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 flavored and pathlike co-, bi-and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in topology, number theory and physics. In particular, we can precisely give conditions the invertibility of characters needed for renormalization in the formulation of Connes and Kreimer and in the canonical examples these conditions are met. To construct the antipodes directly, we discuss formal localization constructions and quantum deformations. These allow to construct and naturally explain the appearance Brown style coactions. Using previous results, we can tie all the relevant coalgebras to categorical constructions and the bialgebra structures to Feynman categories.