Non‐commutative amoebas

Abstract

The group of isometries of the hyperbolic space H 3 $ \mathbb {H}^3$ is the 3-dimensional group PSL 2 ( C ) $\operatorname{PSL}_2(\mathbb {C})$ , which is one of the simplest non-commutative complex Lie groups. Its quotient by the subgroup SO ( 3 ) ⊂ PSL 2 ( C ) $\operatorname{SO}(3)\subset \operatorname{PSL}_2(\mathbb {C})$ naturally maps it back to H 3 $ \mathbb {H}^3$ . Each fiber of this map is diffeomorphic to the real projective 3-space R P 3 ${\mathbb {R}}{\mathbb {P}}^3$ . The resulting map PSL 2 ( C ) → H 3 $\operatorname{PSL}_2(\mathbb {C})\rightarrow \mathbb {H}^3$ can be viewed as the simplest non-commutative counterpart of the map Log : ( C × ) n → R n $\operatorname{Log}:(\mathbb {C}^\times )^n\rightarrow \mathbb {R}^n$ from the commutative complex Lie group ( C × ) n $(\mathbb {C}^\times )^n$ with the Lagrangian torus fibers that can be considered as a Liouville–Arnold type integrable system. Gelfand, Kapranov and Zelevinsky have introduced amoebas of algebraic varieties V ⊂ ( C × ) n $V\subset (\mathbb {C}^\times )^n$ as images Log ( V ) ⊂ R n $\operatorname{Log}(V)\subset \mathbb {R}^n$ . We define the amoeba of an algebraic subvariety of PSL 2 ( C ) $\operatorname{PSL}_2(\mathbb {C})$ as its image in H 3 $ \mathbb {H}^3$ . The paper surveys basic properties of the resulting hyperbolic amoebas and compares them against the commutative amoebas R n $\mathbb {R}^n$ .