[Paper Review] Uniqueness of ad-invariant metrics

Diego Conti, Viviana del Barco+1 more|arXiv (Cornell University)|Mar 30, 2021

Geometry and complex manifolds32 references4 citations

TL;DR

This paper investigates the uniqueness of ad-invariant metrics on Lie algebras up to automorphisms, establishing that a complex Lie algebra has a unique such metric if and only if its cotangent algebra does. The key contribution is proving that uniqueness of the metric on the cotangent algebra forces the original Lie algebra to be solvable, and providing sufficient conditions via the Nikolayevsky derivation and a new metric counterpart; uniqueness holds for irreducible solvable Lie algebras of dimension ≤6 and real nilpotent ones of dimension ≤10.

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 $\leq 10$.

Motivation & Objective

• To determine when a Lie algebra admits a unique ad-invariant metric up to automorphisms and scaling.
• To establish a connection between the uniqueness of ad-invariant metrics on a Lie algebra and its cotangent algebra.
• To identify sufficient conditions for uniqueness using the Nikolayevsky derivation and a newly introduced metric counterpart.
• To classify cases where uniqueness holds, particularly for irreducible solvable and nilpotent Lie algebras.
• To explore the behavior of the solitary property under reductions, complexifications, and cotangent constructions.

Proposed method

• Introduce the concept of a 'solitary' metric, where every self-adjoint ad-invariant map is the self-adjoint part of a derivation.
• Prove that a Lie algebra is solitary if and only if its cotangent algebra is solitary, linking the uniqueness problem to the cotangent construction.
• Use the double extension procedure to analyze the structure of Lie algebras with ad-invariant metrics.
• Apply the Nikolayevsky derivation to derive sufficient conditions for uniqueness, particularly when all eigenvalues are positive.
• Employ the notion of 'nice bases' with σ-diagonal metrics to compute the Nikolayevsky derivation explicitly in low-dimensional cases.
• Use cohomogeneity-one actions of the automorphism group to classify cases with one or two metrics up to scaling.

Experimental results

Research questions

• RQ1 When is the space of ad-invariant metrics on a Lie algebra unique up to automorphisms and scaling?
• RQ2 How does the uniqueness of ad-invariant metrics on a Lie algebra relate to that on its cotangent algebra?
• RQ3 What conditions on the Nikolayevsky derivation or a new metric counterpart ensure uniqueness of ad-invariant metrics?
• RQ4 For which classes of Lie algebras—especially solvable or nilpotent—does uniqueness of the ad-invariant metric hold?
• RQ5 Can the solitary property be preserved under complexification, reduction, or cotangent extension?

Key findings

• A complex Lie algebra has a unique ad-invariant metric up to automorphisms if and only if its cotangent algebra has a unique such metric.
• Uniqueness of the ad-invariant metric on a Lie algebra implies that the algebra is solvable, though the converse does not hold, as shown by constructing solvable Lie algebras with a one-parameter family of inequivalent metrics.
• For irreducible solvable Lie algebras of dimension ≤6 or real nilpotent Lie algebras of dimension ≤10, the ad-invariant metric is unique up to automorphisms and scaling.
• The metric Nikolayevsky derivation coincides with the standard Nikolayevsky derivation on nice Lie algebras, and when all eigenvalues are positive, uniqueness of the metric is guaranteed.
• The solitary property is preserved under complexification and reduction: a real Lie algebra is solitary if and only if its complexification is, and a reducible algebra is solitary iff its factors are.
• The cotangent algebra of any Lie algebra admits an ad-invariant metric, and the solitary property of the original algebra is equivalent to that of its cotangent algebra.