The Hopf Algebra of Heisenberg-Weyl Diagrams

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

a LIPN - UMR 7030 CNRS - Université Paris 13 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,

31-12-2008 15:47

Abstract. It has been found by many authors that the structure constants of the Heisenberg-Weyl algebra (HW) could be computed through rook placement numbers. We here give a diagrammatic realization of this formula in terms of graphs which constitute an algebra. It is also possible to keep track of the number cancellations in Wick’s formula and this provides at once a q-analogue B̃_{q} of the previous algebra. It tuns out that the latter can be very naturally endowed with the structure of a Hopf algebra. Returning to HW, it is well known that it has a 3-dimensional L_{HW} Lie subalgebra which generates (as an algebra) HW. We construct a Hopf arrow between the enveloping algebra U(L_{HW}) and B̃_{q}. It is there, in our sense, that resides the obstruction of HW to have finite dimensional non-zero representation.

Contents

1 Introduction 3

Partially supported by ??.

2 Construction of the diagram algebras B̃ and B̃_{q}. 3 2.1 Heisenberg-Weyl graphs and their matching . . . . . . . . . . . . . . . . 3 2.2 Multiplication of graphs and structure of the algebra B̃. . . . . . . . . . . 5 2.3 Deformed case, the algebra B̃_{q}. . . . . . . . . . . . . . . . . . . . . . . . 6

3 Ado’s theorem, the Lie algebra L_{HW} and its enveloping algebra. TODO : This paragraph will be put as a remark. Replace it by physical motivations for central elements. 6

4 The arrow φ_{2} : B̃ → U(L_{HW}). 7

5 Diagram of algebras 8

6 Comultiplications and Hopf Algebra structures. 8 6.1 Comultiplications and section coefficients. . . . . . . . . . . . . . . . . . 8 6.2 Comultiplication and Hopf Algebra structure on U(L_{HW}). . . . . . . . . 10 6.3 Partitions of a HW -graph and the comultplication. . . . . . . . . . . . . 11

7 Endnotes 12 1. Introduction

Since the advent of Hopf algebras in Combinatorial Physics, when facing an algebra, one always asks oneself whether its structure could be (more or less naturally) be enriched to a Hopf algebra one. For one of the simplest in Quantum Physics, the Heisenberg-Weyl (HW) algebra (with complex coefficients), the answer is “no” since the counit provides one with a one-dimensional representation HW and it is well-known (and easily checked) that this algebra has no non-zero finite dimensional representation. The paper is devoted ...

B̃_{φ_{2}}  _{φ_{3}} U(L_{HW})  _{φ_{1}}  HW

2. Construction of the diagram algebras B̃ and B̃_{q}.

2.1. Heisenberg-Weyl graphs and their matching

As usual in graph theory, a graph Γ is given by the data of two sets Γ^{(1)}, the edges, Γ^{(0)}, the vertices and two functions (head and tail) h, t : Γ^{(1)} → Γ^{(0)}. We are intesrested here by graphs with some edges having a “‘free tail” or a “free head” but not both, so we allow the functions h, t to be partially defined with the condition that

dom(t) ∪ dom(h) = Γ^{(1)} (1)

(every edge has at least a tail or a head, maybe both). The edges a with no origin i. e. such that a ∉ dom(t) (resp. with no endpoint i. e. such that a ∉ dom(h)) will be called “ingoing edges” and denoted Γ^{−} (resp. “outgoing edges”, denoted Γ^{+}). A (proper) cycle is a sequence of edges a_{1}a_{2} ⋯ a_{n} (n ≥ 2) such that, for all k < n, h(a_{k}) = t(a_{k+1}) and h(a_{n}) = t(a_{1}). We will call HW -graph a finite graph with (possibly) ingoing and outgoing edges and no cycle.

Figure TODO : Heisenberg-Weyl Graphs

One can then see that a HW -graph has a natural partition in three parts :

• the outgoing edges dom(t)−dom(h) and its associated subgraph Γ^{+} (with restriction of the tail mapping and void head mapping)

• the ingoing edges dom(h)−dom(t) and its associated subgraph Γ^{−} (with restriction of the head mapping and void tail mapping)

• the inner edges dom(t) ∩ dom(h) and its associated subgraph Γ^{(in)} (with restriction of the tail and head mappings everywhere defined)

Now, in order to define algebra structures on these graphs, one has to define what is a en two HW -graphs. In general set theory a matching or one-to-one correspondence is a set of pairs

m ⊂ pr_{1}(m) × pr_{2}(m) (2)

such that two different pairs have different components i. e. for c_{1}, c_{2} ∈ m with c_{i} = (x_{i}, y_{i}) one has

c_{1} ≠ c_{2} ⟹ (x_{1} ≠ x_{2}) and (y_{1} ≠ y_{2}) . (3)

If U, V are finite sets |U| = p, |V | = q the number of matchings m ⊂ U ×V with |m| = i is

( \begin{matrix}p \\ i \end{matrix} ) ( \begin{matrix}q \\ i \end{matrix} ) i! . (4)

This formula will do the job when we construct the arrow φ_{2}. For two HW -graphs Γ_{2}, Γ_{1} with disjoint sets of vertices (i. e. Γ_{2}^{(0)} ∩ Γ_{1}^{(0)} = ∅), we will call matching of Γ_{1} over Γ_{2} any matching between (subsets of) Γ_{2}^{+} and Γ_{1}^{−}, their set will be denoted match(Γ_{2}, Γ_{1}). Any matching of match(Γ_{2}, Γ_{1}) allows the “pluging” of Γ_{2} into Γ_{1} as follows

Figure TODO : a matching between two Heisenberg-Weyl Graphs

The result will be denoted

( \begin{matrix}Γ_{1} \\ m \\ Γ_{2} \end{matrix} ) .

Formally speaking, we need two disjoints graphs in the sense of the remark that condition (1) implies that,

Γ_{2}^{(0)} ∩ Γ_{1}^{(0)} = ∅ ⟹ Γ_{2}^{(1)} ∩ Γ_{1}^{(1)} = ∅ (5)

then

( \begin{matrix}Γ_{1} \\ m \\ Γ_{2} \end{matrix} )^{(1)} = Γ_{1}^{(1)} ∪ Γ_{2}^{(1)} ∪ m − (pr_{1}(m) ∪ pr_{2}(m)) . (6)

Tail and head functions defined accordingly because there is no ambiguity to define them on Γ_{1}^{(1)} ∪ Γ_{2}^{(1)} − (pr_{1}(m) ∪ pr_{2}(m)) and we define tail and head functions on m (a subset, remind it, of Γ_{2}^{+} × Γ_{1}^{−}) by

t(a_{2}, a_{1}) := t(a_{2}) ; h(a_{2}, a_{1}) := h(a_{1}) where (a_{2}, a_{1}) ∈ m . (7)

We have a sort of associativity of the matchings which will be useful later. Let Γ_{i}, i = 1, 2, 3 be three disjoint HW -graphs. Let

m′ ∈ match(Γ_{2}, Γ_{1}) and m′′ ∈ match(Γ_{3}, ( \begin{matrix}Γ_{1} \\ m′ \\ Γ_{2} \end{matrix} ))

then m′′ can be partitioned in m′′_{3,2} = m′′ ∩ (Γ_{3}^{+} × Γ_{2}^{−}) and m′′_{3,1} = m′′ ∩ (Γ_{3}^{+} × Γ_{1}^{−}) and one has

( \begin{matrix}( & \begin{matrix}Γ_{1} \\ m′ \\ Γ_{2} \end{matrix} & ) \\ & m′′ \\ & Γ_{3} \end{matrix} ) = ( \begin{matrix}Γ_{1} \\ m′ ∪ m′′_{31} \\ ( & \begin{matrix}Γ_{2} \\ m′′_{32} \\ Γ_{3} \end{matrix} & ) \end{matrix} ) (8)

2.2. Multiplication of graphs and structure of the algebra B̃.

Following the general conventions in graph theory, an isomorphism of graphs between Γ_{1} = (Γ_{1}^{(1)}, Γ_{1}^{(0)}, t_{1}, h_{1}) and Γ_{2} = (Γ_{2}^{(1)}, Γ_{2}^{(0)}, t_{2}, h_{2}) is the data of two one-to-one mappings α^{(1)} : Γ_{1}^{(1)} → Γ_{2}^{(1)} and α^{(0)} : Γ_{1}^{(0)} → Γ_{2}^{(0)} such that the following commutative diagrams of partially defined functions hold

Γ_{1}^{(1)} ^{t_{1}}_{α^{(1)}} Γ_{1}^{(0)}_{α^{(0)}} Γ_{2}^{(1)} _{t_{2}} Γ_{2}^{(0)}  and  Γ_{1}^{(1)} ^{h_{1}}_{α^{(1)}}  Γ_{1}^{(0)}_{α^{(0)}} Γ_{2}^{(1)} _{h_{2}} Γ_{2}^{(0)} .  (9)

It is straightforward to check that the class of HW -graphs is closed under isomorphisms. We will call B, the set (it can be shown to be denumerable) of classes of HW -graphs “up to isomorphisms”. The algebra B̃ is the vector space ℂB. The multiplication table in B̃ is given by

Γ̃_{1} ∗ Γ̃_{2} = ∑_{m∈match(Γ_{2},Γ_{1})} \begin{matrix} \\ ( & \begin{matrix}Γ_{1} \\ m \\ Γ_{2} \end{matrix} & ) \end{matrix}  (10)

where Γ_{1}, Γ_{2} are representatives of the classes Γ̃_{1}, Γ̃_{2} with disjoint set of vertices (i. e. Γ_{1}^{(0)} ∩ Γ_{2}^{(0)} = ∅). It can be checked easily that the result is independent of the choice of disjoint representatives. It is straightforward that, for all Γ, Γ̃ ∗ ˜∅ = ˜∅ ∗ Γ̃ = Γ̃. Hence ˜∅ (which, in fact is {∅}) is neutral for the law and will be denoted here 1B̃_{q}. Let us prove now that the law is associative. To this end, take three disjoint HW -graphs Γ_{i}, i = 1, 2, 3 (in this way Γ_{3} will be disjoint of any matching between Γ_{1} and Γ_{2} and, likewise, Γ_{1} is disjoint of any matching between Γ_{2} and Γ_{3}). Then one has, using the denotations of formula (8)

(Γ̃_{1} ∗ Γ̃_{2}) ∗ Γ̃_{3} = ∑_{m′∈match(Γ_{2},Γ_{1})} ∑_{m′′∈match(Γ_{3},( \begin{matrix}Γ_{1} \\ m′ \\ Γ_{2} \end{matrix} ))} ( \begin{matrix}( & \begin{matrix}Γ_{1} \\ m′ \\ Γ_{2} \end{matrix} & ) \\ & m′′ \\ & Γ_{3} \end{matrix} ) =

∑_{m′∈match(Γ_{2},Γ_{1})} ∑_{m′′∈match(Γ_{3},( \begin{matrix}Γ_{1} \\ m′ \\ Γ_{2} \end{matrix} ))} ( \begin{matrix}Γ_{1} \\ m′ ∪ m′′_{31} \\ ( & \begin{matrix}Γ_{2} \\ m′′_{32} \\ Γ_{3} \end{matrix} & ) \end{matrix} ) =

∑_{m′∈match(Γ_{2},Γ_{1})} ∑_{m′′∈match(Γ_{3},( \begin{matrix}Γ_{1} \\ m′ \\ Γ_{2} \end{matrix} ))} ( \begin{matrix}Γ_{1} \\ m′ ∪ m′′_{31} \\ ( & \begin{matrix}Γ_{2} \\ m′′_{32} \\ Γ_{3} \end{matrix} & ) \end{matrix} )  (11)

the result comes from the fact that the correspondence (m′, m′′) ↦ (m′ ∪ m′′_{31}, m′′_{32}) between pairs of matchings in match(Γ_{2}, Γ_{1}) × match(Γ_{3}, ( \begin{matrix}Γ_{1} \\ m′ \\ Γ_{2} \end{matrix} )) and pairs

in match(( \begin{matrix}Γ_{2} \\ m′′_{32} \\ Γ_{3} \end{matrix} ), Γ_{1}) × match(Γ_{3}, Γ_{2}) is one-to-one. Then the sum amounts to

∑_{m′′∈match(Γ_{3},Γ_{2})} ∑_{m′∈match(( \begin{matrix}Γ_{2} \\ m′′ \\ Γ_{3} \end{matrix} ),Γ_{1})} ( \begin{matrix}& Γ_{1} \\ & m′ \\ ( & \begin{matrix}Γ_{2} \\ m′′ \\ Γ_{3} \end{matrix} & ) \end{matrix} ) = Γ̃_{1} ∗ (Γ̃_{2} ∗ Γ̃_{3}) .  (12)

2.3. Deformed case, the algebra B̃_{q}.

As a byproduct TODO give the formula.

3. Ado’s theorem, the Lie algebra L_{HW} and its enveloping algebra. TODO : This paragraph will be put as a remark. Replace it by physical motivations for central elements.

It is well known that, in characteristic 0, HW admits no finite dimensional proper representation (its admits neither no representation in a Banach unital algebra). But one observes that the subspace L = ℂa ⊕ ℂa^{+} ⊕ ℂ1_{HW} is closed for the Lie bracket and hence a Lie subalgebra of (HW, [,]). Now, by Ado theorem such a finite-dimensional Lie algebra admits finite dimensional faithful representations. As an example, one can give the following one (noted ρ)

ρ(a) = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) ; ρ(a^{+}) = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) ; ρ(1_{HW}) = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) . (13)

One sees that the image of 1_{HW} is not a unity and, in order not to get confused in our forthcoming development, we will take a copy of the Lie algebra L = ℂa ⊕ ℂa^{+} ⊕ ℂ1_{HW} as L_{HW} = ℂ{b, b^{+}, u} with the multiplication table

\begin{matrix}[↓, →] & b^{+} & b & u \\ b^{+} & 0 & −u & 0 \\ b & u & 0 & 0 \\ u & 0 & 0 & 0 \end{matrix}

Now, we have a morphism of Lie algebras ϕ_{1} : L_{HW} → HW given by

ϕ_{1}(b) = a ; ϕ_{1}(b^{+}) = a^{+} ; ϕ_{1}(u) = 1_{HW} (14)

and, by the universal property of the enveloping algebras this can be extended uniquely as a morphism of AAU φ_{1} : U(L_{HW}) → HW. We can make it explicit, using the basis of Poincaré-Birkhoff-Witt of U(L_{HW}) which is given (using the order b^{+} ≺ b ≺ u as in the multiplication table above) by the products {(b^{+})^{p} b^{q} u^{r}}_{p,q,r∈ℕ}. We then have φ((b^{+})^{p} b^{q} u^{r}) = (a^{+})^{p} a^{q}.

4. The arrow φ_{2} : B̃ → U(L_{HW}).

The algebra B̃ is a ℂ-AAU, and the set of classes of HW graohs B is a basis of it. One defines φ_{2} on this basis by

φ_{2}(Γ̃) := (b^{+})^{|Γ^{+}|}b^{|Γ^{−}|}u^{|Γ^{in}|} . (15)

the result does not depend on the chosen representative Γ̃ as the quantities (|Γ^{+}|, |Γ^{−}|, |Γ^{in}|) are class-invariants. In order to prove that φ_{2} is a morphism of ℂ-AAU, one has to prove that, for two disjoint HW -graphs Γ_{i}, i = 1, 2.

φ_{2}(Γ̃_{1} ∗ Γ̃_{2}) = φ_{2}(Γ̃_{1})φ_{2}(Γ̃_{2}) . (16)

For convenience, set

match(Γ_{2}, Γ_{1}; i) = {m ∈ match(Γ_{2}, Γ_{1})| |m| = i}

it is not difficult to get the two following preliminary results

|match(Γ_{2}, Γ_{1}; i)| = \{ \begin{matrix}( \begin{matrix}|Γ_{2}^{+}| \\ i \end{matrix} ) ( \begin{matrix}|Γ_{1}^{−}| \\ i \end{matrix} )i! & for & i ≤ min(|Γ_{2}^{+}|, |Γ_{1}^{−}|) \\ 0 & for & i > min(|Γ_{2}^{+}|, |Γ_{1}^{−}|) \end{matrix} (17)

and, for m ∈ match(Γ_{2}, Γ_{1}; i)

φ_{2}(\begin{matrix} \\ ( & \begin{matrix}Γ_{1} \\ m \\ Γ_{2} \end{matrix} & ) \end{matrix}) = (b^{+})^{|Γ_{1}^{+}|+|Γ_{2}^{+}|−i}b^{|Γ_{1}^{−}|−i+|Γ_{2}^{−}|}u^{|Γ^{in}|+i} . (18)

Then, one has

φ_{2}(Γ̃_{1} ∗ Γ̃_{2}) = φ_{2}( ∑_{m match(Γ_{2},Γ_{1})} \begin{matrix} \\ ( & \begin{matrix}Γ_{1} \\ m \\ Γ_{2} \end{matrix} & ) \end{matrix}) =

∑_{i=0}^{min(|Γ_{2}^{+}|,|Γ_{1}^{−}|)} ∑_{m match(Γ_{2},Γ_{1};i)} φ_{2}(\begin{matrix} \\ ( & \begin{matrix}Γ_{1} \\ m \\ Γ_{2} \end{matrix} & ) \end{matrix}) =

∑_{i=0}^{min(|Γ_{2}^{+}|,|Γ_{1}^{−}|)} ∑_{m match(Γ_{2},Γ_{1};i)}(b^{+})^{|Γ_{1}^{+}|+|Γ_{2}^{+}|−i} b^{|Γ_{1}^{−}|−i+|Γ_{2}^{−}|}u^{|Γ_{1}^{in}|+|Γ_{2}^{in}|+i} =

∑_{i=0}^{min(|Γ_{2}^{+}|,|Γ_{1}^{−}|)}(b^{+})^{|Γ_{1}^{+}|+|Γ_{2}^{+}|−i} b^{|Γ_{1}^{−}|−i+|Γ_{2}^{−}|}u^{|Γ_{1}^{in}|+|Γ_{2}^{in}|+i} ∑_{m match(Γ_{2},Γ_{1};i)} 1 =

∑_{i=0}^{min(|Γ_{2}^{+}|,|Γ_{1}^{−}|)} ( \begin{matrix}|Γ_{2}^{+}| \\ i \end{matrix} ) ( \begin{matrix}|Γ_{1}^{−}| \\ i \end{matrix} ) i!(b^{+})^{|Γ_{1}^{+}|+|Γ_{2}^{+}|−i} b^{|Γ_{1}^{−}|−i+|Γ_{2}^{−}|}u^{|Γ_{1}^{in}|+|Γ_{2}^{in}|+i} =

(b^{+})^{|Γ_{1}^{+}|} b^{|Γ_{1}^{−}}u^{|Γ_{1}^{in}|}(b^{+})^{|Γ_{2}^{+}|} b^{|Γ_{2}^{−}|}u^{|Γ_{2}^{in}|} =

φ_{2}(Γ̃_{1})φ_{2}(Γ̃_{2})  (19)

5. Diagram of algebras

We have, so far, constructed the following diagram of algebras

B̃_{φ_{2}}  _{φ_{3}} U(L_{HW})  _{φ_{1}}  HW L_{HW}  _{j_{1}}_{nat}

6. Comultiplications and Hopf Algebra structures.

6.1. Comultiplications and section coefficients.

In a pioneering paper [31], G-C Rota introduced the concept of “section coefficients” and gave many combinatorial examples (set partitions, classes of intervals, placements ...). Let us take, for simplicity, the example of binomial coefficients but presented here as “section coefficients”. One has, for n, p, q ∈ ℕ

( \begin{matrix}n \\ p, q \end{matrix} ) = \{ \begin{matrix}\frac{n!}{p! q!} & = & ( \begin{matrix}n \\ p \end{matrix} ) if p + q = n \\ 0 & & otherwise \end{matrix} (20)

in this spirit, these coefficients count the number of ways of splitting a n-set into two disjoint subsets with, respectively p and q elements. Now consider the number of ways of splitting a n-set in three parts with respectively p, q, r elements (of course it will be 0 if p + q + r ≠ n). This can be performed by first splitting the set in two parts and re-splitting the second part in two again or by first splitting the set in two parts and re-splitting the first part. These two procedures provide the “trichotomy identity”

∑_{m∈ℕ}( \begin{matrix}n \\ p, m \end{matrix} ) ( \begin{matrix}m \\ q, r \end{matrix} ) = ∑_{m∈ℕ}( \begin{matrix}n \\ m, r \end{matrix} ) ( \begin{matrix}m \\ p, q \end{matrix} ) (21)

which, in this case, is obvious as the two sums have only one term. Now, one has also an interplay with the sum, called Van der Monde identity [31]

( \begin{matrix}n_{1} + n_{2} \\ p , q \end{matrix} ) = ∑_{{p_{1}+p_{2}=p \atop q_{1}+q_{2}=q}} ( \begin{matrix}n_{1} \\ p_{1}, q_{1} \end{matrix} ) ( \begin{matrix}n_{2} \\ p_{2}, q_{2} \end{matrix} ) . (22)

In general, section coefficients (i|j, k) count the number of ways of splitting an object i into two sub-objects j, k. They may or may not fulfill identities (21) and/or (22) and have a deep relationship with modern algebra. Let us give another example which generalizes the preceding. For a word w = a_{1}a_{2} ⋯ a_{n} and R ⊂ {r_{1}, r_{2}, ⋯ r_{k}} (r_{1} < r_{2} < ⋯ < r_{k}) one defines the subword w[R] as w[R] = a_{r_{1}}a_{r_{2}} ⋯ a_{r_{k}}. For three words u, v, w, one can define the section coefficient

(w | u , v) = #{(R, Y) | R + Y = [1..|w|], w[R] = u, w[Y] = v} (23)

for example, one has

(abab | aa , bb) = 1 ; (abab | ba , ba) = 0 ; (abab | ab , ba) = 1 ; (abab | ab , ab) = 2.

These coefficients can be considered as “word functions”, they are a generalization of the binomial section coefficients as

(a^{n} | a^{p} , a^{q}) = ( \begin{matrix}n \\ p, q \end{matrix} ) .

For these coefficients, one has the trichotomy identity

∑_{m∈A^{∗}}(w | s , m)(m | t , u) = ∑_{m∈ℕ}(w | m , u)(m | s , t) (24)

and the Van der Monde one reads

(w_{1}w_{2} | u , v) = ∑_{{u=u_{1}u_{2} \atop v=v_{1}v_{2}}} (w_{1} | u_{1} , v_{1})(w_{2} | u_{2} , v_{2}) . (25)

The connection with modern algebra is done through the notion of a comultiplication. Taking C the vector space freely generated by the objects under consideration (the basis will be denoted (e_{i})_{i∈Ind}), one can consider the section coefficients as structure constants for a linear mapping (called comultiplication)

Δ : C → C ⊗ C (26)

by

Δ( ∑_{i∈Ind} α_{i}e_{i}) = ∑_{i∈Ind} α_{i}Δ(e_{i}) = ∑_{i∈Ind} ∑_{j,k∈Ind} α_{i}(e_{i} | e_{j} , e_{k}) e_{j} ⊗ e_{k} . (27)

(for the notion of structure constants of a comultiplication see [20]). Calling I the identity mapping C → C, the trichotomy identity reads

(Δ ⊗ I) ∘ Δ = (I ⊗ Δ) ∘ Δ (28)

whereas the Van der Monde identity (providing C be given an algebra law μ : C ⊗C → C) reads

Δ ∘ μ = μ^{⊗ 2} ∘ Δ (29)

where μ^{⊗ 2} is the law on C ⊗ C defined by

μ^{⊗ 2}(x_{1} ⊗ y_{1}, x_{2} ⊗ y_{2}) = μ(x_{1} ⊗ y_{1}) ⊗ μ(x_{2} ⊗ y_{2}) . (30)

Example 6.1 In the case of the “subwords” section coefficients (form. (23)), an alphabet A and a field K being given, the space (C above) freely generated by the words is the algebra K⟨A⟩. It is easily seen that

Δ(w) = ∑_{R+Y =[1..|w|]} w[R] ⊗ w[Y] (31)

6.2. Comultiplication and Hopf Algebra structure on U(L_{HW}).

As it is classical, the enveloping algebra U(L_{HW}) is endowed with a structure of Hopf Algebra by the extra structure given by

• a comultiplication

• a counit

• an antipode.

The comultiplication Δ_{U} is completely determined by the fact that the generators b, b^{+} and u are primitive i. e.

Δ_{U}(x) = x ⊗ 1_{U} + 1_{U} ⊗ x for x = b, b^{+}, u (32)

one then gets at once the coproduct of an element of the PBW basis

Δ_{U}((b^{+})^{p} b^{q} u^{r}) = ∑_{(α,β,γ)≤(p,q,r)} ( \begin{matrix}p \\ α \end{matrix} ) ( \begin{matrix}q \\ β \end{matrix} ) ( \begin{matrix}r \\ γ \end{matrix} ) (b^{+})^{α} b^{β} u^{γ} ⊗ (b^{+})^{p−α} b^{q−β} u^{r−γ} . (33)

It should be noticed that this comultiplication is constructed on the model of (31) as, for a word w = a^{p}b^{q}c^{r}, on has

Δ(a^{p} b^{q} c^{r}) = ∑_{(α,β,γ)≤(p,q,r)} ( \begin{matrix}p \\ α \end{matrix} ) ( \begin{matrix}q \\ β \end{matrix} ) ( \begin{matrix}r \\ γ \end{matrix} ) a^{α} b^{β} c^{γ} ⊗ a^{p−α} b^{q−β} c^{r−γ} . (34)

The counity is given by ϵ_{U}((b^{+})^{p} b^{q} u^{r}) = 0 unless p = q = r = 0 (in which case it equals 1). The antipode is given classically by S(((b^{+})^{p} b^{q} u^{r})) = (−1)^{p+q+r}u^{r} b^{q} ((b^{+})^{p}. This is the standard structure of any enveloping algebra as a Hopf algebra. The question solved in this paper is a construction on B̃ of a structure of Hopf algebra such that φ_{2} be a Hopf arrow.

6.3. Partitions of a HW -graph and the comultplication.

In order to get φ_{5} as a Hopf arrow, we should use a formula of the type (31) and define

• what is the support (where the subset I would be taken)

• what is the notion of restriction in order to define Γ[I] as we have defined w[I].

Let Γ be a HW -graph and R ⊂ Γ^{(1)} (one can think of some edges coloured in red), one defines Γ[R] as the subgraph with lines R, tail and head functions restricted to R (call them t_{R}, h_{R}) and vertices Γ[R]^{(0)} as t_{R}(R) ∪ h_{R}(R). It is easy to check that Γ[R] is a HW -graph. It is also not difficult to check that, if R_{1} ⊂ R_{2} ⊂ Γ^{(1)}, on has the transitivity of restrictions i. e.

Γ[R_{2}][R_{1}] = Γ[R_{1}] . (35)

We define a comultiplication on B̃ by

Δ(Γ̃) = ∑_{R+Y =Γ^{(1)}} Γ̃[R] ⊗ Γ̃[Y] (36)

(some edges are coloured in red and the others in yellow). Of course, one has to check that the result does not depend of the choice of the representative Γ which is straightforward. The proof of coassociativity rests on (35), i. e.

(Δ ⊗ I)Δ(Γ̃) = (Δ ⊗ I)( ∑_{R+Y =Γ^{(1)}} Γ̃[R] ⊗ Γ̃[Y]) =

∑_{R+Y =Γ^{(1)}} ∑_{R_{1}+R_{2}=Γ[R]^{(1)}} { \atop Γ[R][R_{1}]} ⊗ { \atop Γ[R][R_{2}]} ⊗ Γ̃[Y]  (37)

but, by construction Γ[R]^{(1)} = R and then, using (35), one has

(Δ ⊗ I)Δ(Γ̃) = ∑_{R+Y =Γ^{(1)}} ∑_{R_{1}+R_{2}=R} { \atop Γ[R_{1}]} ⊗ { \atop Γ[R_{2}]} ⊗ Γ̃[Y] =

∑_{R_{1}+R_{2}+Y =R} { \atop Γ[R_{1}]} ⊗ { \atop Γ[R_{2}]} ⊗ Γ̃[Y]  (38)

which is the same as (I ⊗ Δ)Δ(Γ̃). One has now to prove that ΔB̃ → B̃ ⊗ B̃ is a morphism of algebras. Let Γ_{i}, i = 1, 2 be two disjoint HW -graphs, one has

Δ(Γ̃_{1}) ∗^{⊗^{2}} Δ(Γ̃_{2}) = ( ∑_{R_{1}+Y_{1}=Γ_{1}^{(1)}} { \atop Γ_{1}[R_{1}]} ⊗ { \atop Γ_{1}[Y_{1}]}) ∗^{⊗^{2}} ( ∑_{R_{2}+Y_{2}=Γ_{2}^{(1)}} { \atop Γ_{2}[R_{2}]} ⊗ { \atop Γ_{2}[Y_{2}]}) =

∑_{{R_{1}+Y_{1}=Γ{(1) \atop 1} \atop R_{2}+Y_{2}=Γ{(1) \atop 2}}} ({ \atop Γ_{1}[R_{1}]} ∗ { \atop Γ_{2}[R_{2}]}) ⊗ ({ \atop Γ_{1}[Y_{1}]} ∗ { \atop Γ_{2}[Y_{2}]}) =

∑_{{R_{1}+Y_{1}=Γ{(1) \atop 1} \atop R_{2}+Y_{2}=Γ{(1) \atop 2}}} ∑_{{m_{R}∈match(Γ_{2}[R_{2}],Γ_{1}[R_{1}]) \atop m_{Y} ∈match(Γ_{2}[Y_{2}],Γ_{1}[Y_{1}])}} \begin{aligned}& \\ ( & \begin{matrix}Γ_{1}[R_{1}] \\ m_{R} \\ Γ_{2}[R_{2}] \end{matrix} & ) \end{aligned} ⊗ \begin{aligned}& \\ ( & \begin{matrix}Γ_{1}[Y_{1}] \\ m_{R} \\ Γ_{2}[Y_{2}] \end{matrix} & ) \end{aligned} =

∑_{m∈match(Γ_{2},Γ_{1})} ∑_{R+Y =( \begin{matrix}Γ_{1} \\ m \\ Γ_{2} \end{matrix} )^{(1)}} ( \begin{matrix}Γ_{1} \\ m \\ Γ_{2} \end{matrix} ) [R] ⊗ ( \begin{matrix}Γ_{1} \\ m \\ Γ_{2} \end{matrix} ) [Y]  (39)

as the correspondence

(m, R, Y) ↦ (R_{1}, Y_{1}, R_{2}, Y_{2}, m_{R}, m_{Y}) (40)

is one to one between the sets

\{(m, R, Y) such that (m ∈ match(Γ_{2}, Γ_{1})) and (R + Y = ( \begin{matrix}Γ_{1} \\ m \\ Γ_{2} \end{matrix} )^{(1)})\} (41)

and

{(R_{1}, R_{2}, Y_{1}, Y_{2}, m_{R}, m_{Y}) such that (R_{1} + Y_{1} = Γ_{1}^{(1)}, R_{2} + Y_{2} = Γ_{2}^{(1)})
(m_{R} ∈ match(Γ_{2}[R_{2}], Γ_{1}[R_{1}]), m_{Y} ∈ match(Γ_{2}[Y_{2}], Γ_{1}[Y_{1}]))}  (42)

7. Endnotes

Formula (11) : Strictly speaking, the (double) sum in the LHS of this formula is indexed by the set

I_{1} := ⋃_{m′∈match(Γ_{2},Γ_{1})}\{m′\} × match(Γ_{3}, ( \begin{matrix}Γ_{1} \\ m′ \\ Γ_{2} \end{matrix} ))  (43)

and the RHS of this formula is indexed by the set

I_{2} := ⋃_{m′′∈match(Γ_{3},Γ_{2})} match(( \begin{matrix}Γ_{2} \\ m′′ \\ Γ_{3} \end{matrix} ) , Γ_{1}) × \{m′′\} .  (44)

The (one-to-one) correspondence I_{1} → I_{2} is given by (m′, m′′) ↦ (m′ ∪ m′′_{31}, m′′_{32}) and its inverse by (m′, m′′) ↦ (m′_{21}, m′′ ∪ m′_{31}) with m_{ij} = m ∩ (Γ_{i}^{+} × Γ_{j}^{−}).

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