Fibonacci Lie algebra revisited

TL;DR

This paper explores the properties and growth behavior of the Fibonacci Lie algebra, providing new bounds, non-PI results, and geometric insights, along with homology and Euler characteristic analysis.

Contribution

It offers new bounds on growth, nilpotency, and homology, and introduces geometric methods to analyze the Fibonacci Lie algebra's structure and properties.

Findings

Polynomial growth is non-uniform.

Fibonacci Lie algebra is not PI.

Chaotic behavior of Euler characteristic.

Abstract

We describe old and prove new results on properties of the Fibonacci Lie algebra in a self-contained exposition. First, we study the growth of this algebra in more details. So, we show that the polynomial behaviour of the growth function in not uniform. We establish bounds on the growth of its universal enveloping algebra. We find bounds on nilpotency indices for elements of the Fibonacci restricted Lie algebra. We prove that the Fibonacci Lie algebra is not PI. Our approach is also based on geometric ideas. The Fibonacci Lie algebra is $\mathbb Z^2$-graded and its homogeneous components belong to a strip. We illustrate the results by computing the initial components and show positions of the homogeneous components in the strip. We also discuss properties and conjectures on related associative algebras and (restricted) Poisson algebras. Second, we prove infiniteness results on homology and Euler characteristic of arbitrary finitely generated graded Lie algebras of subexponential growth. In case of the Fibonacci Lie algebra, we find bounds on the homology groups and the Euler characteristic, and determine respective positions on plane. The computation of the initial part of the Euler characteristic of the Fibonacci Lie algebra shows that its behaviour is really chaotic. Finally, we formulate results and conjectures on homology groups and infinite presentation.