The Controlling L_∞ -Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples

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_∞ -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 _∞ -algebra. We realize Kotov and Strobl’s construction of an L_∞ -algebra from an embedding tensor, as a functor from the category of homotopy embedding tensors to that of Leibniz _∞ -algebras, and a functor further to that of L_∞ -algebras.