The Controlling L ∞ -Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples
The Controlling L ∞ -Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples
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发表时间：
2020
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通讯作者：
Y. Sheng;Rong Tang;Chenchang Zhu
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作者：
Y. Sheng;Rong Tang;Chenchang Zhu
本文首先构造了嵌入张量的控制代数和fi-Leibniz三元组的控制代数，证明了它们分别是分次李代数和L∞-代数。然后，我们介绍了嵌入张量和Lie-Leibniz三元组的表示和上同调，并证明了存在一个连接各种上同调的长正合列。作为应用，我们利用第二上同调群对fi亚极小变形和中心扩张进行了分类。最后，我们引入了同伦嵌入张量的概念，它将诱导一个Leibniz∞-代数。我们实现了Kotov和Strobl从嵌入张量构造L∞-代数，作为同伦嵌入张量范畴到Leibniz∞-代数范畴的函子，以及进一步到L∞-代数范畴的函子。
: 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.