Inductive limit violates quasi-cocommutativity

TL;DR

This paper demonstrates that the inductive limit of a specific system of quasi-cocommutative C*-bialgebras fails to preserve quasi-cocommutativity, revealing limitations in the structural closure properties of this algebraic category.

Contribution

It provides a counterexample showing that the inductive limit of quasi-cocommutative C*-bialgebras is not necessarily quasi-cocommutative, highlighting a key structural limitation.

Findings

Inductive limits can break quasi-cocommutativity in C*-bialgebras.

The category of quasi-cocommutative C*-bialgebras is not closed under inductive limits.

Counterexamples challenge assumptions about stability of algebraic properties.

Abstract

We show that the inductive limit of a certain inductive system of quasi-cocommutative C$^{*}$-bialgebras is not quasi-cocommutative. This implies that the category of quasi-cocommutative C$^{*}$-bialgebras is not closed with respect to the inductive limit.