Some graded Lie algebra structures associated with Lie algebras and Lie algebroids

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.