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arXiv · 2610.25056

Classification of Hom--Lie structures on finite-dimensional simple Lie algebras

Jiacheng Sun·Chi Zhang·Haoran Zhu
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Abstract

We classify the Hom--Lie structures on finite-dimensional simple Lie algebras over an algebraically closed field of characteristic different from \(2\) and \(3\). Except for the three-dimensional simple Lie algebra and the Zassenhaus algebras, every such structure is a scalar multiple of the identity. In the two exceptional cases, we determine the full Hom--Lie space: it has dimension six for the three-dimensional algebra, while for a Zassenhaus algebra it consists precisely of the multiplication operators arising from the corresponding divided-power algebra. This proves a conjecture of Makhlouf and Zusmanovich.

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Jiacheng Sun, Chi Zhang, Haoran Zhu. 2026-10-18. Classification of Hom--Lie structures on finite-dimensional simple Lie algebras. https://arxiv.org/abs/2610.25056

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