Stability and rigidity of 3-Lie algebra morphisms

Jun Jiang, Yunhe Sheng, Geyi Sun
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Abstract

In this paper, first we use the higher derived brackets to construct an $L_\infty$-algebra, whose Maurer-Cartan elements are $3$-Lie algebra morphisms. Using the differential in the $L_\infty$-algebra that govern deformations of the morphism, we give the cohomology of a $3$-Lie algebra morphism. Then we study the rigidity and stability of $3$-Lie algebra morphisms using the established cohomology theory. In particular, we show that if the first cohomology group is trivial, then the morphism is rigid; if the second cohomology group is trivial, then the morphism is stable. Finally, we study the stability of $3$-Lie subalgebras similarly.
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3-Lie 代数形态的稳定性和刚性
在本文中,我们首先利用高导出括号构造了一个$L_\infty$-代数,其毛勒-卡尔坦元素是$3$-Lie代数态。利用$L_\infty$-代数中支配态变形的微分,我们给出了$3$-Lie代数态的同调。然后,我们利用已建立的同调理论研究了 3 美元李代数变形的刚性和稳定性。我们特别指出,如果第一同调群是微不足道的,那么态是刚性的;如果第二同调群是微不足道的,那么态是稳定的。最后,我们以类似的方法研究了 $3$-Lie 子代数的稳定性。
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