In this paper, we introduce the notion of Koszul–Vinberg–Nijenhuis (KVN) structures on a left-symmetric algebroid as analogues of Poisson–Nijenhuis structures on a Lie algebroid, and show that a KVN-structure gives rise to a hierarchy of Koszul–Vinberg structures. We introduce the notions of [Formula: see text]-structures, pseudo-Hessian–Nijenhuis structures and complementary symmetric [Formula: see text]-tensors for Koszul–Vinberg structures on left-symmetric algebroids, which are analogues of [Formula: see text]-structures, symplectic-Nijenhuis structures and complementary [Formula: see text]-forms for Poisson structures. We also study the relationships between these various structures.

中文翻译：

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左对称代数上的 Koszul-Vinberg 结构和相容结构

在本文中，我们介绍了左对称代数上的 Koszul-Vinberg-Nijenhuis (KVN) 结构的概念，作为李代数上的 Poisson-Nijenhuis 结构的类似物，并表明 KVN 结构产生了 Koszul 的层次结构——温伯格结构。我们介绍了左对称代数上 Koszul-Vinberg 结构的 [Formula: see text]-结构、pseudo-Hessian-Nijenhuis 结构和互补对称 [Formula: see text]-张量，它们是 [Formula: see文本]-结构，辛-Nijenhuis 结构和互补 [公式：见文本]-泊松结构的形式。我们还研究了这些不同结构之间的关系。