Non-degenerate bilinear forms and left-symmetric structures

Abstract

Chu proved that a symplectic structure $\omega$ on an even-dimensional Lie algebra induces a left-symmetric structure which is defined by $\omega(x\scriptstyle{\Delta}_{\omega} y,z)=-\omega(y,[x,z])$. El Bourkadi and Mansouri proved that a cosymplectic structure on an odd-dimensional Lie algebra $\mathfrak{g}$ induces the left-symmetric structure, which is defined by using a linear isomorphism from $\mathfrak{g}$ to $\mathfrak{g}^*$ associated with the cosymplectic structure. In this paper, for a non-degenerate bilinear form $\phi$ on a Lie algebra $\mathfrak{g}$, we give a necessary and sufficient condition for the product $\scriptstyle{\Delta}_{\phi}$ on $\mathfrak{g}$ defined by $\phi(x\scriptstyle{\Delta}_{\phi} y,z)=-\phi(y,[x,z])$ to be a left-symmetric structure. We also prove that the left-symmetric structure $\scriptstyle{\Delta}_{\phi}$ is complete if and only if the Lie algebra $\mathfrak{g}$ is unimodular. Moreover, we formulate the notion of double extension for non-degenerate bilinear forms and prove that a certain class of non-degenerate bilinear forms is obtained by double extension.