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

Abstract

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 $$L_\infty $$ 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$$_\infty $$ ∞ -algebra. We realize Kotov and Strobl’s construction of an $$L_\infty $$ L ∞ -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 $$L_\infty $$ L ∞ -algebras.

Nonabelian embedding tensors

“…Remark 2.4. When (h, [−, −] h ) is an abelian Lie algebra, we obtain that T is a usual embedding tensor [16,28]. In addition, when ρ is the trivial action of g on h, T is a Lie algebra homomorphism from h to g.…”

“…In general, deformation theory was developed for algebras over binary quadratic operads by Balavoine [2]. The deformation theory of embedding tensors were studied in [28]. In the associative algebra context, the deformation theory of averaging operators was studied in [6] via the derived brackets [15,31], and the deformation theory of averaging associative algebras was studied in [32].…”

Maurer-Cartan characterizations and cohomologies of compatible Lie algebras

“…The condition (10) actually means that ϕ is an embedding tensor. See [20] for more details about embedding tensors.…”

The moment map for the variety of Leibniz algebras

“…A Leibniz algebra is a vector space with a multiplication such that every left multiplication operator is a derivation, which was at first introduced by Bloh ( [3]) and later independently rediscovered by Loday in the study of cohomology theory (see [18,19]). Leibniz algebras play an important role in different areas of mathematics and physics [5,8,11,16,17,22,23,24], and we refer to [7] for a nice survey of Leibniz algebras.…”