On pseudo-Euclidean Novikov algebras

Abstract

A flat pseudo-Euclidean Lie algebra is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. We observe that, in many classes of flat pseudo-Euclidean Lie algebras, the left-symmetric product is actually of Novikov. This leads us to study pseudo-Euclidean Novikov algebras (A,〈,〉), that is, Novikov algebra A endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. We show that a Lorentzian Novikov algebra (A,〈,〉) must be transitive. This implies that the underlying Lie algebra AL must be unimodular. In geometrical terms, the left-invariant metric on a corresponding Lie group is geodesically complete. In this case, we show that the restriction of the product to [AL,AL]⊥ is trivial. Using the double extension process, we prove that (A,〈,〉) is a pseudo-Euclidean Novikov algebra such that [AL,AL] is Lorentzian if and only if AL splits as AL=[AL,AL]⊥⊕[AL,AL] where [AL,AL] and [AL,AL]⊥ are abelian, [AL,AL] is Lorentzian and adx is skew-symmetric for any x∈[AL,AL]⊥. This also implies, in this case, that A is transitive, AL is unimodular and two-solvable. This result solves the problem for Lorentzian Novikov algebras such that [AL,AL] is non-degenerate. If [AL,AL] is degenerate, we show that such Lorentzian Novikov algebra is obtained by the double extension process from a flat Euclidean algebra, and we give some applications in low dimensions.