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arXiv · 2009.11096

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

Yunhe Sheng·Rong Tang·Chenchang Zhu
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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.

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Yunhe Sheng, Rong Tang, Chenchang Zhu. 2020-09-23. The controlling $L_\infty$-algebra, cohomology and homotopy of embedding tensors and Lie-Leibniz triples. https://doi.org/10.1007/s00220-021-04032-y

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