Characterization of 9-dimensional Anosov Lie algebras

TL;DR

This paper classifies 9-dimensional Anosov Lie algebras, identifying a unique complex 3-step algebra and characterizing 2-step algebras with specific properties, expanding understanding beyond dimension 8.

Contribution

It provides the first classification results for 9-dimensional Anosov Lie algebras, including uniqueness and characterization results for different step cases.

Findings

Existence of a unique complex 3-step 9-dimensional Anosov Lie algebra.

Characterization of 2-step real 9-dimensional Anosov Lie algebras without abelian factors.

Discovery of infinitely many nonisomorphic complex Anosov Lie algebras.

Abstract

The classification of all real and rational Anosov Lie algebras up to dimension 8 is given by Lauret and Will. In this paper we study 9-dimensional Anosov Lie algebras by using the properties of very special algebraic numbers and Lie algebra classification tools. We prove that there exists a unique (up to a Lie algebra isomorphism) complex 3-step Anosov Lie algebra of dimension 9. In the 2-step case, we prove that a 2-step real 9-dimensional Anosov Lie algebra with no abelian factor must have a 3-dimensional derived algebra and we characterize these Lie algebras in terms of their Pfaffian forms. Among these Lie algebras, we have found a family of infinitely many complex nonisomorphic Anosov Lie algebras.