Generalizations of the Sweedler dual

Hans E. Porst*, Ross Street

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    5 Citations (Scopus)

    Abstract

    As left adjoint to the dual algebra functor, Sweedler’s finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler’s construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring R the left adjoint of the dual algebra functor on the category of R-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf R-algebras, provided that R is noetherian and absolutely flat.

    Original languageEnglish
    Pages (from-to)619-647
    Number of pages29
    JournalApplied Categorical Structures
    Volume24
    Issue number5
    DOIs
    Publication statusPublished - 1 Oct 2016

    Keywords

    • Hopf algebra
    • coalgebra
    • Sweedler dual
    • finite dual coalgebra
    • monoidal functor

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