The main objects of this thesis are graded Lie algebras associated with a Lie algebra or a Lie algebroid such as the Frölicher-Nijenhuis algebra, the Kodaira-Spencer algebra and the newly constructed Gelfand-Dorfman algebra and generalized Nijenhuis-Richardson algebra. Main results are summarized as follows: We introduce a derived bracket which contains the Frölicher-Nijenhuis bracket as a special case and prove an interesting formula for this derived bracket. We develop a rigorous mechanism for the Kodaira-Spencer algebra, reveal its relation with R-matrices in the sense of M. A. Semenov-Tian-Shansky and construct from it a new example of the knit product structures of graded Lie algebras. For a given Lie algebra, we construct a new graded Lie algebra called the Gelfand-Dorfman algebra which provides for r-matrices a graded Lie algebra background and includes the well-known Schouten-Nijenhuis algebra of the Lie algebra as a subalgebra. We establish an anti-homomorphism from this graded Lie algebra to the Nijenhuis-Richardson algebra of the dual space of the Lie algebra, which sheds new light on our understanding of Drinfeld's construction of Lie algebra structures on the dual space with r-matrices. In addition, we generalize the Nijenhuis-Richardson algebra from the vector space case to the vector bundle case so that Lie algebroids on a vector bundle are defined by this generalized Nijenhuis-Richardson algebra. We prove that this generalized Nijenhuis-Richardson algebra is isomorphic to both the linear Schouten-Nijenhuis algebra on the dual bundle of the vector bundle and the derivation algebra associated with the exterior algebra bundle of this dual bundle. A concept of a 2 n-ary Lie algebroid is proposed as an application of these isomorphisms.
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机译:本文的主要目的是与李代数或李代数相关的分级李代数,例如Frölicher-Nijenhuis代数,Kodaira-Spencer代数和新构建的Gelfand-Dorfman代数以及广义Nijenhuis-Richardson代数。主要结果总结如下:我们引入了一个包含Frölicher-Nijenhuis括号的派生括号作为特例,并证明了这个派生括号的有趣公式。我们为Kodaira-Spencer代数建立了严格的机制,揭示了其与M. A. Semenov-Tian-Shansky意义上的R矩阵的关系,并据此构造了分级Lie代数的针织产品结构的新示例。对于给定的李代数,我们构造了一个新的分级李代数,称为Gelfand-Dorfman代数,它为r矩阵提供了一个分级李代数背景,并包括著名的李代数的Schouten-Nijenhuis代数作为子代数。我们建立了从该梯度Lie代数到Lie代数对偶空间的Nijenhuis-Richardson代数的反同态性,这为我们对Drinfeld在带r矩阵的对偶空间上构造Lie代数结构的理解提供了新的思路。此外,我们将Nijenhuis-Richardson代数从向量空间情况推广到向量束情况,以便向量束上的Lie代数由该广义Nijenhuis-Richardson代数定义。我们证明了这种广义的Nijenhuis-Richardson代数与向量束对的双束上的线性Schouten-Nijenhuis代数以及与该对偶束的外部代数束相关的导数都同构。提出了2 n元李代数的概念作为这些同构的应用。
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