Uniqueness of ad-invariant metrics

Abstract: An ad-invariant metric on a Lie algebra is a nondegenerate symmetric bilinear form for which inner derivations are skew-symmetric. These are the algebraic counterparts of bi-invariant metrics on Lie groups.

It is known that a positive definite ad-invariant metric can only be defined on compact semisimple Lie algebras, direct sum with an abelian factor. On compact simple Lie algebras, every ad-invariant metric is a multiple of the Killing form which, in addition, is invariant under the Lie algebra automorphisms.

In the pseudo-Riemannian context ad-invariant metrics appear on more general Lie algebras such as semisimple (non-compact), or solvable. For non-semisimple Lie algebras, the orbit space of ad-invariant metrics under the action of the automorphism group has not been systematically described yet.

In this talk, we will discuss characteristics of Lie algebras possessing a unique ad-invariant metric up to automorphisms (and sign). In particular, we will introduce the concept of "solitary" metrics on Lie algebras, which aims to encode the property of being a unique ad-invariant metric. As we will see, this is actually a property of a Lie algebra rather than of the metric itself.

This characterization of uniqueness allowed us to show that Lie algebras admitting a unique ad-invariant metric are necessarily solvable. In addition, we show that many low dimensional Lie algebras carrying ad-invariant metrics are solitary.

Time permitting, generalizations of the solitary conditions will be discussed.

The talk is based on joint works with Diego Conti and Federico A. Rossi (Milano Bicocca).

differential geometry

Audience: researchers in the topic

( paper )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Riemannian geometry.

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