Cyclic theory of Lie algebroids

In this thesis we study the cyclic theory of universal enveloping algebras of Lie algebroids.
Lie algebroids are geometrical objects that encode infinitesimal symmetries, and the concept encompasses many classical objects from geometry, such as Poisson manifolds, foliations and actions of Lie algebras on manifolds. The study of geometrical objects is in many cases equivalent to the study of the algebras of functions on these objects, and this observation led to the field of noncommutative geometry, where one studies noncommutative algebras, that are not necessarily related to geometrical objects, with techniques from geometry. For each Lie algebroid one can define a noncommutative algebra, called the universal enveloping algebra, which generalizes the algebra of differential operators on a manifold and the universal enveloping algebra of a Lie algebra. In this thesis we show that the cyclic theory of this algebra is equal to the Poisson (co)homology of the dual of the Lie algebroid, which in turn is equal to the Lie algebroid cohomology with values in the symmetric algebra of the adjoint representation up to homotopy (twisted by a line bundle). Moreover, we define a trace-density map from the cyclic theory of the universal enveloping algebra to the de Rham complex of the Lie algebroid, which generalizes known results for the tangent bundle of a manifold. We use a Čech resolution of the de Rham complex, which makes the construction suitable for holomorphic Lie algebroids as well. Both the calculation of the cyclic theory of the universal enveloping algebra as well as the construction of the trace-density map is based on the Poincaré–Birkhoff–Witt theorem for Lie algebroids, which we therefore prove first.