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Uniqueness of ad-invariant metrics

Diego Conti, Viviana del Barco, Federico A. Rossi

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 $\leq 10$.

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 admitting an ad-invariant metric. We prove that uniqueness of the metric on a complex Lie algebra is equivalent to uniqueness of ad-invariant metrics on the cotangent Lie algebra ; 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 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 or real nilpotent of dimension .

Paper Structure

This paper contains 12 sections, 56 theorems, 136 equations, 1 table.

Table of Contents

  1. Lie algebras with ad-invariant metrics
  2. The uniqueness problem for ad-invariant metrics
  3. The real, the complex, the solitary
  4. Gradings adapted to the metric
  5. Gradings and the solitary condition
  6. The metric Nikolayevsky derivation
  7. The metric Nikolayevsky derivation in the ad-invariant case
  8. Cotangent Lie algebras
  9. The solitary condition for the canonical metric
  10. The Nikolayevsky derivation on $T^*\mathfrak{g}$
  11. Double extensions and uniqueness
  12. The solitary condition in low dimension

Key Result

Lemma 1.2

Let $\mathfrak{g}$ be a nonabelian Lie algebra with an ad-invariant metric $g$. If the center of $\mathfrak{g}$ is not contained in its commutator, then $(\mathfrak{g},g)$ is reducible as a metric Lie algebra as $\mathfrak{g}=\mathfrak{m}\oplus\tilde{\mathfrak{g}}$, where $\mathfrak{m}$ is an abelia

Theorems & Definitions (134)

  • Remark 1.1
  • Lemma 1.2
  • proof
  • Corollary 1.3
  • Lemma 1.4: ZhZh01
  • Proposition 1.5
  • proof
  • Proposition 1.6
  • proof
  • Corollary 1.7
  • ...and 124 more