In this paper, we first construct the controlling algebras of embedding tensors and Lie–Leibniz triples, which turn out to be a graded Lie algebra and an \(L_\infty \)-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie–Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz\(_\infty \)-algebra. We realize Kotov and Strobl’s construction of an \(L_\infty \)-algebra from an embedding tensor, as a functor from the category of homotopy embedding tensors to that of Leibniz\(_\infty \)-algebras, and a functor further to that of \(L_\infty \)-algebras.

在本文中，我们首先构造嵌入张量和Lie–Leibniz三元组的控制代数，这分别是一个渐进的Lie代数和一个（（L_ \ infty \）-代数。然后，我们介绍了嵌入张量和Lie-Leibniz三元组的表示形式和同构关系，并证明存在连接各种同构关系的长序列。作为应用程序，我们使用第二个同调类对无穷大变形和中心扩展进行分类。最后，我们介绍了同伦嵌入张量的概念，该张量将引发莱布尼兹\（_ \ infty \）-代数。我们认识到Kotov和Strobl从嵌入张量构造\（L_ \ infty \）-代数，作为从同构嵌入张量到莱布尼兹的函子\（_ infty \）-代数，以及比\（L_ \ infty \）-代数更远的函子。