A Hopf algebra structure on rational functions
type: Item
instance of: scholarly article
title: A Hopf algebra structure on rational functions
zbMATH Open document ID: 0845.16038
author: Luis Verde-Star
published in: Advances in Mathematics
publication date: 1996-09-02T00:00:00Z
review text: Linearly recursive sequences over a field \(k\) appear as the continuous dual \(P^0\) of the Hopf algebra \(P = k[x]\) of polynomials with \(x\) primitive, e.g., see the paper of \textit{B. Peterson} and the reviewer [Aequationes Math. 20, 1-17 (1980; Zbl 0434.16008)]. \(P^0\) is a Hopf algebra with divided power product. As a vector space, \(P^0\) can be identified with rational functions \(p(x)/q(x)\), where \(q(0)\neq 0\), via the usual power-series expansion. In the paper under review, the author derives the same Hopf algebra structure when \(k\) is the field of complex numbers, but by a different procedure. He defines an inner product on the space \(Q\) of rational functions, and considers its restriction to \(P\times R\), where \(R\) is the space of rational functions \(p(x)/q(x)\), where \(\text{degree }p(x)<\text{degree }q(x)\). Then \(R\) embeds in the full linear dual of \(P\), and the Hopf algebra structure on \(P\) yields a Hopf algebra structure on \(R\). The inner product is then used to identify \(R\) with the space of linearly recursive sequences. There is also a discussion of the case when \(x\) is group-like, which yields the Hadamard product on \(R\).
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 863623
zbMATH Keywords: linearly recursive sequences
zbMATH Keywords: Hopf algebras
zbMATH Keywords: divided power product
zbMATH Keywords: rational functions
zbMATH Keywords: inner products
zbMATH Keywords: Hadamard product
reviewed by: Earl J. Taft
MaRDI profile type: Publication
full work available at URL: aima.1995.1072
OpenAlex ID: W1967470162
DOI: 10.1006/AIMA.1995.1072
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description: scientific article; zbMATH DE number 863623