Pathlike co/bialgebras and their antipodes with applications to bi- and Hopf algebras appearing in topology, number theory and physics (Q2162652)

The authors introduce the notions of colored (Def. 2.14, p. 10), sg-flavored (Def. 3.11, p. 16) and pathlike (Def. 3.19, p. 18) coalgebras over an arbitrary unital commutative ground ring, together with the notion of QT-filtration (Def. 3.1, p. 13) as a generalization of the usual filtrations used for Hopf or bi-algebras. This makes it possible to provide a notion of (color) connectedness, used to characterize under which conditions characters are invertible (with respect to the convolution product), see e.g., Theorem 3.3 (p. 14) and Theorem 3.21 (p. 19). It is also shown (Theorem 4.14, p. 24) how to provide a Hopf algebra from a pathlike bialgebra, so essentially how to get an antipode, either by quotienting by a bi-ideal or by localizing the bialgebra at its group-like elements (that is, by adding formal inverses to them).