Magnetic fields on non-singular 2-step nilpotent Lie groups

TL;DR

This paper investigates the existence and properties of magnetic fields represented by closed 2-forms on non-singular 2-step nilpotent Lie groups, revealing specific obstructions and classifications for such forms.

Contribution

It characterizes when closed 2-forms of type II exist on non-singular Lie algebras and identifies the special cases of H-type Lie groups that admit these forms.

Findings

Only real, complex, and quaternionic Heisenberg groups admit type II forms.

Strong obstructions prevent certain closed 2-forms on non-singular Lie algebras.

Non-existence of uniform magnetic fields under specific conditions.

Abstract

The aim of this work is the study of left-invariant magnetic fields on 2-step nilpotent Lie groups. While the existence of closed 2-forms for which the center is either nondegenerate or in the kernel of the 2-form, is always guaranteed, the existence of closed 2-forms for which the center is isotropic but not in the kernel of the 2-form, is a special situation. These 2-forms are called of type II. We obtain a strong obstruction for the existence on non-singular Lie algebras. Moreover, we prove that the only $H$-type Lie groups admitting closed 2-forms of type II are the real, complex and quaternionic Heisenberg Lie groups of dimension three, six and seven, respectively. We also prove the non-existence of uniform magnetic fields under certain hypotheses. Finally we give a construction of non-singular Lie algebras, proving that in some families of these examples there are no closed 2-form of type II.