Slice Dirac operator over octonions

TL;DR

This paper introduces the slice Dirac operator over octonions, extending the theory of stem functions from quaternions to octonions, and establishes fundamental formulas for functions in its kernel.

Contribution

It develops a new framework for the slice Dirac operator over octonions, including representation, integral, and series expansion formulas, advancing non-commutative analysis.

Findings

Established the representation formula for kernel functions.

Derived the Cauchy integral and Cauchy-Pompeiu formulas.

Developed Taylor and Laurent series expansions for kernel functions.

Abstract

The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative $ O(1) $ case to the non-commutative $ O(3) $ case. For functions in the kernel of the slice Dirac operator over octonions, we establish the representation formula, the Cauchy integral formula (and, more in general, the Cauchy-Pompeiu formula), and the Taylor as well as the Laurent series expansion formulas.