Hopf algebras of endomorphisms of Hopf algebras

In the last decennia two generalizations of the Hopf algebra of symmetric functions have appeared and shown themselves important, the Hopf algebra of noncommutative symmetric functions NSymm and the Hopf algebra of quasisymmetric functions QSymm. It has also become clear that it is important to understand the noncommutative versions of such important structures as Symm the Hopf algebra of symmetric functions. Not least because the right noncommmutative versions are often more beautiful than the commutaive ones (not all cluttered up with counting coefficients). NSymm and QSymm are not truly the full noncommutative generalizations. One is maximally noncommutative but cocommutative, the other is maximally non cocommutative but commutative. There is a common, selfdual generalization, the Hopf algebra of permutations of Malvenuto, Poirier, and Reutenauer (MPR). This one is, I feel, best understood as a Hopf algebra of endomorphisms. In any case, this point of view suggests vast generalizations leading to the Hopf algebras of endomorphisms and word Hopf algebras with which this paper is concerned. This point of view also sheds light on the somewhat mysterious formulas of MPR and on the question where all the extra structure (such as autoduality) comes from. The paper concludes with a few sections on the structure of MPR and the question of algebra retractions of the natural inclusion of Hopf algebras of NSymm into MPR and section of the naural projection of MPR onto QSymm.Comment: 40 pages. Revised and expanded version of a (nonarchived) preprint of 200

Niceness theorems

Many things in mathematics seem lamost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of variables in its many incarnations such as the representing object of the Witt vectors, the direct sum of the rings of representations of the symmetric groups, the free lambda ring on one generator, the homology and cohomology of the classifying space BU, ... . In addition attention is paid to the phenomenon that solutions to universal problems (adjoint functors) tend to pick up extra structure.Comment: 52 page

A universal isomorphism for p-typical formal groups and operations in Brown-Peterson cohomology

AbstractWe construct an abstract isomorphism of p-typical formal groups which is universal for isomorphisms of p-typical formal groups over Z(p)-algebras or characteristic zero rings. Associated to this universal isomorphism is a homomorphism of rings Z[V1, V2, …] → Z[V1, V2, …; T1, T2, …] which (after localization at p) can be identified with the right unit homomorphism ηR: BP∗(pt) → BP∗(BP) of the Hopf-algebra BP∗(BP) of Brown-Peterson (co)homology. We calculate ηR modulo the ideal (T1, T2, …)2. These results are then used to obtain information on some of the operations of Brown-Peterson cohomology

Mathematical knowledge management is needed

In this lecture I discuss some aspects of MKM, Mathematical Knowledge Management, with particuar emphasis on information storage and information retrieval.Comment: Keynote speech at the November, 2003 MKM meeting ar Herriott-Watt, Edinburg, UK. Nine pages, one figur