- The paper rigorously extends Takeuchi's correspondence to Hopf algebras lacking assumptions of faithful flatness or coflatness.
- It adapts classical theorems to establish bijections between right coideal subalgebras and left H-module factor coalgebras with a bijective antipode.
- The study explores module properties such as projectivity and injectivity, providing insights for noncommutative algebra and geometric representation theories.
An Examination of Takeuchi's Correspondence in Hopf Algebras
This paper by Serge Skryabin provides a comprehensive paper of Takeuchi's correspondence, focusing particularly on situations where the conditions of faithful flatness or faithful coflatness are not assumed. The Takeuchi correspondence itself establishes a relationship between right coideal subalgebras and left H-module factor coalgebras within the framework of a Hopf algebra H with bijective antipode. The paper reflects the importance of understanding these structures in the broader context of noncommutative algebra and their applications in various mathematical theories.
The paper recapitulates several foundational results derived by Takeuchi, Masuoka, and others, before expanding on them to include cases where the antipode of the Hopf algebra is bijective. A critical starting point for the discussion is Theorem 1.7, which delineates a bijection between the set of right coideal subalgebras of an algebra H and left H-module factor coalgebras when conditions of faithful (co)flatness apply. This bijection captures the correspondence under broad circumstances but falters when the antipode lacks bijectivity, requiring a narrower focus on the classes involved.
The document is diligent in revisiting central theorems and adapting them for broader applicability. For instance, Theorem 4.6 extends Takeuchi's correspondence to larger classes of algebras, noting how each finite-dimensional comodule embeds as a subcomodule. Such extendibility is notably instructive when analyzing observable subgroups within the algebra k[G] representing an affine algebraic group G.
A substantial part of the paper explores the projectivity and injectivity of Hopf modules across various categorical settings—critical in deciphering the conditions under which H can serve as a projective generator or injective cogenerator in the category of modules or comodules. Ultimately, it offers a characterization of flatness over coideal subalgebras, analogous to cogenerator theories in module categories.
Skryabin further explores conditions under which the flatness of a Hopf algebra over its coideal subalgebras or coflatness over its factor coalgebras implies corresponding faithful versions of these properties. For instance, Theorem 7.4 conjectures that faithful (co)flatness on one side can be inferred under certain dual-sided conditions, thereby bridging gaps between standard and faithful versions of these properties.
The resource extends the discussion to quotient categories and localizations, both central components for understanding localizing subcategories of modules and comodules in algebraic settings. Using intricate interplays between localization techniques and module theory, the paper touches upon how quotient categories can represent significant insights when bridging between coideal subalgebras and their respective coalgebraic factors.
In essence, Skryabin’s work robustly expands the mathematical toolkit available for exploring Hopf algebraic structures, extending classical results to less restricted assumptions, and prompting further research in the structural interdependencies inherent in Hopf modules. As Hopf algebra continues to bear relevance across quantization and geometric representation theories, understanding these underpinning frameworks is crucial for advancements in theoretical insights and practical applications. Future investigations could further explore the outlined questions and dualities to refine our comprehension of the algebraic invariants these structures embody.