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

Abstract

This work has a twofold purpose - the existence study of closed 2-forms, known as magnetic fields, on 2-step nilpotent Lie groups and generating examples for the non-singular family. At the Lie algebra level, 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 left-invariant 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.