Feynman graphs and related Hopf algebras

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

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

E-mail:

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

Abstract. In a recent series of communications we have shown that the reordering problem of bosons leads to certain combinatorial structures. These structures may be associated with a certain graphical description. In this paper, we show that there is a Hopf Algebra structure associated with this problem which is, in a certain sense, unique.

Contents

1 Introduction 2

2 Single and double exponentials 2

2.1 One parameter groups and the connected graph theorem . . . . . . . . . 2 2.1.1 Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1.2 Combinatorial matrices and one-parameter groups . . . . . . . . . 4

2.2 A product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 The double exponential formula . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Monomial expansion of the double exponential formula . . . . . . . . . . 6

3 Diagrammatic expansion of the double exponential formula 9 3.1 Algebra structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Admissible coproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Hopf algebra structures associated with Δ_{W S} and Δ_{BS} 11

4.1 The philosophy of bi- and Hopf algebras thru representation theory … 11 4.2 Bialgebra structures on LDiag and Diag . . . . . . . . . . . . . . . . . . 13

5 Conclusion 14

1. Introduction

In a relatively recent paper Bender, Brody and Meister [3] introduced a special Field Theory described by a product formula (a kind of Hadamard product for two exponential generating functions - EGF) in the purpose of proving that any sequence of numbers could be described by a suitable set of rules applied to some type of Feynman graphs. Inspired by this idea, we have worked out combinatorial consequences of the product and exponential formulas in a recent series of papers [12, 13, 14, 15, 16, 17, 18].

Here, we consider two aspects of the product formula for formal power series applied to combinatorial field theories. Firstly, we remark that the case when the functions involved in the product formula have a constant term (equal to one) is of special interest as often these functions give rise to substitutional groups. The groups arising from the normal ordering problem of boson strings are naturally associated with explicit vector fields, or their conjugates, in the case when there is only one annihilation operator [14, 17]. We also consider one-parameter groups of operators when several annihilators are present. Secondly, we discuss the Feynman-type graph representation resulting from the product formula. We show that there is a correspondence between the packed integer matrices of the theory of noncommutative symmetric functions and the labelled version of these Feynman-type graphs.

We thus obtain a new Hopf algebra structure over the space of matrix quasisymmetric functions that is a natural cocommutative Hopf algebra structure on the space of diagrams themselves which originates from the formal doubling of variables in the product formula.

Aknowledgements We would like here to express our gratitude to Jean-Louis Loday, Jean-Bernard Zuber, Jean-Yves Thibon and Florent Hivert for stimulating interactions on this subject.

2. Single and double exponentials

2.1. One parameter groups and the connected graph theorem

2.1.1. Substitutions The Weyl algebra W is the ℂ-associative algebra (with unit) defined by two generators a and a^{+} and the unique relation [a, a^{+}] = 1. This algebra is of Gelfand-Kirillov dimension 2 and has a basis consisting of the following family {(a^{+})^{k}a^{l}}_{k,l≥0}. It is known that it is impossible to represent faithfully a, a^{+} by bounded operators in a Banach space, but one often uses the representation a ↦ \frac{d}{dx}; a^{+} ↦ x as operators acting “on the line” or, better said, on the space of polynomials ℂ[x]. Through this representation (faithful and coined under the name “Bargmann-Fock”), one sees that we can define a grading on W by the weight function w(a) = −1; w(a^{+}) = 1. A homogeneous operator (under this grading) Ω ∈ W is then of the form

Ω = ∑_{k,l; k−l=e} c(k, l)(a^{+})^{k}a^{l}  (1)

According to whether the excess e is positive or negative, the normal ordering of Ω^{n} reads

N (Ω^{n}) = (a^{+})^{ne} (∑_{k=0}^{∞} S_{Ω}(n, k)(a^{+})^{k}a^{k}) or (∑_{k=0}^{∞} S_{Ω}(n, k)(a^{+})^{k}a^{k}) (a)^{n|e|} (2)

We get combinatorial quantities with two indices i.e. an infinite ℕ × ℕ matrix {S_{Ω}(n, k)}_{n,k≥0} which we will call the generalized Stirling matrix of Ω. In fact, it is easily checked that, if the coefficients c(k, l) of Ω are non-negative integers, so are the entries (S_{Ω}(n, k)) of this matrix.

Let us give some examples of these generalized Stirling matrices. For Ω = a^{+}a, one gets the usual matrix of the Stirling numbers of the second kind

⌈ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 3 & 1 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 7 & 6 & 1 & 0 & 0 & ⋯ \\ 0 & 1 & 15 & 25 & 10 & 1 & 0 & ⋯ \\ 0 & 1 & 31 & 90 & 65 & 15 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix}  (3)

For Ω = a^{+}aa^{+} + a^{+}, we obtain

⌈ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 2 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 6 & 6 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 24 & 36 & 12 & 1 & 0 & 0 & 0 & ⋯ \\ 120 & 240 & 120 & 20 & 1 & 0 & 0 & ⋯ \\ 720 & 1800 & 1200 & 300 & 30 & 1 & 0 & ⋯ \\ 5040 & 15120 & 12600 & 4200 & 630 & 42 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix}  (4)

and for w = a^{+}aaa^{+}a^{+}

⌈ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 2 & 4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 12 & 60 & 54 & 14 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 144 & 1296 & 2232 & 1296 & 306 & 30 & 1 & 0 & 0 & ⋯ \\ 2880 & 40320 & 109440 & 105120 & 45000 & 9504 & 1016 & 52 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix}  (5)

Let Ω = ∑_{k,l≥0} c(k, l)(a^{+})^{k}a^{l} (finite supported sum) be a general term of W in normal form and let us call a dominant term, the sum of monomials with maximum length k + l. It is not difficult to prove that, if Ω is homogeneous, the dominant term consists of a single monomial c(k_{0}, l_{0})(a^{+})^{k_{0}}a^{l_{0}}. Thus, the dominant term of Ω^{n} must be c(k_{0}, l_{0})^{n}(a^{+})^{nk_{0}}a^{nl_{0}}. Then, for example in the case when e = k_{0} − l_{0} ≥ 0, in the generalized Stirling matrix of Ω, the rightmost non-zero coefficient of the line n has address (n, n.l_{0}) and bears the coefficient c(k_{0}, l_{0})^{n}. All these matrices are row-finite and triangular iff l_{0} = 1 (which means that no monomial possesses more than one a).

Remark 2.1 i) There is a beautiful combinatorial expression of the normal form of w^{n} in case w is a string in a and a^{+}. The normal form of w is

N (w) = ∑_{k≥0} r(B, k)(a^{+})^{r−k}a^{s−k}  (6)

where r(B, k) is the kth rook number of a certain board B constructed after w (see [13, 25], and r = |w|_{a^{+}}; s = |w|_{a} are the number of occurences of a^{+} and a in w. ii) To each matrix M ∈ ℂ^{ℕ×ℕ} of this kind and more generally “row finite” matrices (which means that, for each n, the family (M(n, k))_{k∈ℕ} is finite supported), one can associate a transformation of EGFs (see [14, 17]) f ↦ f̂ such that, if f = ∑_{n≥0} a_{n} \frac{z^{n}}{n!} then f̂ = ∑_{n≥0} b_{n} \frac{z^{n}}{n!} (with b_{n} = ∑_{k≥0} M(n, k)a_{k}). iii) It can be shown that, if no monomial of Ω possesses more than one a, the action of the transformation induced by Ω (through the Bargmann-Fock representation) can be expressed in terms of vector fields or their conjugates, thus the one-parameter group e^{λΩ} acts by substitutions and products [14, 17].

2.1.2. Combinatorial matrices and one-parameter groups One can also draw generalized Stirling matrices from another source, namely from the combinatorial graph theory. Let C be a class of graphs such that

Γ ∈ C iff every connected component of Γ is in C  (7)

For these classes of graphs, one has the exponential formula [9, 23, 21] saying roughly that

EGF(all graphs) = e^{EGF(Connected Graphs)}  (8)

This implies, in particular, that the matrix

M(n, k) = number of graphs with n vertices and having k connected components (9)

is the matrix of a substitution (see [14, 17]). One can prove, using a Zariski-like argument (a polynomial vanishing for every integer vanishes everywhere), that, if M is such a matrix (with identity diagonal) then, all its powers (positive, negative and fractional) are substitution matrices and form a one-parameter group of substitutions, thus coming from a vector field on the line which can be computed. But no nice combinatorial principle seems to emerge. For example, beginning with the Stirling substitution z ↦ e^{z} − 1, we know that there is a unique one-parameter group of substitutions s_{λ}(z) such that, for λ integer, one has the value (s_{2}(z) ↔ partition of partitions)

s_{2}(z) = e^{e^{z}−1} − 1 ; s_{3}(z) = e^{e^{e^{z}−1}−1} − 1 ; s_{−1}(z) = ln(1 + z)  (10)

but we have no nice description of this group nor of the vector field generating it.

2.2. A product formula

The Hadamard product of two sequences (a_{n})_{n≥0}; (b_{n})_{n≥0} is given by the pointwise product (a_{n}b_{n})_{n≥0}. We can at once transfer this law on EGFs by

( ∑_{n≥0} a_{n} \frac{x^{n}}{n!} ) ⊙_{exp} ( ∑_{n≥0} b_{n} \frac{x^{n}}{n!} ) := ∑_{n≥0} a_{n}b_{n} \frac{x^{n}}{n!}  (11)

In the following, we will omit the subscript (in ⊙_{exp}) as this will be the only kind of Hadamard product under consideration.

But, it is not difficult to check that the family

(\frac{(y \frac{∂}{∂x})^{n}}{n!} \frac{x^{m}}{m!} )_{n,m∈ℕ}  (12)

is summable in ℂ[[x, y]] (the space of formal power series in x and y) as we have

\frac{(y \frac{∂}{∂x})^{n}}{n!} \frac{x^{m}}{m!} = \{ \begin{matrix}0 & if n > m \\ \frac{y^{n}x^{m−n}}{n!(m−n)!} & otherwise \end{matrix}  (13)

and therefore, for F (x) = ∑_{n≥0} a_{n} \frac{x^{n}}{n!} and G(x) = ∑_{n≥0} b_{n} \frac{x^{n}}{n!} one gets the product formula

(F ⊙ G)(x) := F (y \frac{∂}{∂x})G(x)|_{x=0} = ∑_{n≥0} a_{n}b_{n} \frac{y^{n}}{n!}  (14)

With this product, the set of series forms a commutative associative algebra with unit, which is actually the product algebra ℂ^{ℕ}.

2.3. The double exponential formula

The case F (0) = G(0) = 1 will be of special interest in our study. Every series with constant term 1 can be represented by an exponential exp(∑_{n≥1} L_{n} \frac{x^{n}}{n!} ) which can be expanded using Bell polynomials and Faà di Bruno coefficients. Let us now recall some facts about these combinatorial notions. We still consider the alphabet 𝕃 = {L_{1}, L_{2}, ⋯ } = {L_{i}}_{i≥1}, then the complete Bell polynomials [7] are defined by

exp(∑_{m≥1} L_{m} \frac{x^{m}}{m!} ) = ∑_{n≥0} Y_{n}(𝕃)\frac{x^{n}}{n!}  (15)

We will denote alternatively Y_{n}(L_{1}, ⋯ L_{n}) for Y_{n}(𝕃) as this polynomial is independent from the subalphabet (L_{m})_{m>n}. We know [7] that

Y_{n}(𝕃) = Y_{n}(L_{1}, ⋯ L_{n}) = ∑_{||α||=n}((α))𝕃^{α} = ∑_{||α||=n}((α))L_{1}^{α_{1}}L_{2}^{α_{2}} ⋯ L_{n}^{α_{n}}  (16)

where α = (α_{1}, α_{2}, ⋯ α_{n}) is an integral vector, ||α|| := ∑_{j=1}^{m} jα_{j}, 𝕃^{α} = L_{1}^{α_{1}}L_{2}^{α_{2}} ⋯ L_{n}^{α_{n}} is the multiindex standard notation and

((α)) = \frac{||α||!}{(1!)^{α_{1}}(2!)^{α_{2}} ⋯ (n!)^{α_{n}}(α_{1})! ⋯ (α_{n})!}  (17)

is the Faà di Bruno coefficient [8, 20] which will be interpreted, in the next section, as enumerating structures called set partitions. Combining (15) and (16) one gets

exp( ∑_{m≥1} L_{m} \frac{(y \frac{∂}{∂x})^{m}}{m!} )exp( ∑_{n≥1} V_{n} \frac{x^{n}}{n!} )|_{x=0} = ∑_{k≥0} \frac{y^{k}}{k!} ( ∑_{||α||=||β||}((α))((β))𝕃^{α}𝕍^{β}) (18)

Formula (18) will be called in the sequel the double exponential formula.

2.4. Monomial expansion of the double exponential formula

In this paragraph, we will use unordered and ordered set partitions. By an unordered partition P of the set X we mean a finite subset P ⊂ (𝔓(X) − {∅}) (𝔓(X) is the set of all subsets of X [7]) such that

⋃_{Y ∈P} Y = X and (Y_{1}, Y_{2} ∈ P, Y_{1} ≠ Y_{2} ⟹ Y_{1} ∩ Y_{2} = ∅)  (19)

this explains why without any convention the classical Stirling number of second the kind S(0, 0) equals 1. The elements of P are called blocks. Following Comtet ([8] p 39), we will say that a partition P is of type α = (α_{1}, α_{2}, ⋯, α_{m}) iff there is no j-block for j > m and α_{j} j-block(s) for each j ≤ m. This implies in particular that the set X is of cardinality ||α|| := ∑_{j=1}^{m} jα_{j}.

Here one can see easily that the number of blocks of a partition of type α is |α| = ∑_{j=1}^{m} α_{j}. An ordered partition of type α of the set X is just a partition in which the blocks are labelled from 1 to |α|. In other words, one could say that an ordered partition is a list of subsets and an unordered partition is a set of subsets. To every ordered partition P = (B_{1}, B_{2}, ⋯, B_{|α|}) corresponds an unordered one Φ_{p}(P) = {B_{1}, B_{2}, ⋯, B_{|α|}} where Φ_{p} is the “forgetful” function which forgets the order. Now to a pair (P^{(1)}, P^{(2)}) of ordered partitions of the same set (call it X)

P^{(1)} = (B_{1}^{(1)}, B_{2}^{(1)}, ⋯, B_{k_{1}}^{(1)})  P^{(2)} = (B_{1}^{(2)}, B_{2}^{(2)}, ⋯, B_{k_{2}}^{(2)})  (20)

one can associate the intersection matrix IM_{o}(P^{(1)}, P^{(2)}) such that the entry of address (i, j) is the number of elements of the intersection of the block i of the first partition and the block j of the second. For example with partitions of X = {1, 2, 3, 4, 5, 6}, and specifying

P^{(1)} = ({1, 2, 5}, {3, 4, 6}) P^{(2)} = ({1, 2}, {3, 4}, {5, 6}),

one gets

{1, 2} {3, 4} {5, 6}
{1, 2, 5} 2  0  1
{3, 4, 6} 0  2  1

and hence the matrix

(\begin{matrix}2 & 0 & 1 \\ 0 & 2 & 1 \end{matrix})  (21)

Formally, IM_{o}(P^{(1)}, P^{(2)}) is the matrix of size k_{1} × k_{2} such that

IM_{o}(P^{(1)}, P^{(2)})[i, j] = card(B_{i}^{(1)} ∩ B_{j}^{(2)})  (22)

The matrices obtained in such a way form the set of packed matrices defined in [19] as, indeed, one sees that every packed matrix can be obtained through the matching procedure illustrated above. If we consider now a pair of unordered partitions (Q^{(1)}, Q^{(2)}), we cannot associate to them a single matrix but rather a class of matrices obtained from the preimages of (Q^{(1)}, Q^{(2)}) under Φ_{p} × Φ_{p}. In a compact formulation, the set of matrices so obtained is

{IM_{0}(P^{(1)}, P^{(2)})}_{Φ_{p}(P^{(1)})=Q^{(1)}; Φ_{p}(P^{(2)})=Q^{(2)}}  (23)

For example, with

(Q^{(1)}, Q^{(2)}) = ( {{1, 2, 5}, {3, 4, 6}} , {{1, 2}, {3, 4}, {5, 6}} )  (24)

one gets the 12 preimages (P_{i}^{(1)}, P_{j}^{(2)}), where P_{i}^{(1)} are among the two preimages of Q^{(1)} and P_{j}^{(2)} are among the 6 preimages of Q^{(2)}. Explicitely

P_{1}^{(1)} = ({1, 2, 5}, {3, 4, 6})P_{2}^{(1)} = ({3, 4, 6}, {1, 2, 5})

are the preimages of Q^{(1)} and

P_{1}^{(2)} = ({1, 2}, {3, 4}, {5, 6})P_{2}^{(2)} = ({1, 2}, {5, 6}, {3, 4})
P_{3}^{(2)} = ({3, 4}, {1, 2}, {5, 6})P_{4}^{(2)} = ({3, 4}, {5, 6}, {1, 2})
P_{5}^{(2)} = ({5, 6}, {1, 2}, {3, 4})P_{6}^{(2)} = ({5, 6}, {3, 4}, {1, 2})

are the preimages of Q^{(2)}. The set of matrices so obtained reads

IM_{u}(M) = \{ (\begin{matrix}2 & 0 & 1 \\ 0 & 2 & 1 \end{matrix}) , (\begin{matrix}2 & 1 & 0 \\ 0 & 1 & 2 \end{matrix}) , (\begin{matrix}1 & 2 & 0 \\ 1 & 0 & 2 \end{matrix}) ,
(\begin{matrix}0 & 2 & 1 \\ 2 & 0 & 1 \end{matrix}) , (\begin{matrix}0 & 1 & 2 \\ 2 & 1 & 0 \end{matrix}) , (\begin{matrix}1 & 0 & 2 \\ 1 & 2 & 0 \end{matrix}) \}.  (25)

This is the orbit of one of them under permutation of lines and columns. The correspondence which, to a pair of unordered partitions, associates a class of matrices (under permutations of lines and columns) will be denoted IM_{u}.

Thus, one gets a commutative diagram of mappings

Pairs of ordered partitions ⟶^{Φ_{p}×Φ_{p}} Pairs of unordered partitions
IM_{o}↓  ↓^{IM_{u}}
Packed matrices  ⟶^{Class} Classes of packed matrices
Dg_{o}↓  ↓^{Dg_{u}}
Labelled diagrams  ⟶^{Φ_{d}}  Diagrams

(26)

The scheme presented above shows how to associate to a pair of ordered (resp. unordered) set partitions, a packed matrix (resp. a class of packed matrices). The packed matrices can be alternatively represented by labelled diagrams which are bipartite multigraphs built from two sets of vertices being a column of white spots (WS) and column of black spots (BS) as shown below.

[figure]
Labelled diagram of the matrix (\begin{matrix}2 & 0 & 1 \\ 0 & 2 & 1 \end{matrix})

Let us explain how to associate to a (drawn) diagram a packed matrix. The white (resp. black) spots are labelled from 1 to r (resp. 1 to c) from top to bottom and the number of lines from the i-th white spot to the j-th black spot is exactly the entry a_{ij} of the matrix. Conversely, a packed matrix of dimension r × c being given, one draws r white spots (resp. c black spots) and (with the labelling as above) join the i-th white spot to the j-th black by a_{ij} lines. This gives exactly the one-to-one correspondence between (drawn) diagrams and packed matrices.

In the sequel, we set Diag_{u} := Dg_{u} ∘ IM_{u} and Diag_{o} := Dg_{o} ∘ IM_{o} for the mappings which associate diagrams to pairs of partitions. Now, the multiplicity of a diagram D is the number of pairs (P^{(1)}, P^{(2)}) of unordered partitions such that Dg_{u}(IM_{u}(P^{(1)}, P^{(2)})) = D. Let us call bitype of a diagram D the pair (α(P^{(1)}, α(P^{(2)}) where Dg_{u}(IM_{u}(P^{(1)}, P^{(2)})) = D (remark that it does not depend on the choosen premiage inside the formula) and we will refer it as the bitype (α(D), β(D))) of D. In a similar way α(D) (resp. β(D)) will be called the left (resp. the right) type of D. The product formula now reads

exp( ∑_{m≥1} \frac{L_{m}}{m!} (y \frac{∂}{∂x})^{m} )exp( ∑_{n≥1} \frac{V_{n}}{n!} x^{n})|_{x=0} =

∑_{n≥0} \frac{y^{n}}{n!} ( ∑_{{D diagram \atop |D|=n}} mult(D)𝕃^{α(D)}𝕍^{β(D)})= ∑_{D diagram}\frac{mult(D)}{|D|!}m(D, 𝕃, 𝕍, y) (27)

with

m(D, 𝕃, 𝕍, y) := 𝕃^{α(D)}𝕍^{β(D)}y^{|D|}.  (28)

3. Diagrammatic expansion of the double exponential formula

The main interest of the expansion (27) is that we can impose (at least) two types of rules on the diagrams

• on the diagrams themselves (selection rules) : on the outgoing degrees, ingoing degrees, total or partial weights (the graph is supposed oriented from white to black spots)

• on the set of diagrams (composition and decomposition rules): product and coproduct on the space of diagrams.

We have already such a structure on the space of monomials (i.e. the polynomials). The (usual) product of polynomials is well known and amounts to the addition of the multidegrees. The (usual) coproduct is given by the substitution of a “doubled” variable to each variable [4, 20]. For example, with P = x^{2}y^{3}, we first form (x_{1} + x_{2})^{2}(y_{1} + y_{2})^{3}, expand and then separate (on the left) the “1” labelled variables and (on the right) the “2” labelled. As

P = x_{1}^{2}y_{1}^{3} + 3x_{1}^{2}y_{1}^{2}y_{2} + 3x_{1}^{2}y_{1}y_{2}^{2} + x_{1}^{2}y_{2}^{3} + 2x_{1}y_{1}^{3}x_{2} + 6x_{1}y_{1}^{2}x_{2}y_{2} +
6x_{1}y_{1}x_{2}y_{2}^{2} + 2x_{1}x_{2}y_{2}^{3} + y_{1}^{3}x_{2}^{2} + 3y_{1}^{2}x_{2}^{2}y_{2} + 3y_{1}x_{2}^{2}y_{2}^{2} + x_{2}^{2}y_{2}^{3}  (29)

one gets, with Δ the coproduct operator,

Δ(P) = x^{2}y^{3} ⊗ 1 + 3x^{2}y^{2} ⊗ y + 3x^{2}y ⊗ y^{2} + x^{2} ⊗ y^{3}+
2xy^{3} ⊗ x + 6xy^{2} ⊗ xy + 6xy ⊗ xy^{2} + 2x ⊗ xy^{3}+
y^{3} ⊗ x^{2} + 3y^{2} ⊗ x^{2}y + 3y ⊗ x^{2}y^{2} + 1 ⊗ x^{2}y^{3}.  (30)

The space of polynomials with product and coproduct (and other items like neutrals, co-neutrals and antipode, which will be made more precise in the next paragraph) is endowed with the structure of a Hopf algebra. The last consideration suggests the following question:

Is it possible to structure the (spaces of) diagrams into a Hopf algebra ? Is it possible that this structure be compatible, in some sense, with the mapping (D, 𝕃, 𝕍, y) ↦ m(D, 𝕃, 𝕍, y) ?

Answer is yes. To establish it, we have to proceed in three steps.

• First Step : Define the space(s)

• Second Step : Define a product

• Third Step : Define a coproduct

3.1. Algebra structure

First step. — Let Diag_{ℂ} (resp. LDiag_{ℂ}) be the ℂ-vector space freely generated by the diagrams (resp. labelled diagrams) i.e.

Diag_{ℂ} := ⨁_{d diagram} ℂ  LDiag_{ℂ} := ⨁_{d labelled diagram} ℂd  (31)

at this stage, we have a linear mapping (linear arrow) LDiag_{ℂ} ↦ Diag_{ℂ} provided by the linear extension of Φ_{d} and an arrow (linear, by construction) m(., 𝕃, 𝕍, z): Diag_{ℂ} ↦ ℂ[𝕃 ∪ 𝕍 ∪ {z}] provided by the linear extension of m(., 𝕃, 𝕍, z).

Second step. — We remark that, if

d_{1} ⋆ d_{2} = \begin{matrix}d_{1} \\ d_{2} \end{matrix}  (32)

denotes the superposition of the diagrams, then

m(d_{1} ⋆ d_{2}, 𝕃, 𝕍, z) = m(d_{1}, 𝕃, 𝕍, z)m(d_{2}, 𝕃, 𝕍, z).  (33)

The law (32) makes sense as well for labelled and unlabelled diagrams. In the first case, it amounts to computing the blockdiagonal product of packed matrices. Indeed, for M_{1}, M_{2} being packed matrices, one has

Dg_{o} ((\begin{matrix}M_{1} & 0 \\ 0 & M_{2} \end{matrix})) = Dg_{o}(M_{1}) ⋆ Dg_{o}(M_{2}).  (34)

This product yields the product of monomials in the following way. From D a diagram and all the other parameters fixed, with the setting of (28), we get a polynomial.

The product (32) is associative with unit (the empty diagram), it is compatible with the arrow Φ_{d} and so defines the product on Diag which, in turn is compatible with the product of monomials.

Labelled diagrams^{2} ⟶^{Φ_{d}×Φ_{d}} Diagrams^{2} ⟶^{m(?,𝕃,𝕍,z)×m(?,𝕃,𝕍,z)} Monomials^{2}
product↓  ^{product}↓  ^{product}↓
Labelled diagrams ⟶^{Φ_{d}} Diagrams  ⟶^{m(?,𝕃,𝕍,z)}  Monomials

(35)

Remark 3.1 One sees easily that the labelled diagram (resp. diagrams) form monoids thus the spaces LDiag_{ℂ} and Diag_{ℂ} are algebras of these monoids [2, 1].

3.2. Admissible coproducts

For the coproduct on LDiag, we have several possibilities:

(i) split with respect to the white spots (two ways : by intervals and by subsets) (ii) split with respect to the black spots (two ways : by intervals and by subsets) (iii) split with respect to the edges

The discussion goes as follows:

i) (3) does not give a nice identity with the monomials (when applying d ↦ m(d,?,?,?)) nor do (2) and (3) by intervals. ii) (2) and (3) are essentially the same (because of the WS ↔ BS symmetry). In fact (2) and (3) by subsets give a good representation and, moreover, they are appropriate for several physics models.

In the next section, we develop the possibility (1) and (2) by subsets.

4. Hopf algebra structures associated with Δ_{W S} and Δ_{BS}

4.1. The philosophy of bi- and Hopf algebras thru representation theory

Let A is a k-algebra (k is a field as ℝ or ℂ). In this paragraph, we consider associative algebras with unit (AAU). A representation of A is here a pair (V, ρ_{V}) where V is a k-vector space and ρ_{V} : A ↦ End_{k}(V) a morphism of k-algebras (AAU). One can make operations with representations as direct sums and quotient of a representation by a sub-representation (a sub-representation is a subspace which is closed under the action of A). In general, one does not know how to endow the tensor product (of two representations) and the dual (of a representation) with the structure of representation. It is however classical in two cases: groups and Lie algebras. If G is a group, a representation of G is a pair (V, ρ_{V}) where V is a k-vector space and ρ_{V} : G ↦ Aut_{k}(V) a morphism of groups. If G is a Lie algebra, a representation of G is a pair (V, ρ_{V}) where V is a k-vector space and ρ_{V} : G ↦ End_{k}(V) a morphism of Lie algebras (i.e. ρ_{V} ([u, v]) = ρ_{V} (u)ρ_{V} (v) − ρ_{V} (v)ρ_{V} (u)). These two cases enter the scheme of (AAU) as a representation of a group can be extended uniquely as a representation of its algebra kG and a representation of a Lie algebra as a representation of U_{k}(G), its envelopping algebra. These two constructions (kG and U_{k}(G)) are (AAU). For the sake of readibility let us denote in all cases ρ_{V} (g)(u) by g.u (g ∈ G and u ∈ V). If G is a group and V, W two representations, we construct a representation of G on V ⊗ W by

g.(u ⊗ v) = g.u ⊗ g.v  (36)

If G is a Lie algebra and V, W two representations, we construct a representation of G on V ⊗ W by

g.(u ⊗ v) = g.u ⊗ v + u ⊗ g.v  (37)

This can be rephrased in saying that the action of g in the first case (group) is g ⊗ g and in the second (Lie algebra) g ⊗ 1 + 1 ⊗ g (1 is here for the appropriate identity mapping). In the two cases, it amounts to give a linear mapping Δ : A ↦ A ⊗ A which will be called a coproduct. One can show [6] that, if we want that this new operation enjoy “nice” properties (associativity of the tensor product etc...), one has to suppose that this coproduct is a morphism of (AAU) (A ⊗ A has received the structure of - non twisted - tensor product of algebras), that it is coassociative with a counit [6]. Let us make these requirements more precise. The first says that for all x, y ∈ A one has Δ(xy) = Δ(x)Δ(y), the second that the two compositions

A ⟶^{Δ} A ⊗ A ⟶^{Δ⊗1_{A}} A ⊗ A ⊗ A and A ⟶^{Δ} A ⊗ A ⟶^{1_{A}⊗Δ} A ⊗ A ⊗ A  (38)

are equal, the third says that there is a mapping (linear form) ε : A ↦ k such that the compositions

A ⟶^{Δ} A ⊗ A ⟶^{Δ⊗ε} A ⊗ k ⟶^{nat} A and A ⟶^{Δ} A ⊗ A ⟶^{ε⊗Δ} k ⊗ A ⟶^{nat} A  (39)

(where nat is for the natural mappings) are equal to the identity 1_{A}. An algebra (AAU) together with a coproduct Δ and a counity ε which fulfills the three requirements above is called a bialgebra.

If, moreover one wants to have a nice dualization of the representations (i.e. nice structures for the duals V^{∗} = Hom(V, k)), it should exist an element of Hom(A, A) such that the compositions

A ⟶^{Δ} A ⊗ A ⟶^{α⊗1_{A}} A ⊗ A ⟶^{μ} A and A ⟶^{Δ} A ⊗ A ⟶^{1_{A}⊗α} A ⊗ A ⟶^{μ} A  (40)

are equal to e_{A}ε (where e_{A} denotes the unit of A). When a bialgebra possesses such an element (unique), it is called the antipode and the bialgebra a Hopf algebra. For more details and connections to physics, one can consult [6].

One can prove that the bialgebras constructed below possess an antipode and then are Hopf algebras.

4.2. Bialgebra structures on LDiag and Diag

The space spanned by the packed matrices has already received a structure of Hopf algebra, the algebra MQSym [19]. We briefly review the structure of this Hopf algebra. We describe in details Δ_{W S} as the other coproduct is actually got by the same process but applied on the columns instead of the lines. Let M be a packed matrix of dimensions k_{1}×k_{2} for every subset X ∈ [1..k_{1}] we consider the matrix π_{X}(M) := pack(M[X, [1..k_{2}]]), the restriction to the lines of X and then packed (with this restriction to the lines, we only need to perform a horizontal packing). Thus, the coproduct Δ_{W S} reads

Δ_{W S}(M) = ∑_{X+Y =[1..k_{1}]} π_{X}(M) ⊗ π_{Y} (M)  (41)

To avoid confusion we will call the supporting space H_{W S} (= MQSym). We keep the (total) grading of MQSym by the total weight (i.e. the sum of the coefficients) of the matrices. The packed matrices are a linear basis of H_{W S} = MQSym, thus every element expresses uniquely

x = ∑_{M packed} λ_{M}(x)M  (42)

The coproduct above is cocommutative and with counit λ_{∅} where ∅ is the void matrix corresponding to the void diagram. This particular matrix will be denoted 1_{H_{W S}}. For example, with the packed matrix above one has

Δ_{W S}((\begin{matrix}2 & 0 \\ 0 & 2 \\ 1 & 1 \end{matrix})) = (\begin{matrix}2 & 0 \\ 0 & 2 \\ 1 & 1 \end{matrix}) ⊗ 1_{H_{W S}} + (2) ⊗ (\begin{matrix}0 & 2 \\ 1 & 1 \end{matrix}) + (2) ⊗ (\begin{matrix}2 & 0 \\ 1 & 1 \end{matrix}) +

(1 1) ⊗ (\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix}) + (\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix}) ⊗ (1 1) +

(\begin{matrix}2 & 0 \\ 1 & 1 \end{matrix}) ⊗ (2) + (\begin{matrix}0 & 2 \\ 1 & 1 \end{matrix}) ⊗ (2) + 1_{H_{W S}} ⊗ (\begin{matrix}2 & 0 \\ 0 & 2 \\ 1 & 1 \end{matrix})

This coproduct is compatible with the usual coproduct on the monomials for the constant alphabet 𝕍 = 1_{ℕ} defined by V_{n} = 1 for all n ≥ 0. Then, using Sweedler’s notation, for this particular 𝕍, if Δ_{W S}(d) = ∑ d_{(1)} ⊗ d_{(2)}, one has

m(d, 𝕃′ + 𝕃′′, 1_{ℕ}, z) = ∑ m(d_{(1)}, 𝕃′, 1_{ℕ}, z)m(d_{(2)}, 𝕃′′, 1_{ℕ}, z)  (43)

Thus, one sees easily that, with this structure (product with unit, coproduct and the counit), LDiag_{ℂ} is a bialgebra graded in finite dimensions and then a Hopf algebra. The arrow LDiag_{ℂ} ↦ Diag_{ℂ} endows Diag_{ℂ} with a structure of Hopf algebra.

CONTENTS 14

5. Conclusion

The structure of the Hopf algebras LDiag_{ℂ}, Diag_{ℂ}, by a theorem of Cartier, Milnor and Moore [5, 22], is that of envelopping algebras of their primitive elements (Diag_{ℂ}, being commutative, is thus an algebra of polynomials). Moreover, it appears that the structure described above is the starting point for a series of connections with mathematical and physical Hopf algebras. The coproduct Δ_{BS} is the cristallisation (q = 1) of a one-parameter deformation of coproducts (all coassociative) on LDiag_{ℂ} ≃ MQSym, the other end (q = 0) being an infinitesimal coproduct isomorphic to Δ_{MQSym}. Recently, FQSym (a subalgebra of MQSym) has been established by Foissy [10] as a case in a family of Hopf algebras of decorated planar trees which is strongly related to other Hopf algebras like Connes-Kreimer’s and Connes-Moscovici’s [10, 11].

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