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تسجيل مستخدم جديدUniqueness of ad-invariant metrics
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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 $leq 10$.
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