Para-Kähler and pseudo-Kähler structures on Lie–Yamaguti algebras

Pub Date : 2024-02-28 DOI:10.1142/s0219498825502044
Jia Zhao, Yuqin Feng, Yu Qiao
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Abstract

For a pre-Lie–Yamaguti algebra A, by using its sub-adjacent Lie–Yamaguti algebra Ac, we are able to construct a semidirect product Lie–Yamaguti algebra via a representation of Ac. The investigation of such semidirect Lie–Yamaguti algebras leads us to the notions of para-Kähler structures and pseudo-Kähler structures on Lie–Yamaguti algebras, and also gives the definition of complex product structures on Lie–Yamaguti algebras. Furthermore, a Levi–Civita product with respect to a pseudo-Riemannian Lie–Yamaguti algebra is introduced and we explore its relation with pre-Lie–Yamaguti algebras.

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李-山古提代数上的 Para-Kähler 和伪 Kähler 结构
对于前Lie-Yamaguti代数A,通过使用它的子相邻Lie-Yamaguti代数Ac,我们能够通过Ac的表示构造一个半直积Lie-Yamaguti代数。通过对这种半间接Lie-Yamaguti代数的研究,我们可以得出Lie-Yamaguti代数上的准凯勒结构和伪凯勒结构的概念,并给出了Lie-Yamaguti代数上复积结构的定义。此外,我们还引入了关于伪黎曼 Lie-Yamaguti 代数的 Levi-Civita 积,并探讨了它与前 Lie-Yamaguti 代数的关系。
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