Manin Triples for Lie Bialgebroids
Manin Triples for Lie Bialgebroids
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DOI：
10.4310/jdg/1214459842
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发表时间：
1995-08
影响因子：
2.5
通讯作者：
Zhang-Ju Liu;A. Weinstein;P. Xu
中科院分区：
文献类型：
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作者：
Zhang-Ju Liu;A. Weinstein;P. Xu
在狄拉克结构的研究中，T。柯朗介绍了一个括号的直接和向量场和1-形式。这个括号不满足雅可比恒等式，除了在某些子空间上。在本文中，我们系统化的性质，这个括号的定义的柯朗代数体。这个向量丛$E\rightarrow M$上的结构，由$E$的截面上的反对称括号组成，其"Jacobi反常“具有关于丛映射$E\rightarrow TM$和$E$上的对称双线性形式的域的显式表达。当$M$是一个点，定义减少到一个李代数携带不变的非退化对称双线性型。对于$M$上的任何李双代数胚$（A，A^{*}）$（一个由麦肯齐和徐明定义的概念），当$M$是一个点时，在$A\oplus A^{*}$上存在一个自然的Courant代数胚结构，它是李双代数的Drinfel'd double。相反，如果$A$和$A^*$是柯朗代数胚$E$的补迷向子丛，闭于括号下（这样的丛，维数为$E$的一半，称为狄拉克结构），则在$（A，A^{*}）$上存在一个自然的李双代数胚结构，其二重同构于$E$。Manin三元组的理论由此从李代数推广到李代数胚。我们的工作给出了一个新的方法来双哈密顿结构和一个新的方式结合两个泊松结构，以获得第三个。我们还采取了一些尝试性的步骤，推广Drinfel'd的理论的泊松齐性空间从群群胚。
In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does not satisfy the Jacobi identity except on certain subspaces. In this paper we systematize the properties of this bracket in the definition of a Courant algebroid. This structure on a vector bundle $E\rightarrow M$, consists of an antisymmetric bracket on the sections of $E$ whose ``Jacobi anomaly'' has an explicit expression in terms of a bundle map $E\rightarrow TM$ and a field of symmetric bilinear forms on $E$. When $M$ is a point, the definition reduces to that of a Lie algebra carrying an invariant nondegenerate symmetric bilinear form. For any Lie bialgebroid $(A,A^{*})$ over $M$ (a notion defined by Mackenzie and Xu), there is a natural Courant algebroid structure on $A\oplus A^{*}$ which is the Drinfel'd double of a Lie bialgebra when $M$ is a point. Conversely, if $A$ and $A^*$ are complementary isotropic subbundles of a Courant algebroid $E$, closed under the bracket (such a bundle, with dimension half that of $E$, is called a Dirac structure), there is a natural Lie bialgebroid structure on $(A,A^{*})$ whose double is isomorphic to $E$. The theory of Manin triples is thereby extended from Lie algebras to Lie algebroids. Our work gives a new approach to bihamiltonian structures and a new way of combining two Poisson structures to obtain a third one. We also take some tentative steps toward generalizing Drinfel'd's theory of Poisson homogeneous spaces from groups to groupoids.