[Paper Review] On the structure of (co-Frobenius) Hopf algebras

Nicolás Andruskiewitsch, Juan Cuadra|arXiv (Cornell University)|Nov 15, 2010

Algebraic structures and combinatorial models39 references3 citations

TL;DR

This paper introduces the standard filtration—a generalization of the coradical filtration—for Hopf algebras with injective antipode, enabling a structure theorem that deforms any such Hopf algebra as a bosonization of its Hopf coradical by a connected graded diagram in Yetter-Drinfeld modules. The key contribution is a characterization: a Hopf algebra is co-Frobenius if and only if its Hopf coradical is co-Frobenius and its diagram is finite-dimensional, with the standard filtration being finite in this case.

ABSTRACT

We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra is co-Frobenius if and only if its Hopf coradical is so and the diagram is finite dimensional. We also prove that the standard filtration of such Hopf algebras is finite. Finally, we show that extensions of co-Frobenius (resp. cosemisimple) Hopf algebras are co-Frobenius (resp. cosemisimple).

Motivation & Objective

• To generalize the coradical filtration via a new filtration—called the standard filtration—applicable to Hopf algebras with injective antipode.
• To provide a structural decomposition of such Hopf algebras as deformations of bosonizations of their Hopf coradical by a connected graded diagram in Yetter-Drinfeld modules.
• To classify co-Frobenius Hopf algebras by reducing the problem to classifying co-Frobenius Hopf coradicals and finite-dimensional connected graded Hopf algebras in the relevant Yetter-Drinfeld category.
• To establish conditions under which extensions of co-Frobenius or cosemisimple Hopf algebras remain co-Frobenius or cosemisimple, respectively.
• To resolve a conjecture on the finiteness of the coradical filtration for co-Frobenius Hopf algebras by proving the standard filtration is finite in this case.

Proposed method

• Define the standard filtration using iterative wedge operations on the coradical, starting from the Hopf coradical—the subalgebra generated by the coradical.
• Show that the standard filtration is a Hopf algebra filtration when the antipode is injective, allowing the construction of the associated graded Hopf algebra $\operatorname{gr}H$.
• Identify $\operatorname{gr}H$ as a bosonization of the Hopf coradical $H_{[0]}$ by a connected graded Hopf algebra $R$ in the category of Yetter-Drinfeld $H_{[0]}$-modules (the diagram of $H$).
• Use this structure to interpret $H$ as a deformation of $\operatorname{gr}H$ in a suitable cohomological framework.
• Apply the structure theorem to the class of co-Frobenius Hopf algebras, using representation-theoretic characterizations involving finite-dimensional injective hulls of simple comodules.
• Establish a duality between co-Frobenius and cosemisimple properties via extension theory, particularly under faithful coflatness assumptions on the quotient comodule.

Experimental results

Research questions

• RQ1 Under what conditions can a Hopf algebra with injective antipode be decomposed as a deformation of a bosonization of its Hopf coradical by a connected graded diagram?
• RQ2 When is a Hopf algebra co-Frobenius, and how does this property relate to its Hopf coradical and diagram?
• RQ3 Is the standard filtration of a co-Frobenius Hopf algebra finite, and does this imply the finiteness of the coradical filtration?
• RQ4 Under what conditions are extensions of co-Frobenius (or cosemisimple) Hopf algebras themselves co-Frobenius (or cosemisimple)?
• RQ5 How do properties like the existence of nonzero integrals and finite-dimensional injective hulls of comodules interact in extensions of Hopf algebras?

Key findings

• A Hopf algebra with injective antipode is isomorphic to a deformation of the bosonization of its Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the Hopf coradical.
• A Hopf algebra is co-Frobenius if and only if its Hopf coradical is co-Frobenius and its diagram is finite-dimensional.
• The standard filtration of a co-Frobenius Hopf algebra is finite, confirming a conjecture on the finiteness of the coradical filtration in this class.
• An extension $1 \to A \to B \to C \to 1$ of Hopf algebras with $B$ faithfully coflat as a $C$-comodule satisfies: $B$ is co-Frobenius if and only if $A$ and $C$ are co-Frobenius.
• A Hopf algebra $B$ is cosemisimple if and only if both $A$ and $C$ are cosemisimple in such an extension, provided $B$ is faithfully coflat over $C$.
• For a Hopf algebra map $g: K \to H$ with $H$ finitely generated as a right $K$-module, $H^0$ is co-Frobenius if $K^0$ is co-Frobenius, and vice versa if $H$ is flat over $K$.