Uniqueness of ad-invariant metrics
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866929509132926976 |
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| author | Conti, Diego del Barco, Viviana Rossi, Federico A. |
| author_facet | Conti, Diego del Barco, Viviana Rossi, Federico A. |
| contents | We consider Lie algebras admitting an ad-invariant metric, and we study the problem of uniqueness of the ad-invariant metric up to automorphisms. This is a common feature in low dimensions, as one can observe in the known classification of nilpotent Lie algebras of dimension $\leq 7$ admitting an ad-invariant metric. We prove that uniqueness of the metric on a complex Lie algebra $\mathfrak{g}$ is equivalent to uniqueness of ad-invariant metrics on the cotangent Lie algebra $T^*\mathfrak{g}$; a slightly more complicated equivalence holds over the reals. This motivates us to study the broader class of Lie algebras such that the ad-invariant metric on $T^*\mathfrak{g}$ is unique.
We prove that uniqueness of the metric forces the Lie algebra to be solvable, but the converse does not hold, as we show by constructing solvable Lie algebras with a one-parameter family of inequivalent ad-invariant metrics. We prove sufficient conditions for uniqueness expressed in terms of both the Nikolayevsky derivation and a metric counterpart introduced in this paper.
Moreover, we prove that uniqueness always holds for irreducible Lie algebras which are either solvable of dimension $\leq 6$ or real nilpotent of dimension $\leq 10$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_16477 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Uniqueness of ad-invariant metrics Conti, Diego del Barco, Viviana Rossi, Federico A. Differential Geometry 53C30 (Primary) 17B30, 17B40, 53C50 (Secondary) We consider Lie algebras admitting an ad-invariant metric, and we study the problem of uniqueness of the ad-invariant metric up to automorphisms. This is a common feature in low dimensions, as one can observe in the known classification of nilpotent Lie algebras of dimension $\leq 7$ admitting an ad-invariant metric. We prove that uniqueness of the metric on a complex Lie algebra $\mathfrak{g}$ is equivalent to uniqueness of ad-invariant metrics on the cotangent Lie algebra $T^*\mathfrak{g}$; a slightly more complicated equivalence holds over the reals. This motivates us to study the broader class of Lie algebras such that the ad-invariant metric on $T^*\mathfrak{g}$ is unique. We prove that uniqueness of the metric forces the Lie algebra to be solvable, but the converse does not hold, as we show by constructing solvable Lie algebras with a one-parameter family of inequivalent ad-invariant metrics. We prove sufficient conditions for uniqueness expressed in terms of both the Nikolayevsky derivation and a metric counterpart introduced in this paper. Moreover, we prove that uniqueness always holds for irreducible Lie algebras which are either solvable of dimension $\leq 6$ or real nilpotent of dimension $\leq 10$. |
| title | Uniqueness of ad-invariant metrics |
| topic | Differential Geometry 53C30 (Primary) 17B30, 17B40, 53C50 (Secondary) |
| url | https://arxiv.org/abs/2103.16477 |