Split-quaternionic Hermitian Clifford analysis

Abstract

The theory of split-quaternion analysis and split-quaternion geometry is nowadays under full development. In this article, we extend quaternionic Hermitian Clifford analysis to the case of split-quaternions. The split-quaternionic Hermitian Dirac operator is introduced as a self-map of smooth functions defined in domains of with values in the tensor product of the hyperbolic quaternions , the split-quaternions and the Clifford algebra . It is defined by with being the basis of and denoting the twisted Hermitian Dirac operators in the split-quaternionic Clifford algebra whose definition is based on a delicate construction of the split-quaternionic Witt basis. We establish the counterpart of the Cauchy integral formula.

Keywords:

• split-quaternions
• Hermitian Clifford analysis
• Witt basis
• Cauchy integral formula

AMS Subject Classifications:

• 30G35
• 32A26

Additional information

Funding

First author partially supported by the NNSF of China [11071230, 11371337] and RFDP [20123402110068].