The Inhomogeneous Nambu-Poisson Manifolds And The Extension Of Jacobi Algebras

In recent years, the higher-order Courant structure has become a hot research topic of study in Poisson Geometry and mathematical physics. It has important significance in both theoretical research and practical application. In the study of Nambu-Poisson manifolds and Jacobi algebras, it always involves the problems of high-order Courant structure. Many scholars have studied the problems of homogeneous Nambu-Poisson manifolds and Jacobi algebras, also they have made abundant achievements. But these articles are all about the homogeneous Nambu-Poisson manifolds and Jacobi algebras. There is seldom research on the inhomogeneous Nambu-Poisson manifolds and the extension of Jacobi algebras. Therefore, we mainly study the problems of inhomogeneous Nambu-Poisson manifolds and the extension of Jacobi algebras in this paper. The main contents of this paper are as follows:In the first chapter, the research background and development history about higher-order Courant structure theory are introduced, then domestic and overseas scholars’ research achievements about the homogeneous Nambu-Poisson manifolds and Jacobi algebras are analyzed and summarized. The main contents of the paper are introduced.In the second chapter, the definitions and properties of Nambu-Poisson manifold and the abelian extension of Leibniz algebras are introduced, which lay a good foundation for theoretical research and practical application in subsequent chapters.In the third chapter, using the definitions and properties of the Nambu-Poisson manifold, the problems of inhomogeneous Nambu-Poisson manifold are discussed. Firstly, the basic definitions and properties of inhomogeneous Nambu-Poisson structure, the Hamilton vector field cluster and inhomogeneous Nambu-Poisson tensor are obtained. Secondly, we define a bracket operator{·,·) on Ω1(M), and then according to the definition and properties of inhomogeneous Nambu-Poisson manifold, we can obtain the relationship between the bracket{·,·) and Schouten bracket [Pi, Pi]. Finally, through the above results, we construct a Leibniz algebroid canonically associated with the inhomogeneous Nambu-Poisson manifold and prove it.In the fourth chapter, according to the definitions and properties of the Abelian extension of Leibniz algebras, the problems of abelian extension of the Lie algebra TM(?)C∞(M) are proposed. Firstly, the bracket is defined on TM(?)C∞(M)(?)Ω1(M) by giving a linear map. Moreover, it is proved that TM(?)C∞(M)(?)Ω1(M) is a Lie algebra. And we obtain an extension of Leibniz algebra TM(?)C∞(M). Secondly, by using the definition of the representation of Leibniz algebra, it is proved that Ω2(M) is the representation of Leibniz algebras TM(?)C∝(M)(?)Ω1(M).Finally, we define a bracket on TM(?)C∞(M)(?)Ω1(M)(?)Ω2(M). By direct calculation, we prove that TM(?)C∞(M) (?)Ω1(M)Ω2(M) is a Lie algebra. Then we can obtain an extension of the Lie algebra TM(?)C∞(M)(?)Ω1(M).In the last chapter, the summary of this paper is given and the problems for further study are put forward.