Connections on lie Algebroids and the Weil-Kostant Theorem

Από το τεύχος 44 του περιοδικού Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

The Weil ? Kostant theorem characterizes those alternating (real ? valued) 2 ? forms which are curvature forms of connections in U (1) ? bundles. We present in this talk an overview of a corresponding result for arbitrary Lie groups which was proved by K.C.H. Mackenzie in 1987, using the notion of Lie algebroids. A Lie algebroid is a vector bundle whose module of sections has a Lie algebra bracket and a vector bundle morphism to the tangent space of the base manifold which preserves the Lie brackets. The main aims of this talk are two: First, to demonstrate that the theory of Lie algebroids is a suitable environment in which one can do connection theory (which is this speaker?s main research interest), and second to show how one can use Lie algebroids to tackle non ? abelian problems which often arise in the process of quantization and elsewhere.

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