Projective superflows. III. Finite subgroups of $U(2)$

TL;DR

This paper classifies all 2-dimensional complex superflows with symmetry groups as finite subgroups of U(2), extending previous work on real superflows to the complex case, including both irreducible and reducible types.

Contribution

It provides a complete classification of 2D complex superflows with finite U(2) symmetry groups, a significant extension of prior real superflow classifications.

Findings

Classified all 2D complex superflows with finite U(2) symmetry groups.

Included both irreducible and reducible superflows.

Extended the classification from real to complex superflows.

Abstract

Let $X\in\mathbb{R}^{n}$ or $\mathbb{C}^{n}$. For $\phi:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ (respectively, $\phi:\mathbb{C}^{n}\mapsto\mathbb{C}^{n}$) and $t\in\mathbb{R}$ (respectively, $\mathbb{C}$), we put $\phi^{t}=t^{-1}\phi(Xt)$. A projective flow is a solution to the projective translation equation $\phi^{t+s}=\phi^{t}\circ\phi^{s}$, $t,s\in\mathbb{R}$ or $\mathbb{C}$. The projective superflow is a projective flow with a rational vector field which, among projective flows with a given symmetry, is, up to a homothety, unique and optimal. In the first and the second part of this work we classified real $2$ and $3-$dimensional supeflows over $\mathbb{R}$. In this third part we classify all $2-$dimensional complex superflows; that is, whose group of symmetries are finite subgroups of $U(2)$. This includes both irreducible and reducible superflows.