Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics

TL;DR

This paper develops an algebraic framework for pathlike and colored co-, bi-, and Hopf algebras, enabling the analysis of antipodes and invertibility of characters in contexts like topology, number theory, and physics.

Contribution

It introduces a comprehensive algebraic theory for specialized Hopf algebras, including conditions for antipode invertibility and applications to various mathematical and physical structures.

Findings

Provided conditions for antipode invertibility in relevant bialgebras

Connected coalgebra structures to Feynman categories

Applied the theory to topology, number theory, and physics contexts

Abstract

We develop an algebraic theory of colored, semigrouplike-flavored and pathlike co-, bi- and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in combinatorics, topology, number theory and physics. In particular, we can precisely give conditions for the invertibility of characters that is needed for renormalization in the formulation of Connes and Kreimer. These are met in the relevant examples. In order to construct antipodes, we discuss formal localization constructions and quantum deformations. These allow to define and explain the appearance of Brown style coactions. Using previous results, we can interpret all the relevant coalgebras as stemming from a categorical construction, tie the bialgebra structures to Feynman categories, and apply the developed theory in this setting.