Pseudo-Euclidean Novikov algebras of arbitrary signature

Abstract

A pseudo-Euclidean Novikov algebra $(g,•,〈,〉)$ is a Novikov algebra $(g,•)$ endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. If $〈,〉$ is of signature $(1,n-1)$ then $(g,•,〈,〉)$ is called a Lorentzian Novikov algebra. In (H. Lebzioui, 2020 [11]), the author studied Lorentzian Novikov algebras and showed that a Lorentzian Novikov algebra is transitive. In this paper, we study pseudo-Euclidean Novikov algebras in the general case, where $〈,〉$ is of arbitrary signature. We show that a pseudo-Euclidean Novikov algebra of arbitrary signature must be transitive and the associated Lie algebra is two-solvable. This implies that a flat left-invariant pseudo-Riemannian metric on a corresponding Lie group is geodesically complete. We show that if $(g,•,〈,〉)$ is a pseudo-Euclidean Novikov algebra such that $g•g$ is non-degenerate then the underlying Lie algebra is a Milnor Lie algebra; that is $g=b\oplus b^{\perp }$, where $b$ is a sub-Lie algebra, $b^{\perp }$ is a sub-Lie ideal and ${\mathrm{ad}}_b$ is $〈,〉$-skew symmetric for any $b\in b$. If $g•g$ is degenerate, then we show that we can obtain the pseudo-Euclidean Novikov algebra through a double extension process starting from a Milnor Lie algebra. Finally, as applications, we classify all pseudo-Euclidean Novikov algebras of dimension ≤5 such that $g•g$ is degenerate.