[Paper Review] The cyclic theory of Hopf algebroids

Niels Kowalzig, Hessel Posthuma|arXiv (Cornell University)|Apr 30, 2009

Homotopy and Cohomology in Algebraic Topology3 references4 citations

TL;DR

This paper develops a systematic cyclic cohomology and homology theory for Hopf algebroids using their module categories and cyclic duality, establishing structure theorems for commutative and cocommutative cases and computing the theories explicitly for Lie-Rinehart algebras and étale groupoids, where it recovers known results such as the van Est isomorphism and Lie algebra cohomology via jet algebras.

ABSTRACT

We give a systematic description of the cyclic cohomology theory of Hopf algebroids in terms of its associated category of modules. Then we introduce a dual cyclic homology theory by applying cyclic duality to the underlying cocyclic object. We derive general structure theorems for these theories in the special cases of commutative and cocommutative Hopf algebroids. Finally, we compute the cyclic theory in examples associated to Lie-Rinehart algebras and étale groupoids.

Motivation & Objective

• To establish a systematic framework for cyclic cohomology of Hopf algebroids using their module categories.
• To introduce a dual cyclic homology theory via cyclic duality applied to the cocyclic object.
• To derive general structure theorems for commutative and cocommutative Hopf algebroids.
• To compute the cyclic theory in geometric examples, including Lie-Rinehart algebras and étale groupoids.
• To relate the results to known invariants such as Lie algebra cohomology and groupoid homology via the van Est isomorphism.

Proposed method

• Construct the Hopf-cyclic cohomology via the standard cocyclic object associated to the category of modules over a Hopf algebroid.
• Apply cyclic duality to the underlying cocyclic object to define a dual cyclic homology theory.
• Use the Hopf-Galois map to relate the cyclic duality to the antipode structure in the Hopf algebroid.
• Employ the bar and cobar resolutions to describe Hochschild theory with coefficients in the context of Hopf algebroids.
• Utilize coinvariant and invariant constructions to relate the cyclic theories to geometric invariants.
• Leverage the jet space construction of Lie-Rinehart algebras to define the associated Hopf algebroid and compute its cyclic homology.

Experimental results

Research questions

• RQ1 How can cyclic cohomology be systematically defined for Hopf algebroids using their module categories?
• RQ2 What is the structure of the dual cyclic homology theory obtained via cyclic duality?
• RQ3 How do the cyclic theories simplify in the commutative and cocommutative cases?
• RQ4 What is the relationship between the cyclic homology of étale groupoids and their groupoid homology?
• RQ5 How does the cyclic theory of Lie-Rinehart algebras recover Lie algebra cohomology via jet algebras?

Key findings

• The cyclic cohomology of the jet algebra of a Lie-Rinehart algebra is isomorphic to the Lie algebra cohomology of the underlying Lie algebra with trivial coefficients.
• For an étale groupoid, the cyclic homology of its jet algebra is isomorphic to the groupoid homology, generalizing the van Est isomorphism.
• The cyclic homology of the jet algebra of a Lie algebroid at the unit is isomorphic to the Lie algebra cohomology of the corresponding Lie algebra.
• The cyclic cohomology of the coordinate ring of an affine variety is isomorphic to the product of its algebraic de Rham cohomology groups.
• The dual cyclic homology of the Hopf algebroid associated to a Lie groupoid recovers the differential cohomology of the groupoid.
• The construction of the Hopf algebroid from a Lie algebroid via jet spaces provides a consistent framework for computing cyclic invariants in noncommutative geometry.