SOME HOPF ALGEBRAS OF WORDS

Abstract

A number of integral Hopf algebras have been studied that have, as their underlying modules, the free ${\bf Z}$-module generated by finite words in a certain alphabet. For example, the tensor algebra, the rings of quasisymmetric functions and of noncommutative symmetric functions, the Solomon descent algebra, the Malvenuto-Reutenauer algebra and the homology and cohomology of $\Omega\Sigma{\bf C} P^{\infty}$ are all of this type. Some of these are known to be isomorphic or dual to each other, some are known only to be rationally isomorphic, some have been stated in the literature to be isomorphic when they are only rationally isomorphic.

This paper is, in part, an attempt to find order in this chaos of word Hopf algebras. We consider three multiplications on such modules, and their dual comultiplications, and clarify which of these operations can be combined to obtain Hopf structures. We discuss when the results are isomorphic, integrally or rationally, and study the resulting structures. We are not attempting a classification of Hopf algebras of words, merely an organization of some of the Hopf algebras of this type that have been studied in the literature.