Hopf Algebras of Diagrams

Abstract

We investigate several Hopf algebras of diagrams related to Quantum Field Theory of Partitions and whose product comes from the Hopf algebras WSym or WQSym respectively built on integer set partitions and set compositions. Bases of these algebras are indexed either by bipartite graphs (labelled or unlabbeled) or by packed matrices (with integer or set coefficients). Realizations on biword are exhibited, and it is shown how these algebras fit into a commutative diagram. Hopf deformations and dendriform structures are also considered for some algebras in the picture.

A three-parameter Hopf deformation of the algebra of Feynman-like diagrams*

“…Bidendriformity of the algebra LDIAG(q c , q s ) can also be established through a bi-word realization providing yet another (statistical) interpretation of the (q c , q s ) deformation [18]. We will now make clear the relations between the (q c , q s ) deformation and Euler-Zagier sums.…”

“…[ ] which will, therefore, be denoted by "1 (MON + (X)) * " or simply "1" when the context is clear). We will return to this construction (called shifting [18]) later. The alphabet of a list is the set of variables occurring in the list.…”

“…It has been proved recently that LDIAG(q c , q s , q t ) is a tridendriform Hopf Algebra [22] and that LDIAG(1, q s , q t ) is a homomorphic image of the algebra of planar decorated trees of Foissy [20,21]. Bidendriformity of the algebra LDIAG(q c , q s ) can also be established through a bi-word realization providing yet another (statistical) interpretation of the (q c , q s ) deformation [18].…”

Realizations of Hopf algebras of graphs by alphabets

“…Some combinatorial Hopf algebras admit a polynomial realization, which gives an efficient way to prove the existence of the coproduct and more structures, see [8,19,15,7,17]. Let us explicit a well-known example.…”

Some Open Problems in Combinatorial Physics

“…These classes have been shown [2,3] to be in one to one correspondence with Feynman-Bender diagrams [1] which are bicoloured graphs with p (= card(P 1 )) black spots, q (= card(P 2 )) white spots, no isolated vertex and integer multiplicities. We denote the set of such diagrams by diag [8,9]. Then, the correspondence goes as showed below.…”