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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


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Volume 18, Issue 2

Issues

Pseudo-metric 2-step nilpotent Lie algebras

Christian Autenried / Kenro Furutani
  • Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, 2641 Yamazaki, Noda, Chiba (278-8510), Tokyo, Japan
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/ Irina Markina / Alexander Vasiľev
Published Online: 2018-03-27 | DOI: https://doi.org/10.1515/advgeom-2017-0051

Abstract

The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that a 2-step nilpotent Lie algebra is isomorphic to its standard pseudo-metric form, that is a 2-step nilpotent Lie algebra endowed with some standard non-degenerate scalar product compatible with the Lie bracket. This choice of the standard pseudo-metric form allows us to study the isomorphism properties. If the elements of the centre of the standard pseudo-metric form constitute a Lie triple system of the pseudo-orthogonal Lie algebra, then the original 2-step nilpotent Lie algebra admits integer structure constants. Among particular applications we prove that pseudo H-type algebras have bases with rational structure constants, which implies that the corresponding pseudo H-type groups admit lattices.

Keywords: Nilpotent Lie algebra; nilmanifold; H-type Lie algebra; non-degenerate scalar product; isomorphism; Lie triple system; lattice; rational space

MSC 2010: Primary: 15A66; 17B30; 22E25

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About the article


Received: 2015-08-12

Revised: 2016-01-27

Accepted: 2016-05-03

Published Online: 2018-03-27

Published in Print: 2018-04-25


Citation Information: Advances in Geometry, Volume 18, Issue 2, Pages 237–263, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0051.

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