IJPAM: Volume 55, No. 4 (2009)

VARIOUS $S$-ADIC SYMMETRIC FUNCTIONS
AND SMOOTH FORMAL GROUPS

E.J. Ditters
Faculteit der Exacte Wetenschappen
Vrije Universiteit
1083, de Boelelaan, Amsterdam, 1081 HV, THE NETHERLANDS
e-mail: ejd@few.vu.nl


Abstract.This note (expanded version of an oral exposition, Plovdiv, 2007) continues the program started in [#!Di72!#]: find a description of not necessarily commutative smooth formal group theory reducing to the known theories in the commutative case. The graded Leibniz Hopf algebra $\SZ$, better known as the Hopf algebra of noncommutative symmetric functions seems to be a natural tool for these purposes: $\SZ$ is a generator for the category of smooth formal groups. By Cartier duality, the graded dual Hopf algebra $\QSym$ is a cogenerator in the dual category $\FGL$. For formal groups over a field containing $\QQ$, we refind the theory as exposed in [#!Ser!#]. For commutative formal groups our results are identical with those found in 1967 by P. Cartier, [#!Car67a!#] and [#!Car67b!#] - modulo isomorphism of the Hopf algebra $\Lambda$ of commutative symmetric functions with the Hopf algebra $W$ of the Witt vectors. The methods of Cartier: curves and their operators, Witt vectors, typification all have their counterpart for $\SZ$ and its graded dual $\QSym$. We present a new (and unique) $\ZZ_S$-module basis for the primitive elements in the $S$-localizations of $\SZ^{(J)}$ and free polynomial generators for their graded duals.

Received: June 27, 2009

AMS Subject Classification: 05E05, 14L05, 16W30

Key Words and Phrases: SDP-Hopf algebra, Lie algebra, smooth formal group (law), co(ntra)variant bialgebra, sequence of divided powers, primitive element, pure primitive, curve, symmetric (quasisymmetric and noncommutative symmetric) function, composition, Lyndon composition

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 55
Issue: 4