Workshop on Lie algebroids and Lie groupoids in Differential Geometry

李群胚作为李群概念的推广，已经被研究了几十年，以体现丛结构的对称性，以及只在局部定义的对称性。20世纪80年代中后期，一些研究人员--Karasev，Weinstein，S.Zakrzewski--独立地发现辛群胚(具有相容辛结构的李群胚)为Poisson流形提供了全局模型。2000年，Cattneo和Feld利用Poisson-Sigma模型证明了，只要对辛群胚的概念稍作修改，任何Poisson流形都可以积分为辛群胚。这使Weinstein研究Poisson流形的量子化的原始方法重新焕发生机，他以一种与群结构相容的方式量子化了相应的辛群群。Gerbe理论通常被表示为丛理论的高阶形式。在这一点上，它类似于多重李群胚的概念。众所周知，将群的概念加倍只能得到一个阿贝尔群，但双重和多重群胚的概念可以追溯到20世纪60年代的Ehresmann，并导致了丰富的理论。自20世纪80年代末以来，Mackenzie发展了双重和多重Lie群胚的Lie理论。Moerdijk，Laurent，Ping Xu和其他人的工作已经努力将这两种方法联系起来；可以利用多重谎言理论的细菌理论的表述将是一个相当大的进步。超数学进入了李代数体理论，由于Vaintrob的观察，流形上的李代数体结构是相应超流形上的同调向量场(具有一定的奇偶反转)。这一观察结果已被Th广泛推广。Voronov，他用交换的同调向量场表述了Mackenzie的双李代数体的概念。Mackenzie的概念并不容易使用，而超改写给出了一个快速发展的前景，李代数体和Poisson流形具有双重关系，即流形上的对偶Lie代数体具有Poisson结构，而流形上的Poisson结构诱导其余切丛上的Lie代数体结构。利用这些关系，Nguyen Tien Zung得到了一般李代数体的标准型和线性化定理，推广了已有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.