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 $L_{\infty }$-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.

32 pages, comments are welcome

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Cited by in corpus (4)

• Nonabelian embedding tensors
• Cohomology and deformations of nonabelian embedding tensors between Lie triple systems
• Deformations and cohomologies of embedding tensors on 3-Lie algebras
• Nonabelian embedding tensors on 3-Lie algebras and 3-Leibniz-Lie algebras