Weak Bialgebras of Fractions

TL;DR

This paper develops a method to construct fractions of Weak Bialgebras using an 'almost central' set of group-like elements, enabling new algebraic structures linked to fusion categories.

Contribution

It introduces the 'almost central' condition for denominators in Weak Bialgebras and constructs their fractions, expanding the class of Weak Bialgebras with applications to fusion categories.

Findings

Constructed algebra of fractions for Weak Bialgebras using 'almost central' denominators.

Identified conditions under which the algebra of fractions exists and retains Weak Bialgebra structure.

Produced new Weak Bialgebras related to SL2-fusion categories.

Abstract

We construct the algebra of fractions of a Weak Bialgebra relative to a suitable denominator set of group-like elements that is `almost central', a condition we introduce in the present article which is sufficient in order to guarantee existence of the algebra of fractions and to render it a Weak Bialgebra. The monoid of all group-like elements of a coquasi-triangular Weak Bialgebra, for example, forms a suitable set of denominators as does any monoid of central group-like elements of an arbitrary Weak Bialgebra. We use this technique in order to construct new Weak Bialgebras whose categories of finite-dimensional comodules relate to SL2-fusion categories in the same way as GL(2) relates to SL(2).