Endomorphisms of Hopf Algebras and a Little Bit of Control

Abstract

Firstly this paper summarizes some of the the relations between control theory and coalgebra and Hopf algebra theory, particularly the fact that the cofree coalgebra over a finite rank free module consists precisely of the realizable power series in noncommuting variables. The second half of the paper concerns generalizations of the Hopf algebra of permutations which in turn generalizes the Hopf algebras of noncommutative symmetric functions and the Hopf algebra of quasisymmetric functions.