In this paper, first, we study linear deformations of a Lie–Yamaguti algebra and introduce the notion of a Nijenhuis operator. Then we introduce the notion of a product structure on a Lie–Yamaguti algebra, which is a Nijenhuis operator E satisfying E2 = Id. There is a product structure on a Lie–Yamaguti algebra if and only if the Lie–Yamaguti algebra is the direct sum of two subalgebras (as vector spaces). There are some special product structures, each of which corresponds to a special decomposition of the original Lie–Yamaguti algebra. In the same way, we introduce the notion of a complex structure on a Lie–Yamaguti algebra. Finally, we add a compatibility condition between a product structure and a complex structure to introduce the notion of a complex product structure on a Lie–Yamaguti algebra.

中文翻译：

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Lie-Yamaguti 代数上的 Nijenhuis 算子、积结构和复结构

在本文中，我们首先研究了 Lie-Yamaguti 代数的线性变形，并引入了 Nijenhuis 算子的概念。然后我们介绍了 Lie-Yamaguti 代数上的乘积结构的概念，它是 Nijenhuis 算子乙令人满意的乙2 = ID. Lie-Yamaguti 代数上存在乘积结构当且仅当 Lie-Yamaguti 代数是两个子代数（作为向量空间）的直接和。有一些特殊的乘积结构，每一个都对应于对原始 Lie-Yamaguti 代数的特殊分解。以同样的方式，我们在 Lie-Yamaguti 代数上引入了复结构的概念。最后，我们在Lie-Yamaguti代数上加入了积结构和复结构之间的相容条件来引入复积结构的概念。