Pseudo-metric 2-step nilpotent Lie algebras

TL;DR

This paper introduces a metric framework for 2-step nilpotent Lie algebras, showing they can be represented in a standard form that facilitates analysis of their isomorphism properties and lattice structures.

Contribution

It establishes that all 2-step nilpotent Lie algebras can be expressed in a standard pseudo-metric form, enabling new insights into their isomorphisms and lattice existence.

Findings

Any 2-step nilpotent Lie algebra is isomorphic to a standard pseudo-metric form.

Pseudo H-type algebras have bases with rational structure constants.

Pseudo H-type groups admit lattices due to rational structure constants.

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.