Uniqueness of ad-invariant metrics
Abstract
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 $\leq 10$.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2021
- DOI:
- 10.48550/arXiv.2103.16477
- arXiv:
- arXiv:2103.16477
- Bibcode:
- 2021arXiv210316477C
- Keywords:
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- Mathematics - Differential Geometry;
- 53C30 (Primary) 17B30;
- 17B40;
- 53C50 (Secondary)
- E-Print:
- 35 pages, 1 table