Splittings of bicommutative Hopf algebras.
type: Item
instance of: scholarly article
title: Splittings of bicommutative Hopf algebras.
zbMATH Open document ID: 1066.16041
author: Nitu Kitchloo
author: Gerd Laures
author: W. Stephen Wilson
published in: Journal of Pure and Applied Algebra
publication date: 2004-11-18T00:00:00Z
review text: Let \(A\to B\to C\) be a short exact sequence in the category of bicommutative biassociative Hopf algebras with an exhaustive primitive filtration. If \(A\) is cofree as a coalgebra and \(C\) is polynomial (free) as an algebra, then the sequence splits as Hopf algebras. This is the main result of this paper, and its proof relies heavily on the theory of Morava Dieudonné modules. A Morava Dieudonné module is a \(\mathbb{Z}/(p^n-1)\) graded group \(M_*\) with a \(\mathbb{Z}[V,F]/(VF-p)\)-module structure where \(F\colon M_t\to M_{pt}\) and \(V\colon M_t\to M_{t/p}\). A morphism \(M_*\to N_*\) is a graded module map, hence the set of Morava Dieudonné modules forms a category. The full subcategory consisting of modules with \(V\)-torsion is equivalent to the category of Morava (i.e., evenly graded) Hopf algebras via a functor \(m_*\colon(\text{Morava Hopf algebras})\to(\text{Morava}\) Dieudonné modules with \(V\)-torsion). The main result is thus proved by translating the problem into these Dieudonné modules. An application, in fact the motivation for this study, is found in the study of the fibration \(K(\mathbb{Z},3)\to BO\langle 8\rangle\to BSpin\), leading to the study of the short exact sequence obtained by applying \(K(2) _*\) to each part, where \(K(2)_*\) is the homology theory known as Morava \(K\)-theory. The paper concludes by showing that the group of homomorphisms \(m_*(K(2)_*BSpin)\to K\) is a nontrivial \(2\)-complete torsion-free \(\mathbb{Z}_2\)-module, and hence the space of splittings for the above short exact sequence is uncountable.
reviewed by: Alan Koch
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 55Q51
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 57T05
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16E20
zbMATH DE Number: 2116175
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Morava Dieudonné modules
zbMATH Keywords: splittings
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.04.002
OpenAlex ID: W2111099320
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DOI: 10.1016/J.JPAA.2004.04.002
description: scientific article; zbMATH DE number 2116175