A method of realization of bialgebras, and Hopf algebras associated to some realizations

TL;DR

This paper presents a general method for constructing bialgebras and Hopf algebras from basic mathematical data, including conditions for their existence, uniqueness, and applicability to non-finitely generated operator algebras.

Contribution

It introduces a novel construction method for bialgebras and Hopf algebras from simple data, expanding applicability to non-finitely generated cases.

Findings

Constructed bialgebras with coproduct and counity from general data.

Established conditions for when these bialgebras admit unique Hopf algebra structures.

Described a broader abstract condition for the existence of associated Hopf algebras.

Abstract

This article introduces a method, which starting from simple and quite general mathematical data, allows to construct linear algebras of operators which are, each of them, endowed with a bialgebra structure (coproduct and counity). Moreover under some explicit and natural conditions on theses mathematical data we obtain linear algebras of operators with the following property: each of them, is either a Hopf algebra, or its bialgebra structure determines a more abstract Hopf algebra associated with it. Finally, we describe a more general abstract condition for theses bialgebras to admit a unique associated Hopf algebra. The presentation is adapted to the cases where the algebras of linear operators are not finitely generated. This article is restricted to the exposition of the method of construction and to the proofs of existence and uniqueness of the structures associated with each algebra of linear operators that are constructed. Nevertheless, it may be noticed that the ideals of relations associated with bialgebras that are obtained determine an algebraic domain which is of theoretical interest .