Endomorphisms of the binomial coalgebra.
type: Item
instance of: scholarly article
title: Endomorphisms of the binomial coalgebra.
zbMATH Open document ID: 1064.16034
author: Mériem Héraoua
author: Alain Salinier
published in: Journal of Pure and Applied Algebra
publication date: 2004-08-19T00:00:00Z
review text: Let \(R\) be a commutative unital ring and \(B_1\) a free \(R\)-module with an infinitely countable base \(e_n\), \(n\geqslant 0\). Then \(B_1\) is a coalgebra with comultiplication \[ \Delta(e_n)=\sum_{i=0}^n{n\choose i}e_i\otimes e_{n-i} \] and counit \(\varepsilon(e_n)=\delta_{n0}\). For any prime \(p\) define a function \(s_p\) on the set of non-negative integers such that \(s_p(0)=0\) and \(s_p(lp+m)=s_p(l)+m\) where \(0\leqslant m<p\). If \(a\) is a non-negative integer then denote by \(P_a\) the submodule of \(B_1\) spanned by all \(e_n\) with \(s_p(n)=a\). If \(\varphi\) is an \(R\)-module endomorphism of \(R_1\) and \(x\in P_a\) then put \[ \varphi(a)=\sum_{b\geqslant 0}\varphi_a^b(x),\quad \varphi_a^b(x)\in P_b. \] The collection of maps \(\varphi_a^b(x)\) defines a map \(\Theta\) from \(R\)-coalgebra endomorphisms of \(B_1\) to \(\prod_{l\geqslant 1}\hom(P_l,P_1)\), namely, \(\Theta(\phi)=(\phi^1_l)_{l\geqslant 1}\). If \(R\) is a reduced ring of prime characteristic \(p\) then \(\Theta\) is bijective. Let \(HR=R^{\mathbb{N}\cup 0}\) with Hurwitz multiplication \((gh)_n=\sum_{k=0}^ng_kh_n\). There is a bijective map from the set of continuous endomorphisms from \(HR\) to \(\prod_{l\geqslant 1}\hom(P_l,P_1)\) sending \(\chi\) to \(\Theta(\varphi)\) where \(\varphi\) is the endomorphism of the coalgebra \(B_1\) which is adjoint to \(\chi\).
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 05A40
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 2092912
zbMATH Keywords: binomial coalgebras
zbMATH Keywords: coalgebra endomorphisms
zbMATH Keywords: continuous endomorphisms
zbMATH Keywords: Hurwitz multiplication
zbMATH Keywords: algebras of formal Hurwitz series
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.03.002
OpenAlex ID: W1970248369
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DOI: 10.1016/J.JPAA.2004.03.002
description: scientific article; zbMATH DE number 2092912