From Hermitean Clifford Analysis to Subelliptic Dirac Operators on Odd Dimensional Spheres and Other CR Manifolds

P Cerejeiras,J Ryan,U Kähler

doi:10.1007/978-3-030-00049-3_3

Abstract

We show that the two Dirac operators arising in Hermitian Clifford analysis are identical to standard differential operators arising in several complex variables. We also show that the maximal subgroup that preserves these operators are generated by translations, dilations and actions of the unitary n-group. So the operators are not invariant under Kelvin inversion. We also show that the Dirac operators constructed via two by two matrices in Hermitian Clifford analysis correspond to standard Dirac operators in euclidean space. In order to develop Hermitian Clifford analysis in a different direction we introduce a sub elliptic Dirac operator acting on sections of a bundle over odd dimensional spheres. The particular case of the three sphere is examined in detail. We conclude by indicating how this construction could extend to other CR manifolds.

Highlights

Clifford analysis started as an attempt to generalize one variable complex analysis to n-dimensional euclidean space
It has since evolved into a study of the analyst, geometry and applications of Dirac operators over euclidean space, spheres, real projective spaces, conformally flat spin manifolds and spin manifolds with applications to representation theory arising from mathematical physics, classical harmonic analysis and many other topics
An almost complex structure is introduced and two associated projection operators are applied to this complex vector space. This splits this space into two n-dimensional complex spaces. When these operators are applied to the euclidean Dirac operator it splits into two differential operators

Summary

Introduction

Clifford analysis started as an attempt to generalize one variable complex analysis to n-dimensional euclidean space. It has since evolved into a study of the analyst, geometry and applications of Dirac operators over euclidean space, spheres, real projective spaces, conformally flat spin manifolds and spin manifolds with applications to representation theory arising from mathematical physics, classical harmonic analysis and many other topics. The Euclidean Dirac operator is a conformally invariant operator In particular it is invariant under Kelvin inversion. We show that these Hermitian Dirac operators have a narrower range of invariance. We show they are no longer invariant under Kelvin inversion. We conclude by illustrating how this construction might carry over to other CR manifolds

Preliminaries on Hermitean Clifford Analysis

Hermitean Clifford Analysis and matrix differential operators

Conformal transformations in the Hermitean case

Realization on S3

Subelliptic Dirac operator