Extensions, cohomology and classification for Lie algebroids and Lie groupoids

Abstract

This thesis studies the cohomology and classifies extensions (short exactsequences) of transitive Lie algebroids and transitive Lie groupoids. Suchextensions are related to geometric prequantization, but are also independently interesting from the connection theory point of view.General extensions of Lie algebroids were classified by Mackenzie using ageneralisation of the Eilenberg-MacLane cohomology for the classification ofextensions of Lie algebras. Mackenzie saw such extensions as lifts of a certaintype of crossed module. We apply this point of view here to PBG-algebroids,a notion which allows an extension of Lie algebroids to be replaced by asingle Lie algebroid with a group action. PBG-algebroids moreover haveconnections which respect the group action. Here they are called isometablicto distinguish them from the usual notion of an equivariant connection.We give a PBG form of the classification theorem of Mackenzie, andhence show that PBG-algebroids admit trivialisations which also respectthe group action. These are called isometablic trivialisations and they giverise to the local existence of flat isometablic connections. Moreover, theylead to a new classification for extensions, of Cech-type, which also respectsthe group action of the corresponding PBG-algebroid.Extensions of Lie groupoids correspond to groupoids with a PBG structure as well. The local flat isometablic connections, when integrated, giverise to isometablic sections of the PBG-groupoid and a localisation of thegroup action on the gauge group bundle which satisfies a certain cocyclecondition. The transition functions are isometablic with respect to this localisation and it is shown that this data classifies PBG-groupoids. Moreover,the isometablic transition data are reformulated so as to reflect the natureof the PBG-groupoid as an extension and at the same time differentiate tothe Cech type cohomology that classifies PBG-algebroids. Lastly, crossedmodules of PBG-groupoids are studied, as well as their relation with crossedmodules of PBG-algebroids.