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,

Abstract. It has been found by A. Varvak (TODO improve) and others 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 cancellation 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 representations.

Contents

1 Introduction 3

2 Construction of the diagram algebras B̃ and B̃_{q}. 3 2.1 Heisenberg-Weyl graphs and their matching . . . . . . . . . . . . . . . . 3 2.2 Multiplcation of graphs and structure of the algebra B̃_{q}. . . . . . . . . . 5 2.3 Specialization to q = 1: the algebra B̃. . . . . . . . . . . . . . . . . . . . 6 3 Ado’s theorem, the Lie algebra L_{HW} and its envelopping algebra. 6 4 The arrow φ_{4} : B̃_{q} → U(L_{HW}). 7 5 Endnotes 7 1. Introduction

Since the advent of Hopf algebras in Combinatorial Physics, when facing an algebra, one always ask 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, the answer is “no” since the counit provides one with a one-dimesional 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_{q}_{φ_{3}}  _{φ_{4}} B̃  _{φ_{5}}_{φ_{2}}  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” (resp. “outgoing edges”). A cycle is a sequence of edges a_{1}a_{2} ⋯ a_{n} such that, forall 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 matching between 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 between Γ_{2} and Γ_{1} any matching between 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} )

which can be formally defined by

( \begin{matrix}Γ_{1} \\ m \\ Γ_{2} \end{matrix} )^{(1)} = Γ_{1}^{(1)} ∪ Γ_{2}^{(1)} ∪ m − (pr_{1}(m) ∪ pr_{2}(m)) ; (5)

with tail and head functions defined accordingly: remark that condition (1) implies that,

Γ_{2}^{(0)} ∩ Γ_{1}^{(0)} = ∅ ⟹ Γ_{2}^{(1)} ∩ Γ_{1}^{(1)} = ∅ (6)

so, 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}) . (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 partionned 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. Multiplcation of graphs and structure of the algebra B̃_{q}.

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 staightforward 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 (resp. B̃_{q}) is the vector space ℂB (resp. the module ℂ[q]B). The multiplication table in B̃_{q} is given by

Γ̃_{1} ∗ Γ̃_{2} = ∑_{m matchings Γ_{1}→Γ_{2}} q^{|m|}\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, forall Γ, Γ̃ ∗ ˜∅ = ˜∅ ∗ Γ̃ = Γ. 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 (we need two and a third which is disjoint of the results of the matchings of the two first). 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} ))} q^{|m′|}q^{|m′′|} ( \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} ))} q^{|m′|}q^{|m′′|} ( \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} ))} q^{|m′∪m′′_{31}|}q^{|m′′_{32}|} ( \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}), then the sum amounts to

∑_{m′′∈match(Γ_{3},Γ_{2})} ∑_{m′∈match(( \begin{matrix}Γ_{2} \\ m′′ \\ Γ_{3} \end{matrix} ),Γ_{1})} q^{|m′′|}q^{|m′|} ( \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. Specialization to q = 1: the algebra B̃.

3. Ado’s theorem, the Lie algebra L_{HW} and its envelopping algebra.

It is well known that 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 propoerty of the envelopping 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_{4} : B̃_{q} → U(L_{HW}).

The algebra B̃_{q} is a ℂ[q]-AAU, but onee can restrict the scalars to ℂ. In this case, it is a ℂ-AAU with basis q^{n}Γ̃ (n in ℕ and Γ choosen in a set of distinguished representatives). One sets

φ_{4}(q^{n}Γ̃) := (b^{+})^{|Γ^{+}|}b^{|Γ^{−}|}u^{n} . (15)

In order to prove that φ_{4} is a morphism of ℂ-AAU,, one has to prove that

φ_{4}(q^{s}Γ̃_{1} ∗ q^{t}Γ̃_{2}) = φ_{4}(q^{s}Γ̃_{1})φ_{4}(q^{t}Γ̃_{2})

which can be rephrased as

φ_{4}(Γ̃_{1} ∗ Γ̃_{2})u^{s+t} = φ_{4}(Γ̃_{1})φ_{4}(Γ̃_{2})u^{s+t}

5. 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} ))  (16)

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′′\}  (17)

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 ∩ match(Γ_{i}, Γ_{j}).

References

[1] Bayen (F.), Flato (M.), Fronsdal (C.), Lichnerowicz (A.) and Sternheimer (D.), Deformation and Quantization, Ann. of Phys. 111 (1978), pp. 61-151.

[2] J. Berstel, C. Reutenauer, Rational series and their languages EATCS Monographs on Theoretical Computer Science, Springer (1988).

[3] C. M. Bender, D. C. Brody, and B. K. Meister, Quantum field theory of partitions, J. Math. Phys. Vol 40 (1999)

[4] P. Blasiak, A. Horzela, K. A. Penson, G. H. E. Duchamp, A.I. Solomon, Boson normal ordering via substitutions and Sheffer-Type Polynomials, Phys. Lett. A 338 (2005)

[5] P. Blasiak, K. A. Penson, A.I. Solomon, A. Horzela, G. H. E. Duchamp, Some useful formula for bosonic operators, Jour. Math. Phys. 46 052110 (2005).

[6] Bourbaki N., Algebra, chapter III, Springer

[7] Bourbaki N., Algebra, chapter VI, Springer

[8] Bourbaki N., Theory of sets, Springer

[9] Cartier P., Séminaire “Sophus Lie”, 2ème année, Faculté des Sciences de Paris (1955-56)

[10] P. Cartier, Fonctions polylogarithmes, nombres polyzeta et groupes pro-unipotents, Séminaire Bourbaki, Asterisque n. 282 (2002)

[11] P. Cartier, A primer of Hopf algebras, Septembre (2006), IHES preprint IHES/M/06/40.

[12] V. Chari, A. Pressley, A guide to quantum groups. Cambridge (1994).

[13] G. H. E. Duchamp, P. Blasiak, A. Horzela, K. A. Penson, A. I. Solomon, Feynman graphs and related Hopf algebras, Journal of Physics: Conference Series, SSPCM’05, Myczkowce, Poland. arXiv : cs.SC/0510041

[14] Duchamp G., Flouret M., Laugerotte E., Luque J-G., Direct and dual laws for automata with multiplicities arXiv : math.CO0607412

[15] Duchamp G., Luque J-G., Congruences Compatible with the Shuffle Product arXiv : math.CO0607419

[16] G. Duchamp, F. Hivert, J. Y. Thibon, Non commutative functions VI: Free quasisymmetric functions and related algebras, International Journal of Algebra and Computation Vol 12, No 5 (2002).

[17] G. H. E. Duchamp, G. Koshevoy, K. A. Penson, C. Tollu, F. Toumazet, Geometric combinatorial twisting and shifting, Séminaire Lotharingien (in preparation).

[18] G. H. E. Duchamp, J. -G. Luque, J. -C. Novelli, C. Tollu, F. Toumazet, Hopf algebras of diagrams, FPSAC07.

[19] G. Duchamp, A.I. Solomon, K.A. Penson, A. Horzela and P. B lasiak, One-parameter groups and combinatorial physics, Proceedings of the Symposium Third International Workshop on Contemporary Problems in Mathematical Physics (COPROMAPH3) (Porto-Novo, Benin, Nov. 2003), J. Govaerts, M. N. Hounkonnou and A. Z. Msezane (eds.), p.436 (World Scientific Publishing 2004)

arXiv: quant-ph/04011262

[20] L. Foissy, Isomorphisme entre l’algèbre des fonctions quasi-symétriques libres et une algèbre de Hopf des arbres enracinés décorés plans, personal communication.

[21] L. Foissy, Les algèbres de Hopf des arbres enracinés decorés, PhD Memoir, Reims University (2002).

[22] L. Foissy, Personal Communication.

[23] M. Hazewinkel, Hopf algebras of endomorphisms of Hopf algebras, (Oct 2004) ArXiv :

math.QA/0410364

[24] M. E. Hoffman, Quasi-shuffle products, J. Algebraic Combin. (2000), 49-68

[25] A. Horzela, P. Blasiak, G. Duchamp, K. A. Penson and A.I. Solomon, A product formula and combinatorial field theory, Proceedings of the XI International Conference on Symmetry Methods in Physics (SYMPHYS-11) (Prague, Czech Republic, June 2004), C. Burdik, O. Navratil, and S. Posta (eds.) (JINR Publishers, Dubna)

arXiv:quant-ph/0409152

[26] D. Kreimer, Knots and Feynman Diagrams, Cambridge Lecture Notes in Physics (2000).

[27] P. Ochsenschläger, Binomialkoeffitzenten und Shuffle-Zahlen, Technischer Bericht, Fachbereich Informatik, T. H. Darmstadt,1981.

[28] Reutenauer C., Free Lie algebras, Oxford University Press (1993).

[29] M. Rosso, Quantum groups and quantum shuffles, Inventiones Math. 133 (1998).

[30] A.I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K.A. Penson, Combinatorial Physics, Normal Order and Model Feynman Graphs, Proceedings of the Symposium ’Symmetries in Science XIII’, Bregenz, Austria, 2003, B. Gruber, G. Marmo and N. Yoshinaga (eds.), p.527 (Kluwer Academic Publishers 2004)

arXiv : quant-ph/0310174

[31] A.I. Solomon, G. Duchamp, P. Blasiak, A. Horzela and K.A. Penson, Normal Order: Combinatorial Graphs Quantum Theory and Symmetries, Proceedings of the 3rd International Symposium P.C. Argyres, T.J. Hodges, F. Mansouri, J.J. Scanio, P. Suranyi, and L.C.R. Wijewardhana (eds.), p.398 (World Scientific Publishing 2004)

arXiv:quant-ph/0402082

[32] A.I. Solomon, G. Duchamp, P. Blasiak, A. Horzela and K. A. Penson, Partition functions and graphs: A combinatorial approach, Proceedings of the XI International Conference on Symmetry Methods in Physics (SYMPHYS-11) (Prague, Czech Republic, June 2004), C. Burdik, O. Navratil, and S. Posta (eds.) (JINR Publishers, Dubna, 2004)

arXiv:quant-ph/0409082

[33] A. I. Solomon, G. Duchamp, P. Blasiak, A. Horzela and K. A. Penson, Hopf algebra structure of a model quantum field theory, Proceedings of the 26th International Colloquium on Group Theoretical Methods in Physics, New York 2006, Editor: S. Catto (2007).

[34] G. Duchamp, A. I. Solomon, P. Blasiak, A. Horzela and K. A. Penson, A multipurpose Hopf deformation of the algebra of Feynman-like diagrams, Proceedings of the 26th International Colloquium on Group Theoretical Methods in Physics, New York 2006, Editor: S. Catto (2007).