[Paper Review] Free Nilpotent Lie Algebras Admitting Ad-Invariant Metrics

Gabriela P. Ovando, Viviana del Barco|arXiv (Cornell University)|Apr 25, 2011

Advanced Topics in Algebra13 references3 citations

TL;DR

This paper determines the precise conditions under which free nilpotent and free metabelian nilpotent Lie algebras admit ad-invariant metrics—showing that only the 2-step nilpotent algebra in 3 generators and the 3-step nilpotent algebra in 2 generators admit such metrics. The authors use structural properties of free nilpotent Lie algebras and the ad-invariance condition to derive these results, and further characterize the automorphism groups of these two algebras, identifying their orthogonal subgroups and derivations via Lie algebra decompositions.

ABSTRACT

In this work we find necessary and sufficient conditions for a free nilpotent or a free metabelian nilpotent Lie algebra to be endowed with an ad-invariant metric. For such nilpotent Lie algebras admitting an ad-invariant metric the corresponding automorphisms groups are studied.

Motivation & Objective

• To determine necessary and sufficient conditions for free nilpotent and free metabelian nilpotent Lie algebras to admit ad-invariant metrics.
• To resolve the open problem of whether non-semisimple Lie algebras can support ad-invariant metrics, focusing on free nilpotent families.
• To characterize the automorphism groups and derivations of the two Lie algebras that admit such metrics, particularly their orthogonal and skew-symmetric components.
• To provide a structural classification of ad-invariant metrics in free nilpotent Lie algebras beyond known recursive constructions.

Proposed method

• Leveraged the definition of ad-invariant metrics via the condition ⟨[x,y],z⟩ + ⟨y,[x,z]⟩ = 0 for all x,y,z in the Lie algebra.
• Used the intrinsic structure of free k-step nilpotent Lie algebras on m generators, particularly their commutator series and dimension formulas.
• Applied the fact that 2-step solvable Lie algebras with ad-invariant metrics are nilpotent and at most 3-step, restricting the search space.
• Analyzed the Lie algebra of derivations of the two candidate algebras, decomposing them into semisimple and solvable parts via matrix representations.
• Identified the action of sl(2,K) and sl(3,K) on the respective algebras and their representations on the derived and center components.
• Used matrix realizations of derivations and automorphisms to classify orthogonal and skew-symmetric derivations, especially in the context of the ad-invariant metric.

Experimental results

Research questions

• RQ1 For which pairs (m,k) does the free k-step nilpotent Lie algebra on m generators admit an ad-invariant metric?
• RQ2 Do the free metabelian k-step nilpotent Lie algebras on m generators admit ad-invariant metrics, and if so, under what conditions?
• RQ3 What is the structure of the automorphism group of a free nilpotent Lie algebra that admits an ad-invariant metric?
• RQ4 How do the derivations and skew-symmetric derivations relate to the ad-invariant metric in such algebras?
• RQ5 Can the Lie algebra of derivations be decomposed into semisimple and solvable components, and what is their action on the algebra?

Key findings

• The free 2-step nilpotent Lie algebra on 3 generators, denoted 𝔫₃,₂, admits an ad-invariant metric.
• The free 3-step nilpotent Lie algebra on 2 generators, denoted 𝔫₂,₃, admits an ad-invariant metric.
• No other free nilpotent Lie algebra on m generators and k steps admits an ad-invariant metric.
• The Lie algebra of derivations of 𝔫₃,₂ is isomorphic to 𝔤𝔩(3,K) ⋉ K⁹, with the semisimple part 𝔰𝔩(3,K) and solvable radical K⁹.
• The set of skew-symmetric derivations with respect to the ad-invariant metric is isomorphic to 𝔰𝔩(3,K) ⋉ K³, where K³ is an abelian ideal.
• The inner derivations of 𝔫₃,₂ are isomorphic to K³, and the orthogonal automorphisms are determined by the skew-symmetric derivations.