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The Fibonacci Lie algebra is not finitely presented

Dessislava H. Kochloukova, Victor Petrogradsky

Abstract

We prove that the Fibonacci Lie algebra and the related just infinite self-similar Lie algebra are not finitely presented.

The Fibonacci Lie algebra is not finitely presented

Abstract

We prove that the Fibonacci Lie algebra and the related just infinite self-similar Lie algebra are not finitely presented.
Paper Structure (7 sections, 14 theorems, 81 equations)

This paper contains 7 sections, 14 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on homological properties of Lie algebras
  3. Basic homological algebra over associative ring $R$ with unity
  4. Basic homological algebra for Lie algebras
  5. $\mathbb{N}$-graded Lie algebras
  6. Properties of the Fibonacci Lie algebra $L$ and $\widetilde{L}$
  7. Homological argument

Key Result

Lemma 2.1

Let $Q$ be a one dimensional Lie algebra and $V$ be a $U(Q)$-module. Then there is a natural isomorphism $\tau_V : H_1(Q, V) \to H^0(Q, V)$. The naturality means that if $f : V \to W$ is a homomorphism of $U(Q)$-modules then for the induced maps $f_1 : H_1(Q, V) \to H_1(Q, W)$ and $f_2: H^0 (Q, V)

Theorems & Definitions (28)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 18 more