Pseudo-metric 2-step nilpotent Lie algebras

Christian Autenried; Kenro Furutani; Irina Markina; Alexander Vasil'ev

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arXiv · 1508.02882

Pseudo-metric 2-step nilpotent Lie algebras

Christian Autenried·Kenro Furutani·Irina Markina·Alexander Vasil'ev

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Abstract

The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that any 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 Lie brackets. This choice of the standard pseudo-metric form allows 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 structural constants, which implies that the corresponding pseudo $H$-type groups admit lattices.

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BibTeX RIS

Christian Autenried, Kenro Furutani, Irina Markina, Alexander Vasil'ev. 2015-08-12. Pseudo-metric 2-step nilpotent Lie algebras. https://arxiv.org/abs/1508.02882

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