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 ≤ 7 admitting an ad-invariant metric. We prove that uniqueness of the metric on a complex Lie algebra g is equivalent to uniqueness of ad-invariant metrics on the cotangent Lie algebra T * 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 * 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 ≤ 6 or real nilpotent of dimension ≤ 10.

Ad-invariant metrics on nonnice nilpotent Lie algebras

“…In particular, real nilpotent Lie algebras with an ad-invariant metric are classified up to dimension 10 in [15]. In [5], we showed that the irreducible Lie algebras appearing in this classification admit a unique ad-invariant metric (up to sign); as a step in the proof, we proved that all of them are nice. Nice Lie algebras possess a strong algebraic structure, which can be encoded in a special kind of directed graph.…”

“…Proof. Nilpotent Lie algebras of dimension ≤ 10 admitting ad-invariant metrics were classified by Kath in [15] and they were shown to be nice in [5] (see Table 1…”

“…For n > 11, we will answer this question in the affirmative in the rest of this section and the next. The proof will make use of the Lie algebra constructed in [5,Example 2.11], which we now recall.…”

“…Consider the 12-dimensional nilpotent irreducible Lie algebra h given by: (0, 0, e 14 , e 12 , e 47 + e 28 + e 39 + e 10,11 This Lie algebra appeared in [5] as an example of a nilpotent Lie algebra admitting more than one adinvariant metric up to automorphisms, but only one up to automorphisms and rescaling. This is linked to the fact that the Nikolayevsky derivation of h is zero, unlike the 11-dimensional example of Proposition 1.1 (see [5,Proposition 2.21]). We will denote by n = h/z(h) the quotient of h by its center z(h), whose structure equations are (0, 0, e 14 , e 12 , −e 17 − e 29 + e 49 , −e 15 + e 28 − e 39 , e 13 − e 29 , e 24 + e 19 , 0).…”

“…The classification of Lie algebras admitting ad-invariant metrics is an open problem, which has been solved when restricted to particular subfamilies [2,8,9,10,15]. In fact, some of these classifications also list all possible ad-invariant metrics, up to isometry; it turns out that many solvable Lie algebras only admit one ad-invariant metric (see [1,5,19] for some results on the uniqueness of ad-invariant metrics).…”