[Paper Review] An index theorem for Lie algebroids

Arie Blom, Hessel Posthuma|arXiv (Cornell University)|Dec 24, 2015

Homotopy and Cohomology in Algebraic Topology21 references3 citations

TL;DR

This paper establishes an index theorem for Lie algebroids by constructing a Fedosov-type resolution of the sheaf of Weyl algebras, enabling a derived character map from periodic cyclic homology to Lie algebroid cohomology. The key result computes the evaluation of this map on the trivial cycle as the Todd–Chern characteristic class, generalizing the algebraic index theorem and extending the Hochschild–Kostant–Rosenberg isomorphism to the Lie algebroid setting.

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.

Motivation & Objective

• To extend the algebraic index theorem to Lie algebroids using noncommutative geometry and deformation quantization.
• To construct a derived character map from the periodic cyclic homology of the universal enveloping algebra of a Lie algebroid to its Lie algebroid cohomology.
• To establish compatibility of this character map with the Hochschild–Kostant–Rosenberg morphism in the context of Lie algebroids.
• To generalize the Riemann–Roch theorem to Lie algebroids by relating cyclic cocycles to characteristic classes via Chern–Weil theory.

Proposed method

• Using the PBW theorem for Lie algebroids to construct a Fedosov-type resolution of the sheaf of Weyl algebras.
• Defining a canonical morphism in the derived category from the periodic cyclic complex to the twisted Chevalley–Eilenberg complex.
• Employing the Chern–Weil homomorphism to relate curvature forms of connections to characteristic classes in Lie algebroid cohomology.
• Extending the Hochschild cocycle τ₂ₙᴴᵒᶜʰ to a cyclic cocycle τ₂ₙʷ via the exponential of the insertion operator ι_π.
• Using the evaluation map ev₁ to relate the cyclic cocycle to characteristic classes in Lie algebra cohomology.
• Applying the local Riemann–Roch theorem to equate the evaluation of the cyclic cocycle with the Todd–Chern class.

Experimental results

Research questions

• RQ1 How can the algebraic index theorem be generalized to the setting of Lie algebroids?
• RQ2 What is the precise relationship between the periodic cyclic homology of the universal enveloping algebra of a Lie algebroid and its Lie algebroid cohomology?
• RQ3 How does the Hochschild–Kostant–Rosenberg morphism extend to Lie algebroids, and what is its compatibility with the character map?
• RQ4 What role do characteristic classes—specifically the Todd and Chern classes—play in the index theorem for Lie algebroids?

Key findings

• A canonical morphism Φ exists in the derived category from the periodic cyclic complex of the universal enveloping algebra to the twisted Chevalley–Eilenberg complex of the Lie algebroid.
• The character map Φ restricts to a commutative diagram involving the Hochschild–Kostant–Rosenberg map, the Todd class, and the Chern class of a locally free module.
• The evaluation of the cyclic cocycle τ₂ₙʷ on the trivial cycle equals the Todd–Chern characteristic class, as stated in the local Riemann–Roch theorem.
• The cyclic cocycle τ₂ₙʷ is invariant under the action of the Lie algebra 𝔤𝔩(n,𝕂)⊕𝔤𝔩(r,𝕂), confirming its basic and invariant nature.
• The construction generalizes the Bressler–Nest–Tsygan index theorem to arbitrary Lie algebroids via a derived character map.
• The method provides a framework for relating noncommutative cyclic homology to commutative Lie algebroid cohomology through characteristic classes.