Nice bases for Lie algebras
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918393876054016 |
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| author | Deré, Jonas Gantois, Jeroen |
| author_facet | Deré, Jonas Gantois, Jeroen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16540 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nice bases for Lie algebras Deré, Jonas Gantois, Jeroen Differential Geometry 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. |
| title | Nice bases for Lie algebras |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2603.16540 |