V-universal Hopf algebras (co)acting on Ω-algebras

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 [Formula: see text], 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 [Formula: see text]-universal measuring coalgebras and [Formula: see text]-universal comeasuring algebras between [Formula: see text]-algebras [Formula: see text] and [Formula: see text], relative to a fixed subspace [Formula: see text] of [Formula: see text]. By considering the case [Formula: see text], we derive the notion of a [Formula: see text]-universal (co)acting bialgebra (and Hopf algebra) for a given algebra [Formula: see text]. 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 [Formula: see text]-universal acting bi/Hopf algebra and the finite dual of the [Formula: see text]-universal coacting bi/Hopf algebra under certain conditions on [Formula: see text] in terms of the finite topology on [Formula: see text].