Workshop on Lie algebroids and Lie groupoids in Differential Geometry

李群胚作为李群概念的扩展已经被研究了几十年，以体现丛结构的对称性和仅局部定义的对称性。在20世纪80年代中后期，一些研究人员- Karasev，Weinstein，S。Zakrzewski -独立发现辛群胚（李群胚与兼容的辛结构）提供了全球模型的泊松流形。在2000年，Cattaneo和Felder使用Poisson sigma模型证明，只要对辛广群的概念稍加修改，任何Poisson流形都可以被整合为辛广群。这振兴原来的方法温斯坦研究量子化的泊松流形量化相应的辛广群的方式兼容的广群结构。格贝理论通常被认为是丛理论的高阶形式。在这一点上，它类似于多重李群胚的概念。众所周知，将群的概念加倍只会得到一个阿贝尔群，但双重和多重广群的概念可以追溯到20世纪60年代的Ehresmann，并导致了丰富的理论。自20世纪80年代后期以来，麦肯齐广泛地发展了双重和多重李群胚的李群理论。Moerdijk，Laurent，Ping Xu和其他人的工作努力将这两种方法联系起来;一个可以利用多重Lie理论的gerbe理论的制定将是一个相当大的进步。超数学进入李代数体理论的观察，由于Vaintrob，流形上的李代数体结构是（有一些宇称反转）相应的超流形上的同调向量场。这一发现被Th. Voronov，他用交换同调向量场的形式提出了麦肯齐的双李代数体的概念。麦肯齐的概念是不容易处理的，超重构给出了一个快速发展的前景，李代数胚和Poisson流形有双重关系，即李代数胚的对偶具有Poisson结构，流形上的Poisson结构诱导其余切丛上的李代数胚结构。利用这些关系，阮天宗得到了一般李代数胚的规范形和线性化定理，推广了已知的哈密顿系统的结果。

Lie groupoids have been studied for several decades as an extension of the concept of Lie group to embody the symmetry properties of bundle structures, and symmetries which are only locally defined. In the mid late 1980s several researchers - Karasev, Weinstein, S. Zakrzewski - independently found that symplectic groupoids (Lie groupoids with a compatible symplectic structure) provide global models for Poisson manifolds. In 2000 Cattaneo and Felder showed, using Poisson sigma models, that with a small modification of the concept of symplectic groupoid, any Poisson manifold could be integrated to a symplectic groupoid. This revitalized the original approach of Weinstein of studying the quantization of Poisson manifolds by quantizing the corresponding symplectic groupoid in a way compatible with the groupoid structure. Gerbe theory is often presented as a higher-order form of bundle theory. In this itresembles the concept of multiple Lie groupoid. It is well-known that doubling the concept of group leads only to a single abelian group, but the concept of double and multiple groupoid goes back to Ehresmann in the 1960s and leads to a rich theory. The Lie theory of double and multiple Lie groupoids has been extensively developed by Mackenzie since the late 1980s. Work of Moerdijk, Laurent, Ping Xu and others has endeavoured to link these two approaches; a formulation of gerbe theory which could take advantage of multiple Lie theory would be a considerable advance. Supermathematics entered Lie algebroid theory with the observation, due to Vaintrob, that a Lie algebroid structure on a manifold is (with some parity reversion) a homological vector field on the corresponding super manifold. This observation has been widely extended by Th. Voronov, who has formulated Mackenzie's notion of double Lie algebroid in terms of commuting homological vector fields. Mackenzie's notion is not easy to work with and the super reformulation gives a prospect of rapid progress,Lie algebroids and Poisson manifolds have a two-fold relationship in that the dual ofa Lie algebroid has a Poisson structure and a Poisson structure on a manifold induces a Lie algebroid structure on its cotangent bundle. Using these relationships, Nguyen Tien Zung has obtained normal form and linearization theorems for general Lie algebroids which extend results known for Hamiltonian systems.