Two interacting coordinate Hopf algebras of affine groups of formal series on a category.
type: Item
instance of: scholarly article
title: Two interacting coordinate Hopf algebras of affine groups of formal series on a category.
zbMATH Open document ID: 1342.16029
DOI: 10.1155/2013/370618
author: Laurent Poinsot
published in: Algebra
publication date: 2015-12-11T00:00:00Z
review text: Let \(R\) be any ring. There are two group structures on formal series in one variable over \(R\); the first one is \(G=1+xRx\) with the product, the second one is \(H=x+x^2Rx\), with the composition. Moreover, \(H\) acts on \(G\) giving a semidirect product. These three groups are affine, so have a Hopf algebra of coordinates. This is generalized to locally finite categories. If \(C\) is such a category, it admits a large algebra, with a Cauchy-type product, and the set \(G\) of formal series in this large algebra such that the constant term is \(1\) is a group for the product. Under technical conditions, a group \(H\) of substitutions also exists. It is proved that both of these groups are affine, so have a coordinate Hopf algebra, which are free commutative algebras. Moreover, the action of \(H\) over \(G\), inducing a semidirect product, induces, at the level of their coordinate Hopf algebras, a smash coproduct.
reviewed by: Loïc Foissy
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16T30
Mathematics Subject Classification ID: 14L17
Mathematics Subject Classification ID: 20J99
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 6519782
zbMATH Keywords: locally finite categories
zbMATH Keywords: coordinate Hopf algebras of affine groups
zbMATH Keywords: formal series
zbMATH Keywords: smash coproducts
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/370618
OpenAlex ID: W2040226707
Wikidata QID: Q58919618
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description: scientific article; zbMATH DE number 6519782