论文标题
兼容$ 3 $ -LIE代数的变形，共同体和阿贝尔扩展
Deformations, cohomologies and abelian extensions of compatible $3$-Lie algebras
论文作者
论文摘要
在本文中，首先，我们给出了兼容$ 3 $ -LIE代数的概念，并构建了一个双分级分级的Lie代数，其Maurer-Cartan元素兼容$ 3 $ -LIE代数。我们还获得了支配兼容$ 3 $ -LIE代数的变形的BidivertientArtientArged Lie代数。然后，我们介绍了一个兼容$ 3 $ -LIE代数的共同体学理论，其本身具有系数，并表明，在同等类别的无限型兼容$ 3 $ -LIE -LIE代数的无限变形之间存在一对一的对应关系。我们进一步研究了兼容$ 3 $ -LIE代数的2阶1参数变形，并在兼容的$ 3 $ -LIE代数上介绍了Nijenhuis操作员的概念，这可能会引起琐碎的变形。最后，我们介绍了一种兼容$ 3 $ -LIE代数的共同体学理论，其系数具有任意表示的系数，并使用第二个共同学组对兼容$ 3 $ -LIE代数的Abelian扩展进行了分类。
In this paper, first we give the notion of a compatible $3$-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible $3$-Lie algebras. We also obtain the bidifferential graded Lie algebra that governs deformations of a compatible $3$-Lie algebra. Then we introduce a cohomology theory of a compatible $3$-Lie algebra with coefficients in itself and show that there is a one-to-one correspondence between equivalent classes of infinitesimal deformations of a compatible $3$-Lie algebra and the second cohomology group. We further study 2-order 1-parameter deformations of a compatible $3$-Lie algebra and introduce the notion of a Nijenhuis operator on a compatible $3$-Lie algebra, which could give rise to a trivial deformation. At last, we introduce a cohomology theory of a compatible $3$-Lie algebra with coefficients in arbitrary representation and classify abelian extensions of a compatible $3$-Lie algebra using the second cohomology group.
