$V$-universal Hopf algebras (co)acting on $\Omega$-algebras

TL;DR

This paper introduces a unified theory of $V$-universal Hopf algebras acting on $ abla$-algebras, generalizing and refining previous universal constructions using the framework of $ abla$-algebras and vector space families.

Contribution

It develops a comprehensive approach to $V$-universal (co)acting Hopf algebras on $ abla$-algebras, unifying various existing theories and providing new existence criteria.

Findings

Established isomorphism between $V$-universal acting and coacting bi/Hopf algebras.

Refined conditions for the existence of universal coacting bi/Hopf algebras.

Unified treatment of algebras, coalgebras, and braided spaces through $ abla$-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 \cite{AGV1} 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 $\Omega$-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 $\Omega$-algebras $A$ and $B$, relative to a fixed subspace $V$ of $\Vect(A,B)$. By considering the case $A=B$, 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 $\End_F(A)$.