The Second Order Pole over Split Quaternions

TL;DR

This paper extends quaternionic analysis to split quaternions, developing representation-theoretic tools and decompositions for a space of functions under the action of a complexified Lie algebra.

Contribution

It introduces split quaternionic analogues of previous results, constructing a function space with Lie algebra action, and decomposing it into irreducible components with explicit projectors.

Findings

Defined a space of functions with Lie algebra action

Decomposed the space into irreducible components

Constructed equivariant projectors onto each component

Abstract

This is an addition to a series of papers [FL1, FL2, FL3, FL4], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we develop split quaternionic analogues of certain results from [FL4]. Thus we introduce a space of functions ${\cal D}^h \oplus {\cal D}^a$ with a natural action of the Lie algebra $\mathfrak{gl}(2,\mathbb H_{\mathbb C}) \simeq \mathfrak{sl}(4,\mathbb C)$, decompose ${\cal D}^h \oplus {\cal D}^a$ into irreducible components and find the $\mathfrak{gl}(2,\mathbb H_{\mathbb C})$-equivariant projectors onto each of these irreducible components.