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⊕b⊥, where b is a sub-Lie algebra, b⊥ is a sub-Lie ideal and adb is 〈,〉-skew symmetric for any b∈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.