An index theorem for Lie algebroids

Arie Blom; Hessel Posthuma

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arXiv · 1512.07863

An index theorem for Lie algebroids

Arie Blom·Hessel Posthuma

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Abstract

We study Lie algebroids from the point of view noncommutative geometry. More specifically, using ideas from deformation quantization, we use the PBW-theorem for Lie algebroids to construct a Fedosov-type resolution for the associated sheaves of Weyl algebras. This resolution enables us to construct a "character map" --in the derived category-- from the sheafified cyclic chain complexes to the Chevalley--Eilenberg complex of the Lie algebroid. The index theorem computes the evaluation of this map on the trivial cycle in terms of the Todd--Chern characteristic class. Finally, we show compatibility of the character map with the Hochschild--Kostant--Rosenberg morphism.

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Arie Blom, Hessel Posthuma. 2015-12-24. An index theorem for Lie algebroids. https://arxiv.org/abs/1512.07863

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