We determine the $$L_\infty $$ -algebra that controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying $$\mathsf {Lie}\mathsf {Rep}$$ pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define the cohomology of relative Rota–Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota–Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a homotopy relative Rota–Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota–Baxter Lie algebras is intimately related to pre-Lie $$_\infty $$ -algebras.

中文翻译：

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相对旋转-巴克斯特李代数的变形与同伦理论

我们确定了控制相对 Rota-Baxter Li 代数变形的 $$L_\infty $$ -代数，并表明它是控制底层 $$\mathsf {Lie}\mathsf { 变形的 dg Lie 代数的扩展Rep}$$ 对由 dg Lie 代数控制相对 Rota-Baxter 算子的变形。因此，我们定义了相对 Rota-Baxter Li 代数的上同调并将其与它们的无穷小变形联系起来。一大类相对 Rota-Baxter Li 代数是从三角形李双代数中获得的，我们在相应的变形复合体之间构建了一个映射。接下来，介绍了同伦相对 Rota-Baxter Li 代数的概念。我们证明一类同伦相对 Rota-Baxter Lie 代数与 pre-Lie $$_\infty $$ -algebras 密切相关。