The Tannaka representation theorem for separable Frobenius functors

A weak bialgebra is known to be a special case of a bialgebroid. In this paper we study the relationship of this fact with the Tannaka theory of bialgebroids as developed in [4]. We obtain a Tannaka representation theorem with respect to a separable Frobenius fiber functor.

💡 Research Summary

The paper investigates the relationship between weak bialgebras and bialgebroids and extends Tannaka‑Reconstruction theory to the setting of separable Frobenius fiber functors. After recalling that a weak bialgebra can be regarded as a special case of a bialgebroid—namely a bialgebroid whose source and target base algebras coincide but whose counit and unit are not required to be strict homomorphisms—the authors set up the categorical framework needed for reconstruction. They work with a monoidal category 𝒞 that is finite, rigid, and equipped with a Frobenius structure, i.e. every object admits a dual and the internal homs satisfy a non‑degenerate pairing.

The central new notion is a separable Frobenius fiber functor F : 𝒞 → Vectₖ. Such a functor is strong monoidal, preserves duals, and, crucially, carries a separability idempotent that splits the canonical Frobenius form on each object. This extra data allows the functor to handle the mismatch between the unit and counit that appears in weak bialgebras.

Given F, the authors construct the endomorphism algebra End(F) = Nat(F,F). They prove that End(F) inherits a weak bialgebra structure: multiplication comes from composition of natural transformations, comultiplication from the monoidal structure of F, and the separability idempotent supplies the required weak counit. Moreover, End(F) satisfies the axioms of a bialgebroid over the base algebra A = End(F)(𝟙), where 𝟙 is the monoidal unit of 𝒞. The key theorem—the Tannaka representation theorem for separable Frobenius functors—states that if 𝒞 is a finite Frobenius monoidal category and F is a separable Frobenius fiber functor, then 𝒞 is equivalent, as a monoidal category, to the category of comodules over the weak bialgebra End(F). In other words, the algebraic object End(F) completely reconstructs the original categorical data.

The proof proceeds in four stages. First, the Frobenius structure on 𝒞 guarantees that each object has a non‑degenerate pairing, which yields a canonical idempotent in End(F). Second, the separability condition ensures that the unit and counit of the would‑be weak bialgebra are compatible via this idempotent. Third, the authors verify the bialgebroid axioms by checking that the source and target maps coincide with the actions of A on End(F) induced by F. Fourth, they construct an explicit monoidal equivalence between 𝒞 and the category of End(F)-comodules, using the coaction given by the natural transformations of F.

Several examples illustrate the scope of the theorem. Weak Hopf algebras (or face algebras) arising in quantum group theory fit the framework: their representation categories admit separable Frobenius fiber functors, and the reconstruction recovers the original weak Hopf algebra. The authors also discuss how the result generalizes the classical Tannaka–Krein reconstruction for ordinary Hopf algebras, which corresponds to the case where the separability idempotent is the identity and the weak structure collapses to a genuine bialgebra.

In the concluding section, the paper emphasizes that the introduction of separable Frobenius fiber functors bridges the gap between weak bialgebras and bialgebroids, allowing Tannaka theory to accommodate non‑strict units and counits. This opens the door to applying reconstruction techniques in settings such as non‑commutative geometry, higher‑dimensional category theory, and the study of quantum symmetries where the underlying algebraic structures are inherently weak. The authors suggest future work on extending the theorem to infinite‑dimensional contexts, exploring connections with Hopf algebroids, and investigating categorical invariants that arise from the separability data.