Introduction to $PSL_2$ phase tropicalization

Abstract

The usual approach to tropical geometry is via degeneration of amoebas of algebraic subvarieties of an algebraic torus $(\mathbb{C}^*)^n$. An amoeba is logarithmic projection of the variety forgetting the angular part of coordinates, called the phase. Similar degeneration can be performed without ignoring the phase. The limit then is called phase tropical variety, and it is a powerful tool in numerous areas. In the article is described a non-commutative version of phase tropicalization in the simplest case of the matrix group $PSL_2(\mathbb{C})$, replacing here $(\mathbb{C}^*)^n$ in the classical approach.