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arXiv · 2510.14610

Infinitely many left-symmetric structures on nilpotent Lie algebras

Naoki Kato
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Abstract

Dekimpe and Ongenae constructed infinitely many pairwise non-isomorphic complete left-symmetric structures on $\mathbb{R}^n$ for $n\geq 6$. In this paper, we construct a family of complete left-symmetric structures on the cotangent Lie algebra $T^*\mathfrak{g}$ 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 $T^{*}\mathfrak{g}$. As an application of this construction, we also obtain infinitely many symplectic structures on $T^{*}\mathfrak{g}$ which are pairwise non-symplectomorphic up to homothety.

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Naoki Kato. 2025-10-16. Infinitely many left-symmetric structures on nilpotent Lie algebras. https://arxiv.org/abs/2510.14610

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