Non-commutative amoebas

TL;DR

This paper explores non-commutative amoebas derived from hyperbolic 3-space isometries, comparing their properties to classical amoebas, and providing a foundational survey of their characteristics.

Contribution

It introduces and surveys the concept of non-commutative amoebas, establishing their basic properties and relation to classical amoebas in a non-commutative setting.

Findings

Non-commutative amoebas are related to hyperbolic 3-space isometries.

They serve as a non-commutative analogue of classical amoebas.

Comparison reveals both similarities and differences with commutative amoebas.

Abstract

The group of isometries of the hyperbolic 3-space is one of the simplest non-commutative complex Lie groups. Its quotient by the maximal compact subgroup naturally maps it back to the hyperbolic space. Each fiber of this map is diffeomorphic to the real projective 3-space. The resulting map can be viewed as the simplest non-commutative counterpart of the amoeba map introduced, in the commutative setting, by Gelfand, Kapranov and Zelevinsky. The paper surveys basic properties of the non-commutative amoebas and compares them against their commutative counterparts.