Abstract

In the present paper we determine the group of automorphisms of pseudo H-type Lie algebras, that are two step nilpotent Lie algebras closely related to the Clifford algebras Cl(Rr,s).

Highlights

  • Throughout this paper, N denotes an integer and N ≥ 3

  • From the viewpoint of standard vector calculus on RN, we study the functional inequality for vector fields together with its improvement, called the Hardy-Leray inequality

  • Some techniques on R3, employed in the previous work, is not allowed in the higher-dimensional case: we cannot use the “cross product” of vectors in general RN, and there is no way to represent every toroidal field in terms of a single-scalar potential

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Summary

Introduction

Throughout this paper, N denotes an integer and N ≥ 3. Some techniques on R3, employed in the previous work, is not allowed in the higher-dimensional case: we cannot use the “cross product” of vectors in general RN , and there is no way to represent every toroidal field in terms of a single-scalar potential To avoid such a difficulty, we derive with a simple proof the spherical zero-mean property of toroidal fields, from which one can deduce such as a Poincare-type estimate. We review gradient or Laplace operators acting on vector fields on R N and derive some basic formulae, in terms of radial-spherical variables. (2.4) to the case of the spherical gradient field ∇f = r−1∇σf , and we get div∇f = r−2∇σ · ∇σf This together with div∇f = △f = r−2△σf gives the first identity of the lemma.

Poloidal-toroidal fields
Proof of main theorem

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