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Fibonacci Lie algebra revisited

Victor Petrogradsky

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.

Fibonacci Lie algebra revisited

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 -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.

Paper Structure

This paper contains 42 sections, 56 theorems, 97 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Growth of algebras
  3. Pivot elements and their relations
  4. Pivot elements
  5. Basic relations
  6. Weight functions and $\mathbf Z^2$-gradings
  7. Weight functions
  8. $\mathbb Z^2$-gradings
  9. Bases of Fibonacci Lie algebra $\mathcal{L}$ and Fibonacci restricted Lie algebra $\mathbf L$
  10. Fibonacci Lie algebra $\mathcal{L}$, and its basis
  11. Fibonacci restricted Lie algebra $\mathbf L$, and its basis
  12. Fibonacci Lie algebra is not PI
  13. Estimates on the weight function
  14. Nillity of the Fibonacci restricted Lie algebra $\mathbf L$
  15. Nillity
  16. ...and 27 more sections

Key Result

Lemma 1

The $q$-dimensions have the following properties:

Figures (4)

  • Figure 1: Standard monomials $W$ of the Fibonacci Lie algebra $\mathcal{L}=\mathop{\mathrm{Lie}}\nolimits(v_1,v_2)$, $p=2$. Point $(a,b)$ is green --- all $W_n\cap \mathcal{L}_{(a,b)}$ are of form $[v_{n-2},W_{n-1}]$, blue --- all are of form $[v_{n-3},W_{n-1}]$, proportionally mixed color --- monomials of both types. The pivot elements are red. The area of circles is proportional to the number of monomials, indicated also by numbers. The red lines pass through the pivot elements. Monomials of the abelian ideal $A$ are below axis $\xi$.
  • Figure 2: Standard monomials $W_N$, where $N=15$, are shown in respective rectangle $\{(\xi,\eta) \mid \lambda^{N-1} < \xi \le\lambda^N, \, -\lambda < \eta< 1\}$. It seems that points belong to two families of parallel lines (Conjecture \ref{['ConWN']})
  • Figure 3: Standard monomials of $W_{\le 15}$ in their rectangles are shown together in one rectangle. The pattern of two families of parallel lines observed for a fixed $W_N$ (Fig. \ref{['Fig2']}) disappears
  • Figure 4: Euler characteristic $\mathbf E({\mathcal{L}},x,y)$. Circle at $(a,b)\in \mathbb Z^2$ is green if the respective coefficient is positive, blue --- otherwise. The area of circles is proportional to the coefficients, indicated also by numbers. The shaded region is the strip for monomials of $\mathcal{L}$. The red lines pass through the pivot elements. The last red line passes through $v_{10}$.

Theorems & Definitions (114)

  • Lemma 1: Pe96
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 104 more