相容准李代数的变形与交换扩张
Deformations and abelian extensions of compatible pre-Lie algebras
摘要：
本文首先给出相容准李代数的表示及其表示。研究相容李代数与相容准李代数之间的关系。我们还构造了一个新的双微分分次李代数，它的Maurer-Cartan元是相容的预李结构.给出了控制相容预李代数变形的双微分分次李代数。然后，我们引入了系数在自身中的相容预李代数的上同调。研究了相容准李代数的无穷小变形，证明了等价的无穷小变形在同一个二上同调群中。我们进一步给出了相容准李代数上的Nijenhuis算子的概念。研究相容准李代数的形式变形。若二阶上同调群$\huaH^2(\g;\g)$是平凡的，则相容准李代数是刚性的.最后，我们给出了系数为任意表示的相容预李代数的一个上同调，并利用这个上同调研究了相容预李代数的交换扩张。我们证明了阿贝尔扩张是由第二上同调群分类的。
英文摘要：
In this paper, we first give the notation of a compatible pre-Lie algebra and its representation. We study the relation between compatible Lie algebras and compatible pre-Lie algebras. We also construct a new bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie algebra which controls deformations of a compatible pre-Lie algebra. Then, we introduce a cohomology of a compatible pre-Lie algebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie algebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group $\huaH^2(\g;\g)$ is trivial, then the compatible pre-Lie algebra is rigid. Finally, we give a cohomology of a compatible pre-Lie algebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie algebras using this cohomology. We show that abelian extensions are classified by the second cohomology group.
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