  A number of problems in theoretical physics share a common nucleus of
combinatoric nature. It is argued here that Hopf algebraic concepts and
techiques can be particularly efficient in dealing with such problems. As a
first example, a brief review is given of the recent work of Connes, Kreimer
and collaborators on the algebraic structure of the process of renormalization
in quantum field theory. Then the concept of $k$-primitive elements is
introduced -- these are particular linear combinations of products of Feynman
diagrams -- and it is shown, in the context of a toy-model, that they
significantly reduce the computational cost of renormalization.
  As a second example, Sorkin's proposal for a family of generalizations of
quantum mechanics, indexed by an integer $k>2$, is reviewed (classical
mechanics corresponds to $k=1$, while quantum mechanics to $k=2$). It is then
shown that the quantum measures of order $k$ proposed by Sorkin can also be
described as $k$-primitive elements of the Hopf algebra of functions on an
appropriate infinite dimensional abelian group.
