Table of Contents
Algebra
Volume 2013, Article ID 370618, 10 pages
http://dx.doi.org/10.1155/2013/370618
Research Article

Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

Université Paris 13, Sorbonne Paris Cité, LIPN, CNRS (UMR 7030), 93430 Villetaneuse, France

Received 19 March 2013; Revised 17 May 2013; Accepted 14 July 2013

Academic Editor: Stefaan Caenepeel

Copyright © 2013 Laurent Poinsot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A locally finite category is defined as a category in which every arrow admits only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some fields may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover, both groups are actually affine groups. In this contribution, we introduce their coordinate Hopf algebras which are both free as commutative algebras. The semidirect product structure obtained from the action of reversible series on invertible series by anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct.