The framework used to prove the multiplicative law deformation of the algebra of Feynman-Bender diagrams is a \textit{twisted shifted dual law} (in fact, twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameters is a perturbation of the shuffle coproduct of Hoffman type which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions.
Deep Dive into Combinatorial Deformations of Algebras: Twisting and Perturbations.
The framework used to prove the multiplicative law deformation of the algebra of Feynman-Bender diagrams is a \textit{twisted shifted dual law} (in fact, twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameters is a perturbation of the shuffle coproduct of Hoffman type which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions.
arXiv:0903.2101v2 [cs.SC] 27 Aug 2010
Combinatorial Deformations of Algebras:
Twisting and Perturbations
G. H. E. Duchamp, C. Tollu, ∗,
K. A. Penson †and G. Koshevoy ‡,
Contents
1
Introduction
2
2
The deformed algebra LDIAG(qc, qs)
3
2.1
Review of the construction of the algebra . . . . . . . . . . . . . . . . . . .
3
2.2
Coding and the recursive definition . . . . . . . . . . . . . . . . . . . . . .
5
3
Colour factors and products
7
4
Special classes of laws
8
4.1
Dual laws
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
4.1.1
Algebras and coalgebras in duality
. . . . . . . . . . . . . . . . . .
8
4.1.2
Duality between grouplike elements and unities
. . . . . . . . . . .
9
4.2
Deformed laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
4.3
Shifted laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
5
Application to the structure of LDIAG(qc, qs)
11
5.1
Associativity of LDIAG(qc, qs) using the previous tools . . . . . . . . . . .
11
5.2
Structure of LDIAG(qc, qs)
. . . . . . . . . . . . . . . . . . . . . . . . . .
12
6
Conclusion
13
Abstract
The framework used to prove the multiplicative deformation of the algebra of
Feynman-Bender diagrams is a twisted shifted dual law (in fact, twisted twice). We
give here a clear interpretation of its two parameters. The crossing parameter is a
deformation of the tensor structure whereas the superposition parameter is a per-
turbation of the shuffle coproduct which, in turn, can be interpreted as the diagonal
restriction of a superproduct. Here, we systematically detail these constructions.
∗LIPN - UMR 7030 CNRS - Universit´e Paris 13 F-93430 Villetaneuse, France
†Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee Universit´e Pierre et Marie Curie, CNRS
UMR 7600 Tour 24 - 2i`eme ´etage, 4 place Jussieu, F 75252 Paris cedex 05
‡Central Institute of Economics and Mathematics (CEMI) Russian Academy of Sciences
1
1
Introduction
In [1], Bender, Brody, and Meister introduced a special field theory, then called “Quantum
Field Theory of Partitions”. This theory is based on a bilinear product formula which
reads
H(F, G) = F
z d
dx
G(x)
x=0
.
(1)
If one develops this formula in the case when F and G are free exponentials, one
obtains a summation over all the (finite) bipartite 1 graphs with multiple edges and no
isolated point [6] (the set of these diagrams will be called diag), a data structure which
is equivalent to classes of packed matrices [8] under permutations of rows and columns.
So, one has a Feynman-type expansion of the product formula
H
exp(
∞
X
n=1
Ln
zn
n! ), exp(
∞
X
n=1
Vn
zn
n! )
!
X
n≥0
zn
n!
X
d∈diag
|d|=n
mult(d)Lα(d)Vβ(d)
(2)
where mult(d) is the number of pairs (P1, P2) of (ordered) set partitions of {1, . . . , n}
which correspond to a diagram d, |d| the number of edges in d and
Lα(d) = Lα1
1 Lα2
2 · · · ; Vβ(d) = V β1
1 V β2
2
· · ·
(3)
is the multiindex notation for the monomials in L∪V where αi = αi(d) (resp. βj = βj(d))
is the number of white (resp. black) spots of degree i (resp. j) in d.
The set diag endowed with disjoint receive the structure of a monoid such that the arrow
d 7→Lα(d)Vβ(d) is a morphism (of monoids) and then, by linear extension, one deduces a
morphism of algebras
C[diag] →Pol(C; L ∪V) .
(4)
where Pol(C; L ∪V) is the Hopf algebra of (commutative) polynomials with complex
coefficients generated by the alphabet L ∪V. For at least three models of Physics, one
can specialize L so that the canonical Hopf algebra structure of Pol(C; L∪V) can be lifted,
through (4). The resulting Hopf algebra (based on C[diag]) has been denoted DIAG. To
our great surprise, this Hopf algebra structure could be lifted at the (noncommutative)
level of the objects themselves instead of classes, resulting in the construction of a Hopf
algebra on (linear combinations of) “labelled diagrams” (the monoid ldiag, see [6]). As
these “labelled diagrams” are in one-to-one correspondence with the packed matrices of
MQSym, we get on the vector space C[ldiag] two (combinatorially natural) structures of
algebra (and co-algebra) and one could raise the question of the existence of a continuous
deformation between the two.
The answer is positive and can be performed through
a three-parameter (two formal, or continuous and one boolean) Hopf deformation2 of
LDIAG called LDIAG(qc, qs, qt) [6] such that
LDIAG(0, 0, 0) ≃LDIAG ; LDIAG(1, 1, 1) ≃MQSym .
(5)
1The (bi)-partition of the vertices is understood ordered. In this case, the term bicoloured can also be
found in the literature.
2This algebra deformation has received recently another realisation in terms of bi-words [9].
2
The rˆole of the two parameters qc, qs (algebra parameters, whereas qt is a coalge-
bra parameter) was discovered just counting crossings and superpositions in the twisted
labelled diagrams (see [6] for details). This simple statistics (counting crossings and su-
perpositions) yields an associative product on the dia
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
