Hopf algebras of diagrams

TL;DR

This paper explores various Hopf algebras constructed from diagrams related to quantum field theory, focusing on their bases, realizations, and algebraic structures like deformations and dendriform structures.

Contribution

It introduces new Hopf algebras of diagrams linked to quantum field theory, detailing their bases, realizations, and algebraic properties, and situates them within a unifying diagram.

Findings

Hopf algebras indexed by bipartite graphs and packed matrices.

Realizations on biword representations are provided.

Analysis of Hopf deformations and dendriform structures.

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.