Nice bases for Lie algebras

TL;DR

This paper investigates the existence and enumeration of nice bases in various classes of Lie algebras, revealing their dependence on algebra structure and underlying field, with explicit counts for low-dimensional cases.

Contribution

It provides new results on the existence and number of nice bases for specific Lie algebra classes, including explicit counts and dependence on the field.

Findings

Number of nice bases computed for Lie algebras up to dimension 3

Existence of nice bases depends on the field of definition

Constructed indecomposable Lie algebras with exactly n nice bases

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.