The lower central series of the free partially commutative group
type: Item
instance of: scholarly article
title: The lower central series of the free partially commutative group
zbMATH Open document ID: 0814.20025
DOI: 10.1007/BF03025778
author: Daniel Krob
author: G. H. E Duchamp
published in: Semigroup Forum
publication date: 1993-03-01T00:00:00Z
full work available at URL: https://eudml.org/doc/135200
review text: Let \(\theta\) be a subset of \(A \times A\). The free partially commutative group \(F(A,\theta)\) is the quotient of the free group generated by \(A\) by the relations \(ab = ba\), \((a,b) \in \theta\). The Magnus transformation \(\mu : F(A,\theta) \to \mathbb{Z} \langle\langle A,\theta \rangle\rangle\) is the group homomorphism which sends \(a\) onto \(1 + a\), where \(\mathbb{Z} \langle\langle A,\theta \rangle\rangle\) is the \(\mathbb{Z}\)-algebra of series in the variables \(a \in A\), subject to the same relations. Let \((F_ n(A,\theta))\) be the lower central series of \(F(A,\theta)\). The direct sum \(\bigoplus_{n \in \mathbb{N}} F_ n(A,\theta)/F_{n + 1}(A,\theta)\) has a natural structure of a Lie algebra, and is isomorphic to the free partially commutative Lie algebra \(L(A,\theta)\). An element \(g\) of \(F(A,\theta)\) is in \(F_ n\) if and only if \(\mu(g) = 1 + O(A^ n)\); \(F(A,\theta)\) has an order compatible with its group structure.
reviewed by: Christophe Reutenauer
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F60
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 126119
zbMATH Keywords: commutator relations
zbMATH Keywords: free partially commutative group
zbMATH Keywords: free group
zbMATH Keywords: Magnus transformation
zbMATH Keywords: lower central series
zbMATH Keywords: free partially commutative Lie algebra
MaRDI profile type: Publication
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OpenAlex ID: W2017461533
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description: scientific article; zbMATH DE number 126119