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

Abstract

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].

Functors between representation categories. Universal modules

“…We use Sweedler's notation with implied summation for both coalgebras (resp. bialgebras), as in ∆(c) = c (1) ⊗ c 2 , and for comodule structures: a right C-comodule structure ρ on a vector space V will be denoted by ρ(v) = v (0) ⊗ v (1) . When we need to be precise, the structures involved will be adorned.…”

“…One of the most general constructions of universal (co)acting bialgebras/Hopf algebras, performed in the setting of Ω-algebras, was introduced in [1] together with generalized duality results. Necessary and sufficient conditions for the existence of universal coacting bialgebras/Hopf algebras are provided, explaining in this general setting the need for assuming finite dimensionality in both Manin and Tambara's papers.…”