Workshop on Lie algebroids and Lie groupoids in Differential Geometry

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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.