[Paper Review] Cohomology and Coquasi-bialgebras in the category of Yetter-Drinfeld Modules

Iván Angiono, Alessandro Ardizzoni+1 more|arXiv (Cornell University)|Sep 16, 2015

Algebraic structures and combinatorial models27 references19 citations

TL;DR

This paper establishes a cohomological criterion for finite-dimensional Hopf algebras with the dual Chevalley property over a field of characteristic zero to be quasi-isomorphic to a Radford-Majid bosonization. It proves that if the third Hochschild cohomology group of the diagram in the Yetter-Drinfeld category vanishes, then the algebra is quasi-isomorphic to a bosonization of a connected bialgebra in that category. The key contribution is a cohomological obstruction condition for bosonization realizability via gauge equivalence.

ABSTRACT

We prove that a finite-dimensional Hopf algebra with the dual Chevalley Property over a field of characteristic zero is quasi-isomorphic to a Radford-Majid bosonization whenever the third Hochschild cohomology group in the category of Yetter-Drinfeld modules of its diagram with coefficients in the base field vanishes. Moreover we show that this vanishing occurs in meaningful examples where the diagram is a Nichols algebra.

Motivation & Objective

• To determine when a finite-dimensional Hopf algebra with the dual Chevalley property is quasi-isomorphic to a Radford-Majid bosonization.
• To identify cohomological conditions under which gauge deformations of coquasi-bialgebras in the Yetter-Drinfeld category yield bialgebras.
• To analyze the relationship between Hochschild cohomology in the Yetter-Drinfeld category and invariants of the cohomology of the bosonized algebra.
• To provide explicit examples where the third Hochschild cohomology group in the Yetter-Drinfeld category vanishes, despite nontrivial cohomology in the bosonized algebra.

Proposed method

• Use of the coradical filtration and associated graded coalgebra structure in the category of Yetter-Drinfeld modules.
• Application of gauge transformations to coquasi-bialgebras in the Yetter-Drinfeld category to study deformation invariance.
• Computation of Hochschild cohomology in the Yetter-Drinfeld category via invariants of the cohomology of the bosonized algebra.
• Leveraging spectral sequences and filtration arguments to relate cohomology of the original algebra to that of its associated graded and bosonized forms.
• Use of braided bialgebras and Nichols algebras as key examples, particularly of Cartan type and non-diagonal type.
• Proof of isomorphism between cohomology in the Yetter-Drinfeld category and invariants of the cohomology of the bosonized algebra via the D-action.

Experimental results

Research questions

• RQ1 Under what cohomological conditions is a finite-dimensional Hopf algebra with the dual Chevalley property quasi-isomorphic to a Radford-Majid bosonization?
• RQ2 How does the vanishing of the third Hochschild cohomology group in the Yetter-Drinfeld category relate to the existence of gauge-equivalent bialgebras?
• RQ3 Can the cohomology of a Nichols algebra in the Yetter-Drinfeld category vanish even when the cohomology of its bosonized algebra is nontrivial?
• RQ4 What role does the D-action play in computing invariants of Hochschild cohomology in the Yetter-Drinfeld category?
• RQ5 In which classes of braided algebras (e.g. Cartan type, non-diagonal type) does the third cohomology group in the Yetter-Drinfeld category vanish?

Key findings

• The paper proves that if H³_YD(D(A), k) = 0, then a finite-dimensional Hopf algebra A with the dual Chevalley property is quasi-isomorphic to a Radford-Majid bosonization E#H, where E is a connected bialgebra in the Yetter-Drinfeld category with gr(E) ≅ D(A).
• For the quantum group Bq of Cartan type, H³_YD(Bq, k) = 0, even though H³(Bq#kΓ, k) may be nontrivial, demonstrating a non-trivial cohomological distinction.
• In the case of the FKn algebra of non-diagonal type for n=3, H³_YD(FK3, k) = 0, showing the criterion applies beyond diagonal braided types.
• The isomorphism H³_YD(B, k) ≅ H³(B, k)^D holds, where D is the group algebra of the group acting, and this isomorphism is used to compute invariants and verify vanishing.
• The associated graded coalgebra of a connected coquasi-bialgebra in the Yetter-Drinfeld category is a connected bialgebra, a key structural result.
• The paper constructs a spectral sequence argument to relate the cohomology of the original algebra to that of its associated graded, enabling the proof of vanishing results in specific cases.