n-李代数与 n-泊松结构

n-Lie Algebra and n-Poisson Structure

本文从高阶角度出发,首先研究 -李代数的结构常数,它是作为李代数的自然推广,是基本乘法运算为 n- 元线性运算的一种代数系统。其次通过定义流形上的n-泊松括号引出 n-泊松结构的定义及性质,得到 n-李代数与 n-泊松结构一一对应关系。最后在向量从上研究余切从上的 n-李代数胚,给出了 n-李代数胚的余态射与 n-泊松映射的关系。

In this paper, we first study the structural constants of n-Lie algebras, which is a natural generalization of Lie algebras and an algebraic system whose basic multiplication operations are linear operations of n-elements. Secondly, the definition and properties of n-Poisson structure are derived by defining the n-Poisson bracket on a manifold and the one-to-one correspondence between n-Lie algebras and n-Poisson structure is obtained. Finally, we study the n-Lie algebras on cotangent bundles, and give the relation between the comorphism of n-Lie algebras and n-Poisson mapping.