An Invitation to Split Quaternionic Analysis

TL;DR

This paper introduces regular functions on split quaternions, establishes two analogues of the Cauchy-Fueter formula, and applies these methods to harmonic analysis on specific Lie groups and spaces.

Contribution

It defines regular functions on split quaternions and proves two new analogues of the Cauchy-Fueter formula, extending quaternionic analysis to split quaternions.

Findings

Established two analogues of the Cauchy-Fueter formula for split quaternionic functions

Applied quaternionic analysis methods to harmonic analysis on SL(2,R) and related spaces

Extended the framework of quaternionic analysis to split quaternions

Abstract

Six years after William Rowan Hamilton's discovery of quaternions, in 1849 James Cockle introduced the algebra of split quaternions. (He called them ``coquaternions.'') In this paper we define regular functions on split quaternions and prove two different analogues of the Cauchy-Fueter formula for these functions. In the paper "Split quaternionic analysis and the separation of the series for SL(2,R) and SL(2,C)/SL(2,R)" joint with Igor Frenkel we naturally apply the methods and formulas of quaternionic analysis to solve the problems of harmonic analysis on SL(2,R) and the imaginary Lobachevski space SL(2,C)/SL(2,R).