[Paper Review] Quantum Groupoids and their Hopf Cyclic Cohomology

Mohammad Hassanzadeh, Bahram Rangipour|arXiv (Cornell University)|Jul 29, 2011

Algebraic structures and combinatorial models14 references3 citations

TL;DR

This paper introduces $\times$-Hopf coalgebras as a new quantization of groupoids, generalizing Hopf algebras and weak Hopf algebras. It develops a Hopf cyclic cohomology theory with coefficients in stable-anti-Yetter-Drinfeld (SAYD) modules, proving that the cyclic complex of a groupoid coalgebra precisely matches the nerve-generated cyclic complex, thus resolving a consistency gap in prior approaches.

ABSTRACT

A new quantization of groupoids under the name of imes-Hopf coalgebras is introduced. We develop a Hopf cyclic theory with coefficients in stable-anti-Yetter-Drinfeld modules for imes-Hopf coalgebras. We use imes-Hopf coalgebras to study coextensions of coalgebras. Finally, equivariant imes-Hopf coalgebra Galois coextensions are defined and applied as functors between categories of stable anti-Yetter-Drinfeld modules over imes-Hopf coalgebras involved in the coextension.

Motivation & Objective

• To address the inconsistency between the Hopf cyclic complex of a groupoid algebra and the cyclic complex from its nerve, which fails to coincide in prior formulations.
• To propose $\times$-Hopf coalgebras as a correct quantization of groupoids by using the coalgebra structure instead of the algebra structure.
• To develop a Hopf cyclic cohomology theory for $\times$-Hopf coalgebras with coefficients in stable-anti-Yetter-Drinfeld (SAYD) modules.
• To define equivariant $\times$-Hopf coalgebra Galois coextensions and show they induce functors between categories of SAYD modules.
• To establish that the cyclic complex of the groupoid coalgebra matches the nerve-generated cyclic complex, achieving consistency in homological invariants.

Proposed method

• Axiomatize $\times$-Hopf coalgebras as generalizations of Hopf algebras and weak Hopf algebras, using bicoalgebroid structures as a foundation.
• Define stable-anti-Yetter-Drinfeld (SAYD) modules over $\times$-Hopf coalgebras via compatible coactions and actions.
• Construct the Hopf cyclic cohomology complex for $\times$-Hopf coalgebras using the standard cyclic operator framework with coefficients in SAYD modules.
• Use the groupoid coalgebra as the primary example, showing its cyclic complex matches the nerve-generated cyclic complex exactly.
• Introduce equivariant $\times$-Hopf coalgebra Galois coextensions as a generalization of Hopf Galois extensions, with structure maps satisfying compatibility conditions.
• Prove that such coextensions induce functors between categories of SAYD modules over the involved $\times$-Hopf coalgebras, extending known results from Hopf algebra theory.

Experimental results

Research questions

• RQ1 Does the Hopf cyclic complex of a groupoid algebra as a $\times$-Hopf algebra coincide with the cyclic complex from the nerve of the groupoid?
• RQ2 Can a new quantization of groupoids be constructed using coalgebraic rather than algebraic structures to achieve consistency in cyclic homology?
• RQ3 Is there a well-defined Hopf cyclic cohomology theory for $\times$-Hopf coalgebras with coefficients in SAYD modules?
• RQ4 Do equivariant $\times$-Hopf coalgebra Galois coextensions induce functors between categories of SAYD modules over the respective coalgebras?
• RQ5 Does the cyclic complex of the groupoid coalgebra match the nerve-generated cyclic complex under the new framework?

Key findings

• The Hopf cyclic complex of a groupoid coalgebra as a $\times$-Hopf coalgebra precisely coincides with the cyclic complex generated by the nerve of the groupoid, resolving a key inconsistency in prior approaches.
• The construction of $\times$-Hopf coalgebras via groupoid coalgebras provides a consistent quantization of groupoids, generalizing both Hopf algebras and weak Hopf algebras.
• A well-defined Hopf cyclic cohomology theory is established for $\times$-Hopf coalgebras with coefficients in stable-anti-Yetter-Drinfeld modules.
• Equivariant $\times$-Hopf coalgebra Galois coextensions are defined and shown to induce functors between categories of SAYD modules over the involved $\times$-Hopf coalgebras.
• The use of coalgebraic structures (rather than algebraic) in the quantization ensures compatibility with the cyclic nerve construction, validating the new framework for groupoid homology.
• The proof relies on coaction compatibility, comodule coalgebra axioms, and properties of the Nakayama automorphism, with explicit verification of the cyclic operator relations via tensor product and comultiplication structures.