We classify all non-affine Hopf algebras H over an algebraically closed field k of characteristic zero that are integral domains of Gelfand–Kirillov dimension two and satisfy the condition Ext1H(k, k) ≠ 0. The affine ones were classified by the authors in 2010 (Goodearl and Zhang, J. Algebra324 (2010), 3131–3168).

A coassociative Lie algebra is a Lie algebra equipped with a coassociative coalgebra structure satisfying a compatibility condition. The enveloping algebra of a coassociative Lie algebra can be viewed as a coalgebraic deformation of the usual universal enveloping algebra of a Lie algebra. This new enveloping algebra provides interesting examples of non-commutative and non-cocommutative Hopf algebras and leads to the classification of connected Hopf algebras of Gelfand–Kirillov dimension four in Wang et al. (Trans. Amer. Math. Soc., to appear).

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.

It is shown that the mod $3$ cohomology of a $1$-connected, homotopy associative mod $3$$H$-space that is rationally equivalent to the Lie group $E_6$ is isomorphic to that of $E_6$ as an algebra. Moreover, it is shown that the mod $3$ cohomology of a nilpotent, homotopy-associative mod $3$$H$-space that is rationally equivalent to $E_6$, and whose fundamental group localized at $3$ is non-trivial, is isomorphic to that of the Lie group $\Ad E_6$ as a Hopf algebra over the mod $3$ Steenrod algebra. It is also shown that the mod $3$ cohomology of the universal cover of such an $H$-space is isomorphic to that of $E_6$ as a Hopf algebra over the mod $3$ Steenrod algebra.

In this note we generalise a result of D. Husemoller to certain bipolynomial Hopf algebras and are able to give Hopf algebra decompositions for these. As an easy consequence of our approach we give a simplified derivation of recent results of P. Hoffman on polynomial generators for these algebras; we also give explicit systems of “Borel generators” for a related family of quotient Hopf algebras considered by Hoffman.