\(R\)-analogue of the Burnside ring of profinite groups and free Lie algebras
type: Item
instance of: scholarly article
title: \(R\)-analogue of the Burnside ring of profinite groups and free Lie algebras
zbMATH Open document ID: 1105.19001
author: Young-Tak Oh
published in: Advances in Mathematics
publication date: 2005-01-11T00:00:00Z
review text: Let \(G\) be a profinite group. The author generalizes results of \textit{A. W. M. Dress} and \textit{C. Siebeneicher} [Adv. Math. 70, 87--132 (1988; Zbl 0691.13026); ibid. 78, 1--41 (1989; Zbl 0691.13020)]. If \(R\) is a finite-dimensional torsionfree special \(\lambda\)-ring, he constructs an \(R\)-analogue of the completeted Burnside ring of profinite groups, say \(\Omega_R(G)\). [In the corrigendum a new definition of \(\Omega_R(G)\) is given.] He remarks that the Lie-module denominator identity for free Lie algebras [\textit{Y.-T. Oh}, Adv. Math. 205, 434--486 (2006; Zbl 1105.19002)] is closely related to the canonical isomorphism between \(\Omega_R(G)\) and the Grothendieck ring of formal power series with coefficients in \(R\) and constant term 1. Some identities in this paper can be interpreted in combinatorial form by the \(R\)-necklace ring of \(G\) [loc. cit. (2006)].
reviewed by: Bernd Richter
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05E99
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 18F30
zbMATH DE Number: 2126443
zbMATH Keywords: Special \(\lambda\)-ring
zbMATH Keywords: Burnside ring
zbMATH Keywords: free Lie algebra
Wikidata QID: Q115362152
MaRDI profile type: Publication
full work available at URL: j.aim.2003.10.004
OpenAlex ID: W2028889053
cites work: Group representations, \(\lambda\)-rings and the \(J\)-homomorphism
cites work: Cohomology of modules in the principal block of a finite group
cites work: Witt vectors and Tambara functors
cites work: Free Lie algebras and formal power series
cites work: The Burnside ring of profinite groups and the Witt vector construction
cites work: The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, \(\lambda\)-rings, and the universal ring of Witt vectors
cites work: Graded Lie superalgebras, supertrace formula, and orbit Lie superalgebras.
cites work: \(\lambda\)-rings and the representation theory of the symmetric group
cites work: Witt vectors and the algebra of necklaces
cites work: Free Lie algebra and lambda-ring structure
cites work: \(\lambda\)-Ringstrukturen auf dem Burnsidering der Permutationsdarstellungen einer endlichen Gruppe
DOI: 10.1016/J.AIM.2003.10.004
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description: scientific article; zbMATH DE number 2126443