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

Abstract

We develop an algebraic theory of colored, semigroup-like-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 are needed for renormalization in the formulation of Connes and Kreimer. These are met in the relevant examples. To construct antipodes, we discuss formal localization constructions and quantum deformations. These allow us 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.