δ ∘ T_{n} = (T_{n} ⊗ T_{n}) ∘ Δ). Coassociativity of Δ_{list}. — One has

(Δ_{list} ⊗ Id) ∘ Δ_{list} = (Δ_{list} ⊗ Id) ∘ (cpt ⊗ cpt) ∘ Δ = ‘ ((Δ_{list} ∘ cpt) ⊗ cpt) ∘ Δ = (Δ_{list} ⊗ cpt) ∘ Δ = (((cpt ⊗ cpt) ∘ Δ) ⊗ cpt) ∘ Δ = (cpt ⊗ cpt ⊗ cpt) ∘ (Δ ⊗ Id) ∘ Δ = (cpt ⊗ cpt ⊗ cpt) ∘ (Id ⊗ Δ) ∘ Δ (cpt ⊗ ((cpt ⊗ cpt) ∘ Δ)) ∘ Δ = (cpt ⊗ Δ_{list}) ∘ Δ = (cpt ⊗ (Δ_{list} ∘ cpt)) ∘ Δ =

(Id ⊗ Δ_{list}) ∘ (cpt ⊗ cpt) ∘ Δ = (Id ⊗ Δ_{list}) ∘ Δ_{list}  (21)

Δ_{list} is a morphism. —

For two lists u, v ∈, let us compute Δ_{list}(u¯∗v). With p = max(IndAlph(u)), one has

Δ_{list}(u¯∗v) = (cpt ⊗ cpt) ∘ Δ(l_{1} ∗ T_{p}(v)) = (cpt ⊗ cpt)(Δ(u) ∗^{⊗2} Δ(T_{p}(v))) = (cpt ⊗ cpt)(Δ(u) ∗^{⊗2} (T_{p} ⊗ T_{p})Δ(v) =

(cpt ⊗ cpt)( ∑_{(1)(2)} u_{(1)} ⊗ u_{(2)}) ∗^{⊗2} (T_{p} ⊗ T_{p})( ∑_{(3)(4)} v_{(3)} ⊗ v_{(4)}) =

(cpt ⊗ cpt)( ∑_{(1)(2)(3)(4)} u_{(1)} ∗ T_{p_{1}}(T_{p−p_{1}}(v_{(3)})) ⊗ u_{(2)} ∗ T_{p_{2}}(T_{p−p_{2}}(v_{(4)}))) (22)

with, for each term in the sum

p_{1} = max(IndAlph(u_{(1)})) ≤ p ; p_{2} = max(IndAlph(u_{(2)})) ≤ p

so, the quantity in (22) is

(cpt ⊗ cpt)( ∑_{(1)(2)(3)(4)} u_{(1)}¯∗(T_{p−p_{1}}(v_{(3)})) ⊗ u_{(2)}¯∗(T_{p−p_{2}}(v_{(4)}))) =

∑_{(1)(2)(3)(4)} cpt(u_{(1)}¯∗(T_{p−p_{1}}(v_{(3)}))) ⊗ cpt(u_{(2)}¯∗(T_{p−p_{2}}(v_{(4)}))) =

∑_{(1)(2)(3)(4)} (cpt(u_{(1)})¯∗cpt(T_{p−p_{1}}(v_{(3)}))) ⊗ (cpt(u_{(2)})¯∗cpt(T_{p−p_{2}}(v_{(4)}))) =

∑_{(1)(2)(3)(4)} (cpt(u_{(1)})¯∗cpt(v_{(3)})) ⊗ (cpt(u_{(2)})¯∗cpt(v_{(4)})) =

( ∑_{(1)(2)} cpt(u_{(1)}) ⊗ cpt(u_{(2)}))¯∗^{⊗2}( ∑_{(3)(4)} cpt(v_{(3)}) ⊗ cpt(v_{(4)})) =

Δ_{list}(u)¯∗^{⊗2}Δ_{list}(v)  (23)

iii) As C_{ldiag} is generated by the image of cpt it is clear that this space is a subcoalgebra of A. Moreover, cpt is a (multiplicative) morphism A ↦ A and thus its image C_{ldiag} is a subalgebra of A. We now supply the missing ingredients to complete the proof of the Hopf algebra structure.

ε is a counit. —

Let l = cpt(l) be a compact list. We remark that, for any list u, one has cpt(u) = [ ] ⟺ u = [ ]. Then, with μ_{l} : k ⊗ A ↦ A the scaling operator

μ_{l}(ε ⊗ Id)Δ_{list}(l) = ∑_{I+J=[1..n]} ε(cpt(l[I]))cpt(l[J]) =

∑_{{I+J=[1..n] \atop I=∅}} ε(cpt(l[I]))cpt(l[J]) + ∑_{{I+J=[1..n] \atop I≠∅}} ε(cpt(l[I]))cpt(l[J]) = cpt(l) + 0 = l  (24)

the proof of the fact that ε is a left counit is similar.

S is the antipode. —

One has C_{ldiag} = k.1 ⊕ ker(ε), let us denote Id^{+} the projection C_{ldiag} ↦ ker(ε) according to this decomposition. Then, for every list l,

∑_{r≥0} ∑_{{I_{1}+I_{2}+...Ir=[1..p] \atop I_{j} ≠∅}}(−1)^{r}cpt(l[I_{1}])cpt(l[I_{2}]) ⋯ cpt(l[I_{r}])

is well defined as the first sum is locally finite. Thus, the operator

∑_{r≥0} ∑_{{I_{1}+I_{2}+...Ir=[1..p] \atop I_{j} ≠∅}}(−1)^{r} \underbrace{(Id^{+} ∗ Id^{+} ∗ ⋯ ∗ Id^{+})}_{r times}

is well defined and is the convolutional inverse of Id.

4.3. Subalgebras of LDIAG

4.3.1. Graphic primitive elements The problem of Graphic Primitive Elements (GPE) is the following. Let H be a Hopf algebra with (linear) basis G, a set of graphs. The GPE are the primitive elements Γ ∈ G which are primitive i.e.

Γ is a GPE ⟺ Γ ∈ G and Δ(Γ) = Γ ⊗ 1 + 1 ⊗ Γ. (25)

It is not difficult to check that, in any case, the subalgebra H^{GPE} generated by these elements is also a sub-coalgebra. We make an extra hypothesis (which is often fulfilled)

1_{H} ∈ G and (Γ ∈ G − {1_{H}} ⟹ ε(Γ) = 0). (26)

Then (if (26) is fulfilled) H^{GPE} is a sub-Hopf algebra as the antipode of the product Γ_{1}Γ_{2} ⋯ Γ_{p} of (GPE) is

S(Γ_{1}Γ_{2} ⋯ Γ_{p}) = (−1)^{p} Γ_{p}Γ_{p−1} ⋯ Γ_{1}. (27)

The following proposition helps to determine LDIAG^{GPE}.

Proposition 4.4 In LDIAG (with basis G = ldiag), the following are equivalent i) d is a GPE ii) d has only one black spot.

Then, the Hopf algebra LDIAG^{GPE} is generated by the product of “one-black-spot” diagrams.

Fig 4. — Graphic Primitive Elements of LDIAG have only one black spot and therefore are coded by the sequence of the ingoing degrees of their white spots (a composition). The first one here has code [1, 2, 3, 1]. The picture shows an element of the monoid generated by Graphic Primitive Elements (a linear basis of LDIAG^{GPE}) which is then coded by a list of compositions, here [[1, 2, 3, 1], [2, 3, 1], [2, 1, 4]].

4.3.2. Level subalgebras

One can also impose limitations on the incoming degrees of the white spots in a way compatible with the coproduct. In this case, one defines an infinity of Hopf-subalgebras of LDIAG which we will call “level subalgebras”. More precisely, given an integer l > 0, one can ask for spaces generated by the diagrams d for which every white spot has an incoming degree ≤ l. This amounts to say that the “white spot type” of every diagram d is of the form

α(d) = (α_{1}, α_{2}, ⋯ α_{k}, 0, 0 ⋯ 0, ⋯) ; ( all the α_{i} ≤ l for i ≤ k and α_{i} = 0 for i > k)

We denote by LDIAG^{≤l} the subspace generated by these diagrams. One has a chain of Hopf algebras

LDIAG^{≤1} ⊂ LDIAG^{≤2} ⊂ ⋯ LDIAG^{≤l} ⊂ LDIAG^{≤l+1} ⊂ ⋯ ⊂ LDIAG (28)

In the next paragraph, we will specially be interested in

LBELL = LDIAG^{≤1} ∩ LDIAG^{GPE}.

4.3.3. BELL and LBELL

The algebras BELL and LBELL were defined in [39].

The algebra LBELL is the intersection LDIAG^{≤1} ∩ LDIAG^{GPE} and since they are subspaces generated by subsets of ldiag, LBELL is generated by diagrams that

• are concatenations of one-black-spot-diagrams

• such that the incoming degree of every white spot is one.

Let d_{k} be the diagram with code [x_{1}, x_{2}, ⋯ x_{k}]. LBELL is generated by concatenations of these diagrams. Indeed, the diagrams d_{k} are a subalphabet of the free monoid ldiag so that they generate a free submonoid which we will denote here lbell.

Fig 5. — An element of lbell, concatenation d_{1}d_{3}d_{2}.

The algebras LDIAG and LBELL are both enveloping algebras. They are generated by their primitive elements which are in general linear combinations of diagrams and not pure diagrams. For an analysis of “graphic primitive elements” see section (4.3.1).

5. The algebra LDIAG(q_{c}, q_{s}, q_{t}) (deformed case)

5.1. Counting crossings (q_{c}) and superpositions (q_{s})

The preceding coding is particularly well adapted to the the deformation we want to construct here. The philosophy of the deformed product is expressed by the descriptive formula∥.

[d_{1}|d_{2}]_{L(q_{c},q_{s})} = ∑_{{cs(?) all crossing and \atop superpositions of black spots}} q_{c}^{nc×weight}q_{s}^{weight×weight}cs([d_{1}|d_{2}]_{L})(29)

where

• q_{c}, q_{s} ∈ ℂ or q_{c}, q_{s} formal. These and other cases may be unified by considering the set of coefficients as belonging to a ring K.

• the exponent of q_{c}^{nc×weight} is the number of crossings of “what crosses” times its weight

∥ Exact definition of the coefficient q_{c}^{nc×weight}q_{s}^{weight×weight} is the result of crossing and shifting processes which will be detailed in paragraph (5.2).

• the exponent of q_{s}^{weight×weight} is the product of the weights of “what is overlapped”

• cs() are the diagrams obtained from [d_{1}|d_{2}]_{L} by the process of crossing and superposing the black spots of d_{2} on to those of d_{1}, the order and distinguishability of the black spots of d_{1} (i.e. d_{2}) being preserved. What is striking is that this law is associative. This result will be established after

the following paragraph.

∗ =  +q_{s}^{2}  + q_{c}^{2} + q_{c}^{2}q_{s}^{6}  + q_{c}^{8}

Fig 5. — Counting crossings and superposings produces an associative law.

Fig 6. — Detail of the fourth monomial (with coefficient q_{c}^{2}q_{s}^{6}), crossings (circles) and superposings (black squares) are counted the same way but with a different variable.

5.2. Modified laws

• Twisting

Proposition 5.1 Let A = (A_{n})_{n∈ℕ} a graded semigroup and A^{∗} the set of lists (denoted by [a_{1}, a_{2}, ⋯ a_{k}]) with letters in A.

For convenience we define the operator ∗ (left append) A × A^{∗} ↦ A^{∗} by

a ∗ [b_{1}, b_{2}, ⋯ b_{n}] := [a, b_{1}, b_{2}, ⋯ b_{n}] (30)

Let q_{c}, q_{s} ∈ k be two elements in a ring k. We define on k < A >= k[A^{∗}] a new law ↑ by

w ↑ 1_{A^{∗}} = 1_{A^{∗}} ↑ w = w

a ∗ u ↑ b ∗ v = a ∗ (u ↑ b ∗ v) + q_{c}^{|a∗u||b|}b ∗ (a ∗ u ↑ v) + q_{c}^{|u||b|}q_{s}^{|a||b|}ab ∗ (u ↑ v) (31)

where the weights (|x| = n if x ∈ A_{n}) are extended additively to lists by

|[a_{1}, a_{2}, ⋯, a_{k}]| = ∑_{i=1}^{k} |a_{i}|

Then the new law ↑ is graded, associative with 1_{A^{∗}} as unit.

Proof — It suffices to prove the identity x ↑ (y ↑ z) = (x ↑ y) ↑ z ; x, y, z being lists (as the two members are trilinear). It is obviously true when one of the factors is the empty list. We show it when the three factors are non-empty (throughout the computation, the law ∗ will have priority over other operators).

(a ∗ u ↑ b ∗ v) ↑ c ∗ w = (a ∗ (u ↑ b ∗ v) + q^{|u||b|}t^{|a||b|}(ab)(u ↑ v) + q^{|a∗u||b|}b(a ∗ u ↑ v)) ↑ c ∗ w = [a ∗ ((u ↑ b ∗ v) ↑ c ∗ w) + q^{(|u|+|b∗v|)|c|}t^{|a||c|}(ac)((u ↑ b ∗ v) ↑ w)

+q^{(|a∗u|+|b∗v|)|c|}c ∗ (a ∗ (u ↑ b ∗ v) ↑ w)] +

[q^{|u||b|}t^{|a||b|}(ab)(u ↑ v ↑ c ∗ w) + q^{|u||b|+(|u|+|v|)|c|}t^{|a||b|}t^{(|a|+|b|)|c|}(abc)(u ↑ v ↑ w) +

q^{|u||b|+(|a∗u|+|b∗v|)|c|}t^{|a||b|}c(((ab)(u ↑ v)) ↑ w)] +

[q^{|a∗u||b|}b((a ∗ u ↑ v) ↑ c ∗ w) + q^{|a∗u||b|+(|a∗u|+|v|)|c|}t^{|b||c|}(bc)(au ↑ v ↑ w) +

q^{|a∗u||b|+(|a∗u|+|b∗v|)|c|}c(b(a ∗ u ↑ v) ↑ w)]  (32)

a ∗ u ↑ (b ∗ v ↑ c ∗ w) = a ∗ u ↑ (b ∗ (v ↑ c ∗ w) + q^{|v||c|}t^{|b||c|}(bc)(v ↑ w) + q^{|b∗v||c|}c(b ∗ v ↑ w)) = [a ∗ (u ↑ b ∗ (v ↑ c ∗ w)) + q^{|u||b|}t^{|a||b|}(ab)(u ↑ v ↑ c ∗ w) + q^{|a∗u||b|}b(a ∗ u ↑ v ↑ c ∗ w)] +

[q^{|v||c|}t^{|b||c|}a ∗ (u ↑ (bc)(v ↑ w)) + q^{|v||c|+|u|(|c|+|b|)}t^{|b||c|+|a|(|b|+|c|)}(abc)(u ↑ v ↑ w) +

q^{|v||c|+|a∗u|(|b|+|c|)}t^{|b||c|}(bc)(a ∗ u ↑ v ↑ w)] +

[q^{|b∗v||c|}a ∗ (u ↑ c(b ∗ v ↑ w)) + q^{(|u|+|b∗v|)|c|}t^{|a||c|}(ac)(u ↑ b ∗ v ↑ w) +

q^{(|a∗u|+|b∗v|)|c|}c ∗ (a ∗ u ↑ b ∗ v ↑ w)]  (33)

in the second expression, one gathers the three terms which we find first in the square brackets and we get

a ∗ (u ↑ b ∗ (v ↑ cw)) + q^{|v||c|}t^{|b||c|}a ∗ (u ↑ (bc) ∗ (v ↑ w)) +
q^{|b∗v||c|}a ∗ (u ↑ c ∗ (b ∗ v ↑ w)) = a ∗ (u ↑ b ∗ v ↑ c ∗ w)  (34)

in the first expression, one gathers the three terms which we find last in the square brackets and we get

q^{(|a∗u|+|b∗v|)|c|}c ∗ (a ∗ (u ↑ b ∗ v) ↑ w) +
q^{|u||b|+(|a∗u|+|b∗v|)|c|}t^{|a||b|}c ∗ (((ab) ∗ (u ↑ v)) ↑ w) +
q^{|a∗u||b|+(|a∗u|+|b∗v|)|c|}c ∗ (b ∗ (a ∗ u ↑ v) ↑ w) =
q^{(|au|+|bv|)|c|}c ∗ (a ∗ u ↑ b ∗ v ↑ w)  (35)

and one finds the 7-term expression

a ∗ (u ↑ b ∗ v ↑ c ∗ w) + q^{|a∗u|}b ∗ (a ∗ u ↑ v ↑ c ∗ w) +
q^{|a∗u|+|b∗v|}c ∗ (a ∗ u ↑ b ∗ v ↑ w) + q^{|u||b|}t^{|a||b|}(ab) ∗ (u ↑ v ↑ c ∗ w) +
q^{(|u|+|b∗v|)|c|}t^{|a||c|}(ac) ∗ (u ↑ b ∗ v ↑ w) +
q^{|v||c|(|b|+|c|)|au|}t^{|b||c|}(bc) ∗ (a ∗ u ↑ v ↑ w)
+q^{|v||c|+|u|(|c|+|b|)}t^{|b||c|+|a|(|b|+|c|)}(abc) ∗ (u ↑ v ↑ w)  (36)
□

The framework with diagrams will need another proposition on shifted laws.

• Shifting

We begin by the “shifting lemma” [20].

Lemma 5.2 Let A be an associative algebra (which law will be denoted by ⋆) and A = ⊕_{n∈ℕ}A_{n} a decomposition of A in direct sum. Let T ∈ End(A) be an endomorphim of the algebra A. We will denote T^{n} = T ∘ T ∘ ⋯ ∘ T the n-th compositional power of T. We suppose that the shifted law

a ¯⋆ b = a ⋆ T^{α}(b) (37)

for a ∈ A_{α} is graded for the decomposition A = ⊕_{n∈ℕ}A_{n}. Then, if the law ⋆ is associative so is the law ¯⋆.

Remark 5.3 The hypothesis that the shifted law given by eq.(37) be graded is automatically satisfied if A = ⊕_{n∈ℕ}A_{n} is a graded algebra and if all the morphisms T_{n} are of degree 0.

This lemma will be applied to the decomposition given by n = sup(Alph(w)) (the highest index of variables appearing in w) and the morphism given by T (x_{i}) = x_{i+1}.

What do these statements mean for us ? Here the graded semigroup is 𝔐𝔒𝔑^{+}(X) and we do not forget the coding arrow φ_{lm} : ldiag → (𝔐𝔒𝔑^{+}(X))^{∗}. The image of φ_{lm} is exactly the set of lists of monomials w = [m_{1}, m_{2}, ⋯, m_{k}] such that the set of variables involved Alph(w) is of the form x_{1} ⋯ x_{l} (the labelling of the white spots is without hole). By abuse of language we will say that a list of monomials “is in ldiag” in this case. It is not difficult to see, from formulas (31,37) that if w_{i}, i = 1, 2 are in ldiag so are all the factors of w_{1}¯↑w_{2}, this defines a new law on K[ldiag] and this algebra will be called LDIAG(q_{c}, q_{s}). The properties of this algebra will be made precise in the following proposition.

Proposition 5.4 Let C_{ldiag} be the subspace of (K < 𝔐𝔒𝔑^{+}(𝕏) >, ¯↑) generated by the codes of the diagrams (i.e. the lists w ∈ 𝔐𝔒𝔑^{+}(𝕏) such that Alph(w) is without hole). Then

i) (C_{ldiag}, ¯↑) is a unital subalgebra of (K < 𝔐𝔒𝔑^{+}(𝕏) >, ¯↑) ii) (C_{ldiag}, ¯↑) is a free algebra. More precisely, for any diagram decomposed in irreducibles d = d_{1}.d_{2} ⋯ d_{k} let

B(d) := φ_{lm}(d_{1})¯↑φ_{lm}(d_{2}) ⋯ ¯↑φ_{lm}(d_{k}) (38)

then α) (B(d))_{d∈ldiag} is a basis of C_{ldiag} β) B(d_{1}.d_{2}) = B(d_{1})¯↑B(d_{2})

As k[ldiag] is isomorphic to C_{ldiag} as a linear space, we denote LDIAG(q_{c}, q_{s}) the new algebra structure of k[ldiag] inherited from C_{ldiag}. one has

LDIAG(0, 0) ≃ LDIAG; LDIAG(1, 1) ≃ MQSym (39)

6. Coproducts

We must now define a parametrized (say, by q_{t}) coproduct such that (LDIAG(q_{c}, q_{s}), ¯↑, 1_{ldiag}, Δ_{q_{t}}, ε) is a graded bialgebra (as in the non-deformed Hopf algebra of [15], the counit ε is just the “constant term” linear form). We will take advantage of the freeness of LDIAG(q_{c}, q_{s}) through the following lemma.

Lemma 6.1 Let 𝕐 be an alphabet, k a ring and k < 𝕐 >= k[𝕐^{∗}] be the free algebra constructed on 𝕐. For every mapping Δ : A → k < 𝕐 > ⊗k < 𝕐 >, we denote Δ̄ : k < 𝕐 >↦ k < 𝕐 > ⊗k < 𝕐 > its extension as a morphism of algebras (k < 𝕐 > ⊗k < 𝕐 > being endowed with its non-twisted structure of tensor product of algebras). Then, in order to be coassociative, it is necessary and sufficient that

(Δ̄ ⊗ I) ∘ Δ and (I ⊗ Δ̄) ∘ Δ (40)

coincide on 𝕐.

The preceding lemma expresses the fact that, for a free algebra, the variety of the possible coproducts is a linear subspace. This will be transparent in formula (43).

We now consider the structure constants of the coproduct of MQSym [18] expressed with respect to the family of free generators

{MS_{P} }_{P ∈PM^{c}}

where PM^{c} is the set of connex packed matrices (similarly, PM is the set of packed matrices).

Δ_{MQSym}(MS_{P}) = ∑_{Q,R∈PM} α_{P}^{Q,R} MS_{Q} ⊗ MS_{R} (41)

For the irreducible diagram d, we set

Δ_{1}(d) = ∑_{d_{1},d_{2}∈irr(ldiag)} α_{φ_{lm}(d)}^{φ_{lm}(d_{1}),φ_{lm}(d_{2})}d_{1} ⊗ d_{2}  (42)

and Δ_{0}(d) = Δ_{W S}(d). Then proposition (6.1) proves that, for q_{t} ∈ {0, 1}

Δ_{t} = \overline{(1 − q_{t})Δ_{0} + q_{t}Δ_{1}} (43)

is a coproduct of graded bialgebra for (LDIAG(q_{c}, q_{s}), ↑, 1_{ldiag}).

We sum up the results

Proposition 6.2 i) With the operations defined above, q_{c}, q_{s} complex or formal and q_{t} boolean (q_{t} ∈ {0, 1}),

LDIAG(q_{c}, q_{s}, q_{t}) := (LDIAG(q_{c}, q_{s}), ¯↑, 1_{ldiag}, Δ_{q_{t}}, ε)

is a Hopf algebra. ii) At parameters (0, 0, 0), one has LDIAG(0, 0, 0) ≃ LDIAG iii) At parameters (1, 1, 1), one has LDIAG(1, 1, 1) ≃ MQSym

7. More on LDIAG(q_{c}, q_{s}, q_{t}) : structure, images and the link with Euler-Zagier sums

It has been proved recently that LDIAG(q_{c}, q_{s}, q_{t}) is a tridendriform Hopf Algebra [25] and that LDIAG(1, q_{s}, q_{t}) is a homomorphic image of the algebra of planar decorated trees of Foissy [24]. Bidendriformity of the algebra LDIAG(q_{c}, q_{s}) can also be established through a bi-word realization providing yet another (statistical) interpretation of the (q_{c}, q_{s}) deformation [20]. We will now make clear the relations between the (q_{c}, q_{s}) deformation and Euler-Zagier sums. According the notation of [31], one has

ζ(s_{1}, ⋯, s_{n}; σ_{1}, ⋯, σ_{n}) = ∑_{0<i_{1}<···<i_{n}} \frac{σ_{1}^{i_{1}} ⋯ σ_{n}^{i_{n}}}{i_{1}^{s_{1}} ⋯ i_{n}^{s_{n}}} (44)

with σ_{i} ∈ {−1, 1} and s_{1} > 1 if σ_{1} = 1. Here we are more interested in the multiplication mechanism, so we extend the notation to formal variables and use, for indices, the biword notation. Hence

ζ_{F P} ( \begin{matrix}m_{1} & ⋯ & m_{n} \\ s_{1} & ⋯ & s_{n} \end{matrix} ) = ∑_{0<i_{1}<···<i_{n}} \frac{m_{1}^{i_{1}} ⋯ m_{n}^{i_{n}}}{i_{1}^{s_{1}} ⋯ i_{n}^{s_{n}}} . (45)

We remark that the indices are taken as words (i.e. lists) with variables located in the semigroup 𝔐𝔒𝔑(Z) × ℕ^{+} with Z = {z_{i}}_{i≥1}. The set of these functions is closed under multplication and will be called below F P (Z), formal polyzeta functions in the variables Z. Hence, the multiplication of these sums fits in the hypotheses of Proposition (5.1) with q_{c} = q_{s} = 1 (quasi-shuffle in [12]). From this, we deduce an arrow

LDIAG(1, 1) → F P (Z). (46)

More precisely, if d is a diagram with code [m_{1}, m_{2} ⋯, m_{p}] we make correspond

ζ_{F P} ( \begin{matrix}m_{1} & ⋯ & m_{n} \\ deg(m_{1}) & ⋯ & deg(m_{n}) \end{matrix} ) (47)

where deg(m_{i}) is the total degree of m_{i}. We will denote ζ_{D2F P} (d) this value (47). One has

ζ_{D2F P} (d_{1})ζ_{D2F P} (d_{2}) = ζ_{D2F P} (d_{1} ↑_{11} d_{2}) (48)

the law ↑_{11} being unshifted and specialized to (q_{c}, q_{s}) = (1, 1). When restricted to “convergent” diagrams (i.e. diagrams with deg(m_{1}) ≥ 2 which form a subalgebra of LDIAG_{u}(q_{c}, q_{s})) and specializing all the variables to 1, we recover the “usual” Euler-Zagier sums by just counting the outgoing degrees of the black spots and the arrow of (46) becomes

d → ζ(deg(m_{1}), ⋯, deg(m_{n})) (49)

(usual Euler-Zagier sums). Denoting the last (49) value ζ_{D2EZ}(d), one has

ζ_{D2EZ}(d_{1})ζ_{D2EZ}(d_{2}) = ζ_{D2EZ}(d_{1} ↑_{11} d_{2}) (50)

8. Concluding remarks

For a diagram d with r black spots, the code [m_{1}, m_{2}, ⋯, m_{r}] can be temporarily seen as a “vector of coordinates” for the given diagram, but we prefer to stick to the structure of lists as, firstly, the dimension of the vector varies with the diagram and secondly, we have to concatenate the codes. The coordinate functions of the diagram d are therefore the family (a_{i})_{i>0} defined by a_{i}(d) = m_{i} for i ≤ r and a_{i}(d) = 0 for i > r. From this perspective the “q_{t}” of our three parameter deformation is a quantization in the sense of Moyal’s deformed products [1] on the algebra of coordinate functions (but without the first order condition; see the introduction of [13]), by the formula

a_{i_{1}} ∗ a_{i_{1}} ⋯ ∗ a_{i_{k}}(d) = μ(a_{i_{1}} ⊗ a_{i_{1}} ⊗ ⋯ ⊗ a_{i_{k}}(Δ_{q_{t}}^{[k]}(d))) (51) where μ is the ordinary multiplication of polynomials. The crossing parameter q_{c} is also a quantization parameter as, for q_{s} = 0, one has

code(d_{1} ∗ d_{2}) = code(d_{1}) ⊔_{q_{c}} T (code(d_{2})) (52)

where T is a suitable translation of the variables and ⊔_{q_{c}} is the quantum shuffle [35] for the braiding on V = ℂ[x_{i}; i ≥ 1] defined by

B(x_{i_{1}}^{α_{1}}x_{i_{2}}^{α_{2}} ⋯ x_{i_{k}}^{α_{k}} ⊗ y_{j_{1}}^{β_{1}}y_{j_{2}}^{β_{2}} ⋯ y_{j_{l}}^{β_{l}}) = q_{c}^{(∑ α_{i})(∑ β_{j})} y_{j_{1}}^{β_{1}}y_{j_{2}}^{β_{2}} ⋯ y_{j_{l}}^{β_{l}} ⊗ x_{i_{1}}^{α_{1}}x_{i_{2}}^{α_{2}} ⋯ x_{i_{k}}^{α_{k}} (53)

Let us add that q_{s} and q_{c} are of different nature as q_{s} is the coefficient of a perturbation of the shuffle product (better seen on the coproduct). This kind of perturbation occurs in various domains as : computer science by means of the infiltration product introduced by Ochsenschläger [33] (see also [17] and [16]), algebra of the Euler-Zagier sums [27] and noncommutative symmetric functions [18]. The mathematics of this dual aspect is of geometrical nature and will be developped in [19].

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