On a class of automorphisms in $\mathbb{H}^2$ which resemble the property of preserving volume

TL;DR

This paper explores quaternionic automorphisms in two variables, extending shears and overshears, defining volume-preserving automorphisms without a quaternionic volume form, and analyzing their properties within Anderson-Lempert theory.

Contribution

It introduces a novel class of quaternionic automorphisms and extends the Anderson-Lempert theory to this non-commutative setting.

Findings

Defined quaternionic shears and overshears in two variables.

Identified a class of volume-preserving automorphisms in quaternionic space.

Provided an example outside the closure of finite compositions of volume-preserving shears.

Abstract

We give a possible extension for shears and overshears in the case of two non commutative (quaternionic) variables in relation with the associated vector fields and flows. We present a possible definition of volume preserving automorphisms, even though there is no quaternionic volume form on $\mathbb{H}^2$ . Using this, we determine a class of quaternionic automorphisms for which the Ander- sen-Lempert theory applies. Finally, we exhibit an example of a quaternionic automor- phism, which is not in the in the closure of the set of finite compositions of volume preserving quaternionic shears.