Tohoku Mathematical Journal
2024
September
SECOND SERIES VOL. 76, NO. 3

Tohoku Math. J.
76 (2024), 317-359

Title UNIQUENESS OF AD-INVARIANT METRICS

Author Diego Conti, Viviana del Barco and Federico A. Rossi

(Received February 25, 2022, revised September 14, 2022)
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 $\leq 10$.

Mathematics Subject Classification. Primary 53C30; Secondary 17B30, 17B40, 53C50.
Key words and phrases. Ad-invariant metric, metric Lie algebras, orbits of the group of automorphisms, cotangent Lie algebras.

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