Courant Algebroids and Poisson Geometry
Courant Algebroids and Poisson Geometry
复制标题
DOI：
10.1093/imrn/rnp048
复制
发表时间：
2008-11
影响因子：
1
通讯作者：
David Li-Bland;E. Meinrenken
中科院分区：
文献类型：
--
作者：
David Li-Bland;E. Meinrenken
给定一个流形M，其作用是二次李代数，使得所有的稳定子代数在中是余迷向的，我们证明了乘积成为M上的Courant代数体。如果M上的双线性形式是分裂的，则M上的横拉格朗日子空间的选择定义了M上的一个双向量场π，如果是Manin三元组，则它是Poisson。通过这种方式，我们恢复了Lu-Yakimov的泊松结构，特别是Evens Lu泊松结构的各种拉格朗日格拉斯曼和de Concini Procesi紧化。这些例子之间的各种泊松映射解释的行为下柯朗态射的拉格朗日分裂。
Given a manifold M with an action of a quadratic Lie algebra , such that all stabilizer algebras are coisotropic in , we show that the product becomes a Courant algebroid over M. If the bilinear form on is split, the choice of transverse Lagrangian subspaces of defines a bivector field π on M, which is Poisson if is a Manin triple. In this way, we recover the Poisson structures of Lu-Yakimov, and in particular the Evens-Lu Poisson structures on the variety of Lagrangian Grassmannians and on the de Concini-Procesi compactifications. Various Poisson maps between such examples are interpreted in terms of the behavior of Lagrangian splittings under Courant morphisms.