Some calculations of \(\mathrm{Lie}(n)^{\max}\) for low \(n\)
type: Item
instance of: scholarly article
title: Some calculations of \(\mathrm{Lie}(n)^{\max}\) for low \(n\)
zbMATH Open document ID: 1152.55003
published in: Journal of Pure and Applied Algebra
publication date: 2008-09-04T00:00:00Z
review text: Let \(V\) be an \(n\)-dimensional vector space over a field \(k\). Then \(\text{Lie}(n)\) is the linear span within the \(n\)-fold tensor product of \(V\) of all \(n\)-fold iterated commutators in which each basis element of \(V\) appears (exactly) once. This is naturally a module over the group ring of the \(n\)-th symmetric group. This module appears in important work of topologists related to the homology of configuration spaces, cf. \textit{F. R. Cohen} [Contemp. Math. 188, 49--55 (1995; Zbl 0849.55015)], and to Goodwillie towers, cf. \textit{G. Arone} and \textit{M. Mahowald} [Invent. Math. 135, No. 3, 743--788 (1999; Zbl 0997.55016)]. The object of this paper is to understand the maximum projective submodule \(\text{Lie}(n)^{max}\) of \(\text{Lie}(n)\) in some cases. The driving motivation here is to understand the factor \(A(V)^{min}\) which appears in a natural coalgebra decomposition of the tensor algebra \(T(V)\) and which was constructed and studied by the authors in \textit{P. Selick} and \textit{J. Wu} [Mem. Am. Math. Soc. 701 (2000; Zbl 0964.55012)]. The factor \(A(V)^{min}\) is the minimum natural coalgebra retract of \(T(V)\) which contains \(V\), and its study depends heavily on properties of the module \(\text{Lie}(n)\) and its submodule \(\text{Lie}(n)^{max}\). Note that when \(n\) is prime to the characteristic of \(k\) then \(\text{Lie}(n)\) is itself projective and \(\text{Lie}(n)^{max}=\text{Lie}(n)\). The focus of this paper is the calculation of \(\text{Lie}(n)^{max}\) for \(n < 10\) and char\((k)=2\). Since \(\text{Lie}(n)^{max}=0\) for \(n=2,4\) (\S1) or when \(n\) is odd (see above), the interesting and difficult task is to understand \(\text{Lie}(6)^{max}\) and \(\text{Lie}(8)^{max}\). Their respective dimensions are found to be \(96\) and \(4224\). This dimension count is just the start of extensive and laborious calculations where the modules in question are decomposed into explicit indecomposable projective submodules; their dimension…
reviewed by: Sadok Kallel
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 5321431
zbMATH Keywords: tensor algebra
zbMATH Keywords: Lie algebra
zbMATH Keywords: symmetric group
Wikidata QID: Q115345599
author: Paul Selick
author: Jie Wu
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.04.011
OpenAlex ID: W2066736287
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DOI: 10.1016/J.JPAA.2008.04.011
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description: scientific article; zbMATH DE number 5321431