Non-abelian tensor and exterior products of multiplicative Lie rings
type: Item
instance of: scholarly article
title: Non-abelian tensor and exterior products of multiplicative Lie rings
zbMATH Open document ID: 1366.18013
author: Guram Donadze
author: N. Inassaridze
author: M. Ladra
published in: Forum Mathematicum
publication date: 2017-05-03T00:00:00Z
review text: The main goal of the paper is the introduction of the notion of non-abelian tensor product of multiplicative Lie rings and to show that this concept generalizes the corresponding notions of non-abelian tensor product of groups in [\textit{R. Brown} and \textit{J.-L. Loday}, Topology 26, 311--335 (1987; Zbl 0622.55009)] and for Lie algebras [\textit{G. J. Ellis}, Glasg. Math. J. 33, No. 1, 101--120 (1991; Zbl 0724.17016)], together with the properties of these tensor products. After that, an analogue of Miller's theorem for free multiplicative Lie rings is proved and a relation between the second homology and the non-abelian exterior square of multiplicative Lie rings is obtained. The paper concludes with a question about the description of the third homology group of a multiplicative Lie ring in terms of non-abelian exterior product. The authors provide the following partial answer to this question: Let \(1 \to \mathfrak{r} \to \mathfrak{f} \to \mathfrak{g} \to 1\) be a free presentation of a multiplicative Lie ring \(\mathfrak{g}\). Assume that \(\mathrm{Ker}(\mathfrak{r}' \wedge \mathfrak{f}' \to \mathfrak{f}' \wedge \mathfrak{f}') = 1\) for all free presentations \(1 \to \mathfrak{r}' \to \mathfrak{f}' \to \mathfrak{f} \to 1\). Then there exists an isomorphism \(HS_3^{\mathrm{mlr}}( \mathfrak{g} ) \cong \text{Ker} (\mathfrak{r} \wedge \mathfrak{f} \to \mathfrak{f} \wedge \mathfrak{f})\), where \(HS_3^{\mathrm{mlr}}( \mathfrak{g} )\) denotes the third strong homology group of \(\mathfrak{g}\).
reviewed by: J. M. Casas
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 18G50
Mathematics Subject Classification ID: 17D99
zbMATH DE Number: 6710803
zbMATH Keywords: multiplicative Lie rings
zbMATH Keywords: non-abelian tensor product
zbMATH Keywords: non-abelian exterior product
zbMATH Keywords: homology
MaRDI profile type: Publication
full work available at URL: forum-2015-0096
OpenAlex ID: W2509264565
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DOI: 10.1515/FORUM-2015-0096
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description: scientific article; zbMATH DE number 6710803