One of the best known examples of a polynomial Hopf algebra is associated to the (graded) ring of symmetric functions. It has the remarkable property that it and its dual are commutative and in fact isomorphic as Hopf algebras. It appears in many contexts including representation theory of symmetric groups and the study of Chern classes and multiplicative sequences. It is in fact the cohomology ring of the classifying space $BU$.

A less well known example is obtained from a non-commutative version of this with commutative dual, the ring of quasi-symmetric functions. Again this appears in many contexts and is actually the cohomology of an $H$-space $\Omega\Sigma \mathbb{C}P^\infty$. It was conjectured in the 1970s that this ring was polynomial and after several incomplete proofs, this Ditters conjecture was proved around 2000 by Hazewinkel.

I will explain some of the algebraic background on Hopf algebras, then discuss a strategy for proving this sort of result for cohomology rings of certain loop spaces, generalising an earlier joint proof with Birgit Richter. The methods used involve the Eileneberg-Moore spectral sequence and various standard topological tools.

Speaker's webpage

http://www.maths.gla.ac.uk/~ajb/