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Nice bases for Lie algebras

Jonas Deré, Jeroen Gantois

Abstract

The concept of a nice basis for a Lie algebra was introduced to study the Ricci curvature on nilpotent Lie groups equipped with a left-invariant metric. Despite the many applications in differential geometry, for example in the construction of Einstein manifolds, very little is known about the existence and number of nice bases on a given Lie algebra. This paper studies this question for three classes of Lie algebras, namely direct sums, almost abelian ones and nilpotent Lie algebras associated to a graph. As an application we compute the number of nice bases for Lie algebras up to dimension $3$, and show that for a general Lie algebra the existence depends on the field over which it is defined. Moreover, for every natural number $n$ we give an indecomposable Lie algebra such that there exists exactly $n$ nice bases up to equivalence.

Nice bases for Lie algebras

Abstract

The concept of a nice basis for a Lie algebra was introduced to study the Ricci curvature on nilpotent Lie groups equipped with a left-invariant metric. Despite the many applications in differential geometry, for example in the construction of Einstein manifolds, very little is known about the existence and number of nice bases on a given Lie algebra. This paper studies this question for three classes of Lie algebras, namely direct sums, almost abelian ones and nilpotent Lie algebras associated to a graph. As an application we compute the number of nice bases for Lie algebras up to dimension , and show that for a general Lie algebra the existence depends on the field over which it is defined. Moreover, for every natural number we give an indecomposable Lie algebra such that there exists exactly nice bases up to equivalence.
Paper Structure (10 sections, 35 theorems, 173 equations, 1 figure)

This paper contains 10 sections, 35 theorems, 173 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Nice bases for direct sum Lie algebras
  3. Almost abelian Lie algebras
  4. Existence of a nice basis
  5. Number of nice bases
  6. Application: nice bases for 3-dimensional Lie algebras
  7. Nice bases for nilpotent Lie algebras associated to graphs
  8. 3-step nilpotent Lie algebras associated to graphs
  9. 4-step nilpotent Lie algebras
  10. Higher nilpotency classes

Key Result

Lemma 2.2

[lemma]lem. UCS/LCS-adapted If a Lie algebra ${\mathfrak g}$ admits a nice basis $\mathcal{B}$, then $Z_i({\mathfrak g})$ and $\gamma_j({\mathfrak g})$ are spanned by $\mathcal{B}\cap Z_i({\mathfrak g})$ and $\mathcal{B}\cap\gamma_j({\mathfrak g})$ respectively.

Figures (1)

  • Figure 1: Eigenvalues of the pre-Einstein derivation of $L_n$

Theorems & Definitions (73)

  • Definition 1.1
  • Definition 1.2
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Remark 2.5
  • Example 2.6
  • Proposition 2.7
  • proof
  • ...and 63 more