Generator coalgebras are not necessarily quasi-coFrobenius

TL;DR

This paper investigates whether generator coalgebras are necessarily quasi-coFrobenius, providing counterexamples in general and characterizations for specific classes like monomial pointed coalgebras.

Contribution

It demonstrates that generator coalgebras are not always quasi-coFrobenius and characterizes when monomial pointed coalgebras are quasi-coFrobenius.

Findings

Counterexamples show generator coalgebras need not be quasi-coFrobenius.

Monomial pointed coalgebras are quasi-coFrobenius under certain conditions.

Characterization of quivers with monomial subcoalgebras that are quasi-coFrobenius.

Abstract

We study the problem of whether a coalgebra that generates its category of left (right) comodules is left (right) quasi-coFrobenius or not. We prove it does not hold in general, by giving a method of constructing counterexamples. This gives a negative answer to a question stated in \cite{kn:coalgen}. We also prove it is true for monomial pointed coalgebras and we characterize the quivers $Q$ such that $\Bbbk Q$ admits a monomial subcoalgebra that is left (right) quasi-coFrobenius.