2020/05/31 by A. L. Agore, Ana Agore, А. С. Гордиенко +2 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.CT #math.QA #math.RA #math.RT #msc:16T05 #msc:16T25 #msc:16W22 #msc:16W50
paper · pdf · doi:10.1142/s0219199721500954
published as Commun. Contemp. Math. 25 (2023), 2150095 (40 pages) · 31 pages; the formulation of Theorem 4.7 was clarified and minor misprints were corrected
openalex created_date 2020/06/05 · openalex publication_date 2021/10/25 · openalex updated_date 2026/07/28 · arxiv created 2026/08/01 · arxiv updated 2026/08/04
We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced [A. L. Agore, A. S. Gordienko and J. Vercruysse, On equivalences of (co)module algebra structures over Hopf algebras, J. Noncommut. Geom., doi: 10.4171/JNCG/428.] bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of A, called [Formula: see text]-algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study V-universal measuring coalgebras and V-universal comeasuring algebras between [Formula: see text]-algebras A and B, relative to a fixed subspace V of [Formula: see text]. By considering the case [Formula: see text], we derive the notion of a V-universal (co)acting bialgebra (and Hopf algebra) for a given algebra A. In particular, this leads to a refinement of the existence conditions for the Manin–Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the V-universal acting bi/Hopf algebra and the finite dual of the V-universal coacting bi/Hopf algebra under certain conditions on V in terms of the finite topology on [Formula: see text].