A multipurpose Hopf deformation of the Algebra of Feynman-like Diagrams

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

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

Abstract. We construct a three parameter deformation of the Hopf algebra LDIAG. This new algebra is a true Hopf deformation which reduces to LDIAG on one hand and to MQSym on the other, relating LDIAG to other Hopf algebras of interest in contemporary physics. Further, its product law reproduces that of the algebra of polyzeta functions.

1 Introduction

The complete journey between the first appearance of a product formula by Bender et al. [1] and their related Feynman-like diagrams to the discovery of a Hopf algebra structure [8] on the diagrams themselves, goes roughly as follows. Firstly, Bender, Brody, and Meister [1] introduced a special field theory which proved to be particularly rich in combinatorial links and by-products [11] (not to mention the link with vector fields and one-parameter groups [7, 10]). Secondly, the Feynman-like diagrams of this theory label monomials which combine naturally in a way compatible with monomial multiplication and co-addition. This is the Hopf algebra DIAG [8]. The (Hopf-)subalgebra of DIAG generated by the primitive graphs is the Hopf algebra BELL described in Solomon’s talk at this conference [12]. Thirdly, the natural noncommutative pull-back of this algebra, LDIAG, has a basis (the labelled diagrams) which is in one-to-one correspondence with that of the Matrix Quasi-Symmetric Functions [13] (the packed matrices of MQSym), but their algebra and co-algebra structures are completely different. In particular, multiplication in MQSym involves a sort of shifted shuffle with overlappings reminiscent of Hoffmann’s shuffle used in the theory of polyzeta functions [2]. The superpositions and overlappings involved there are not present in (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 announce the existence of a Hopf algebra deformation which connects LDIAG to other Hopf algebras relevant to physics (Connes-Kreimer, Connes-Moscovici, Brouder-Frabetti, see [6]) and other fields (noncommutative symmetric functions, Euler-Zagier sums). Acknowledgement : The authors would like to thank Christophe Tollu who checked carefully the (lengthy) computation which proves the associativity of LDIAG(q_{c},q_{s}).

2 Labelled Diagrams and Diagrams

Product formula involves a summation over all diagrams of a certain type [12] a labelled version of which is described below. Labelled diagrams can be identified with their weight functions which are mappings ω : ℕ^{+} × ℕ^{+} → ℕ such that the supporting subgraph

Γ_{ω} = {(i,j) ∈ ℕ^{+} × ℕ^{+} | w(i,j) ≠ 0}  (1)

has specific projections i.e. pr_{1}(Γ_{ω}) = [1..p]; pr_{2}(Γ_{ω}) = [1..q] for some p,q ∈ ℕ (notice that when one of p,q is zero so too is the other and the diagram is empty). These graphs are represented by labelled diagrams as follows

The labelled diagrams form the set ldiag and prescribe monomials through the formula 𝕃^{α(d)}𝕍^{β(d)} where α(d) (resp. β(d)) is the “white spot type” (resp. the “black spot type”) i.e. the multi-index (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). There is a (graphically) natural multiplicative structure on ldiag such that the arrow

m_{(𝕃,𝕍)} : d ↦ 𝕃^{α(d)}𝕍^{β(d)}  (2)

is a morphism. It is clear that one can permute black spots, or white spots, of d without changing the monomial 𝕃^{α(d)}𝕍^{β(d)}. The classes of (labelled) diagrams up to this equivalence (permutations of white - or black - spots among themselves) are naturally represented by unlabelled diagrams and will be denoted diag (including the empty one).

For both types of diagram the product consists of concatenating the diagrams i.e. placing d_{2} on the right of d_{1} [8] (the result, for d_{1},d_{2}, will be denoted [d_{1}|d_{2}]_{D} in diag and [d_{1}|d_{2}]_{L} in ldiag). These products endow diag and ldiag with the structure of monoids, with the empty diagram as neutral element. The corresponding commutative diagram is as follows (where X^{2} means the cartesian square of the set X).

Labelled diagrams^{2} −−−−−⟶^{Unlabelling^{2}} Diagrams^{2} −−−−−−−→^{m_{(𝕃,𝕍)}×m_{(𝕃,𝕍)}} Monomials^{2}
product↓  ^{product}↓  ^{product}↓
Labelled diagrams −−−−−−→^{Unlabelling} Diagrams −−⟶^{m_{(𝕃,𝕍)}}  Monomials

(3)

It is easy to see that the labelled diagram (resp. diagrams) form free monoids. We denote by DIAG and LDIAG the K-algebras of these monoids [8] (K is a field). One can shuffle the product in ldiag, counting crossing and superpositions. The definition of the deformed product is expressed by the diagrammatic formula

and the descriptive formula below.

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

(4)

where

• q_{c},q_{s} are coefficients in K • the exponent of q_{c}^{nc×weight_{2}} is the number of crossings of “what crosses”

times its weight

• the exponent of q_{s}^{weight_{1}×weight_{2}} is the product of the weights of “what is overlapped”

• terms cs([d_{1}|d_{2}]_{L}) are the diagrams obtained from [d_{1}|d_{2}]_{L} by the process of crossing and superposing the black spots of d_{2} on those of d_{1}, the order and identity of the black spots of d_{1} (resp. d_{2}) being preserved.

What is striking is that this law (denoted above ¯↑) is associative. Moreover, it can be shown [3, 4] that this process decomposes into two transformations: twisting and shifting. In fact, specialized to certain parameters, this law is reminiscent of others [2].

Parameters (0,0) (shifted) (1,1) (shifted)  (1,1) (unshifted)
Laws  LDIAG  MQSym  Hoffmann & Euler-Zagier

3 Hopf Deformation

Using a total order on the monomials of ldiag, it can be shown that the algebra LDIAG(q_{c},q_{s}) is free. Thus one may construct a coproduct Δ_{t}; t ∈ K such that LDIAG(q_{c},q_{s},t) = (LDIAG(q_{c},q_{s}),1_{ldiag},Δ_{t},ε,S) is a Hopf algebra. We have the following specializations

(q_{c},q_{s},t) = (0,0,0) (1,1,1) ; ≃  LDIAG MQSym

4 Conclusion

The results which we announced here in this note can be illustrated by the following picture. All details will be given in forthcoming papers [3, 4].

References

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

[2] P. Cartier, Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents, Séminaire Bourbaki, Mars 2001, 53ème année, 2000-2001, 885

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

[4] G. H. E. Duchamp, A.I. Solomon, P. Blasiak, A. Horzela and K.A. Penson, The Hopf Algebra of Feynman-like Diagrams : Twisting and Shifting. (in preparation)

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

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

[7] 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

[8] G. H. E. Duchamp, P. Blasiak, A. Horzela, K. A. Penson, A. I. Solomon, Feynman graphs and related Hopf algebras, J. Phys: Conference Series (30) (2006) 107, Proc of SSPCM’05, Myczkowce, Poland. arXiv : cs.SC/0510041

[9] 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

[10] 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) 108

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

[12] A.I. Solomon, G. H. E. Duchamp, P. Blasiak, A. Horzela, K. A. Penson, Hopf Algebra Structure of a model Quantum Field Theory, This volume.

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