The groups of two by two matrices in double and dual numbers, and associated M\"obius transformations

TL;DR

This paper explores M"obius transformations over the rings of double and dual numbers, extending classical complex analysis to these algebraic structures and classifying their continuous subgroups.

Contribution

It introduces the classification of continuous one-parameter subgroups of general linear and special linear groups over double and dual numbers, which are not fields.

Findings

Classification of continuous one-parameter subgroups over double and dual numbers

Extension of M"obius transformations to non-field rings

New insights into algebraic structures related to these transformations

Abstract

M\"obius transformations have been studied over the field of complex numbers. In this paper, we investigate M\"obius transformations over two rings which are not fields: the ring of double numbers and the ring of dual numbers. We give types of continuous one-parameter subgroups of $\GL{\fpower{\BR}},$ $\SL{\fpower{\BR}},$ $\GL{\BD},$ and $\SL{\BD}.$