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Forum Mathematicum

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Hrsg. v. Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


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Band 29, Heft 3

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Non-abelian tensor and exterior products of multiplicative Lie rings

Guram Donadze / Nick Inassaridze
  • A. Razmadze Mathematical Institute of Tbilisi State University, Tamarashvili Str. 6, 0177 Tbilisi, Georgia; Georgian Technical University, Kostava Str. 77, 0175 Tbilisi, Georgia; and Tbilisi Centre for Mathematical Sciences, Tbilisi, Georgia
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/ Manuel Ladra
  • Korrespondenzautor
  • Department of Algebra, IMAT, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain
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Online erschienen: 18.08.2016 | DOI: https://doi.org/10.1515/forum-2015-0096

Abstract

We define a non-abelian tensor product of multiplicative Lie rings. This is a new concept providing a common approach to the non-abelian tensor product of groups defined by Brown and Loday and to the non-abelian tensor product of Lie rings defined by Ellis. We also prove an analogue of Miller’s theorem for multiplicative Lie rings.

Keywords: Multiplicative lie rings; non-abelian tensor and exterior products; homology

MSC 2010: 18G10; 18G50

References

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    Bak A., Donadze G., Inassaridze N. and Ladra M., Homology of multiplicative Lie rings, J. Pure Appl. Algebra 208 (2007), no. 2, 761–777. Google Scholar

  • [2]

    Brown R. and Ellis G. J., Hopf formulae for the higher homology of a group, Bull. Lond. Math. Soc. 20 (1988), no. 2, 124–128. Google Scholar

  • [3]

    Brown R., Johnson D. L. and Robertson E. F., Some computations of non-abelian tensor products of groups, J. Algebra 111 (1987), no. 1, 177–202. Google Scholar

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    Brown R. and Loday J.-L., Van Kampen theorems for diagrams of spaces, Topology 26 (1987), no. 3, 311–335. Google Scholar

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    Cohen D. E. and Lyndon R. C., Free bases for normal subgroups of free groups, Trans. Amer. Math. Soc. 108 (1963), 526–537. Google Scholar

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    Donadze G., Inassaridze N. and Porter T., N-fold Čech derived functors and generalised Hopf type formulas, J. K-Theory 35 (2005), no. 3–4, 341–373. Google Scholar

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    Donadze G. and Ladra M., More on five commutator identities, J. Homotopy Relat. Struct. 2 (2007), no. 1, 45–55. Google Scholar

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    Ellis G. J., Non-abelian exterior products of groups and exact sequences in the homology of groups, Glasg. Math. J. 29 (1987), no. 1, 13–19. Google Scholar

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    Ellis G. J., Non-abelian exterior products of Lie algebras and an exact sequence in the homology of Lie algebras, J. Pure Appl. Algebra 46 (1987), no. 2–3, 111–115. Google Scholar

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    Ellis G. J., The non-abelian tensor product of finite groups is finite, J. Algebra 111 (1987), no. 1, 203–205. Google Scholar

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    Ellis G. J., A non-abelian tensor product of Lie algebras, Glasg. Math. J. 33 (1991), no. 1, 101–120. Google Scholar

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    Ellis G. J., On five well-known commutator identities, J. Aust. Math. Soc. Ser. A 54 (1993), no. 1, 1–19. Google Scholar

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    Inassaridze H., Non-Abelian Homological Algebra and its Applications, Math Appl. 421, Kluwer, Dordrecht, 1997. Google Scholar

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    Miller C., The second homology group of a group; relations among commutators, Proc. Amer. Math. Soc. 3 (1952), 588–595. Google Scholar

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    Point F. and Wantiez P., Nilpotency criteria for multiplicative Lie algebras, J. Pure Appl. Algebra 111 (1996), no. 1–3, 229–243. Google Scholar

Artikelinformationen


Erhalten: 22.05.2015

Revidiert: 18.07.2016

Online erschienen: 18.08.2016

Erschienen im Druck: 01.05.2017


Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2013-43687-P

Funding Source: Shota Rustaveli National Science Foundation

Award identifier / Grant number: FR/189/5-113/14

Funding Source: Consellería de Cultura, Educación e Ordenación Universitaria, Xunta de Galicia

Award identifier / Grant number: GRC2013-045

The authors were supported by Ministerio de Economía y Competitividad (Spain), grant MTM2013-43687-P (European FEDER support included). The first author was also partially supported by PEIN (USC-India) program. The second author was also supported by Shota Rustaveli National Science Foundation, grant FR/189/5-113/14. The third author was also supported by Xunta de Galicia (European FEDER support included), grant GRC2013-045.


Quellenangabe: Forum Mathematicum, Band 29, Heft 3, Seiten 563–574, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0096.

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