Bialgebras in cointeraction, the antipode and the eulerian idempotent

TL;DR

This paper reviews double bialgebras with two coproducts, focusing on connected cases, and explores their algebraic structures, antipodes, and invariants, with applications to graph theory and quasishuffle bialgebras.

Contribution

It provides a comprehensive review of double bialgebras, introduces new proofs, and applies these concepts to graph invariants and quasishuffle bialgebras.

Findings

Analysis of monoid of characters and their actions

New combinatorial interpretation of chromatic polynomial coefficients

Applications to graphs and quasishuffle bialgebras

Abstract

We give here a review of results about double bialgebras, that is to say bialgebras with two coproducts, the first one being a comodule morphism for the coaction induced by the second one. An accent is put on the case of connected bialgebras. The subjects of these results are the monoid of characters and their actions, polynomial invariants, the antipode and the eulerian idempotent. As examples, they are applied on a double bialgebra of graphs and on quasishuffle bialgebras. This includes a new proof of a combinatorial interpretation of the coefficients of the chromatic polynomial due to Greene and Zaslavsky.