In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce ${𝒪}\mathrm{\text{N}}$-structures on bimodules over pre-Lie algebras. We show that an ${𝒪}\mathrm{\text{N}}$-structure gives rise to a hierarchy of pairwise compatible ${𝒪}$-operators. We study solutions of the strong Maurer–Cartan equation on the twilled pre-Lie algebra associated to an ${𝒪}$-operator, which gives rise to a pair of ${𝒪}\mathrm{\text{N}}$-structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra $𝔤$ are corresponding to ${𝒪}\mathrm{\text{N}}$-structures on the bimodule $({𝔤}^{\ast };{\mathrm{\text{ad}}}^{\ast },-R^{\ast })$, and KVB-structures are corresponding to solutions of the strong Maurer–Cartan equation on a twilled pre-Lie algebra associated to an $𝔰$-matrix.

在本文中，我们研究了 Poisson-Nijenhuis 结构的 pre-Lie 类似物，并介绍了${𝒪}\mathrm{\text{ñ}}$-pre-Lie 代数上的双模结构。我们证明了一个${𝒪}\mathrm{\text{ñ}}$-结构产生成对兼容的层次结构${𝒪}$- 运营商。我们研究斜纹预李代数上的强 Maurer-Cartan 方程的解。${𝒪}$- 运算符，这会产生一对${𝒪}\mathrm{\text{ñ}}$- 自然具有二元性的结构。我们在预李代数上展示了 KVN 结构和 HN 结构$𝔤$对应于${𝒪}\mathrm{\text{ñ}}$-双模块上的结构$({𝔤}^{*};{\mathrm{\text{广告}}}^{*},-R^{*})$, 和 KVB 结构对应于强 Maurer-Cartan 方程在斜纹预李代数上的解$𝔰$-矩阵。