Infinitely many left-symmetric structures on nilpotent Lie algebras

Dekimpe and Ongenae constructed infinitely many pairwise non-isomorphic complete left-symmetric structures on ${\mathbb{R}}^n$ for $n\ge 6$. In this paper, we construct a family of complete left-symmetric structures on the cotangent Lie algebra $T^{\ast }\mathfrak{g}$ of a certain $n$-dimensional almost abelian nilpotent Lie algebra $\mathfrak{g}$ and give a condition under which two left-symmetric structures in this family are isomorphic. As a consequence of this result, we obtain infinitely many pairwise non-isomorphic left-symmetric structures on $T^{\ast }\mathfrak{g}$. As an application of this construction, we also obtain infinitely many symplectic structures on $T^{\ast }\mathfrak{g}$ which are pairwise non-symplectomorphic up to homothety.