Some Classes of Affinely Equivalent Metrics on Lie Algebras

Two geodesically equivalent invariant metrics on a homogeneous space are affinely invariant. This is especially true for left invariant metrics on Lie groups. If a Lie group is connected, it admits a biinvariant metric if and only if its algebra admits an inner product that is ad-invariant, also called a metric. We study the affine equivalence of ad-invariant metrics on real Lie algebras and Norden metrics on underlying real Lie algebras of complex algebras. Our results are illustrated with some examples. We also study some geometric properties of these algebras, in particular perfect algebras, algebras with non-vanishing center and Norden pairs of metrics, which are also ad-invariant.