The controlling $L_\infty$-algebra, cohomology and homotopy of embedding tensors and Lie-Leibniz triples

TL;DR

This paper develops the algebraic framework for embedding tensors and Lie-Leibniz triples, introducing controlling algebras, cohomologies, and homotopy structures, and explores their deformations, extensions, and categorical relationships.

Contribution

It constructs the controlling graded Lie and $L_$-algebras for embedding tensors and Lie-Leibniz triples, and establishes their cohomology theories and homotopy relations.

Findings

Classified infinitesimal deformations using second cohomology.

Established long exact sequences connecting cohomologies.

Realized functorial relationships between homotopy embedding tensors and $L_$-algebras.

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 $L_\infty$-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, to 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.