标题： 预辛代数胚及其应用 显示英文标题

标题： Pre-symplectic algebroids and their applications

摘要： 本文引入了预辛代数胚的概念，并证明了预辛代数胚与辛李代数胚之间存在一一对应关系。 这一结果推广了几何上左对称代数与辛（Frobenius）李代数之间的关系。 尽管预辛代数胚不是左对称代数胚，但它们仍可以被视为辛李代数胚的底层结构。 然后我们研究了正合预辛代数胚，并证明它们由左对称代数胚的第三上同调群分类。 最后，我们研究了para-复数预辛代数胚。 与para-复数预辛代数胚相关联的，是一个伪黎曼李代数胚。 para-复数预辛代数胚中的乘法刻画了相应伪黎曼李代数胚中Levi-Civita联络限制到拉格朗日子代数胚上的特性。 显示英文摘要

摘要： In this paper, we introduce the notion of a pre-symplectic algebroid, and show that there is a one-to-one correspondence between pre-symplectic algebroids and symplectic Lie algebroids. This result is the geometric generalization of the relation between left-symmetric algebras and symplectic (Frobenius) Lie algebras. Although pre-symplectic algebroids are not left-symmetric algebroids, they still can be viewed as the underlying structures of symplectic Lie algebroids. %We study three classes of pre-symplectic algebroids in detail. Then we study exact pre-symplectic algebroids and show that they are classified by the third cohomology group of a left-symmetric algebroid. Finally, we study para-complex pre-symplectic algebroids. Associated to a para-complex pre-symplectic algebroid, there is a pseudo-Riemannian Lie algebroid. The multiplication in a para-complex pre-symplectic algebroid characterizes the restriction to the Lagrangian subalgebroids of the Levi-Civita connection in the corresponding pseudo-Riemannian Lie algebroid.