Feynman Graphs, Rooted Trees, and Ringel-Hall Algebras

Abstract

Abstract. We construct symmetric monoidal categories LRF , LF G of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of LRF , LF G, H LRF , H LF G are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman diagrams. We thus obtain an interpretation of the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs as Ringel-Hall Lie algebras.

Hecke correspondences and Feynman graphs

“…In [7], a categorification of the Hopf algebras U (n T ), U (n FG ) was obtained, by showing that they arise naturally as the Ringel-Hall algebras of certain categories LRF , LF G of labeled rooted forests and Feynman graphs respectively. We briefly recall this 1 notion.…”

“…In this light, the main result of [7] is the construction of categories LRF , LF G such that n LRF ≃ n T , and n LF G ≃ n FG (note however that LRF , LF G are not abelian).…”

“…In the applications considered in this paper, the category C in question will not be abelian, but "nearly" so, and the Ringel-Hall algebra construction goes through without problems (see [7] for a detailed discussion of these issues).…”

“…In this section, we briefly recall the construction of the category LRF of labeled rooted forests from [7]. We begin by reviewing some notions related to rooted trees.…”

“…In this section we review the construction of the category LF G of labeled Feynman graphs following [7]. Our treatment of the combinatorics of graphs is taken from [10].…”