Classification of weak bialgebra structures on path algebras of general quivers

Classify all weak bialgebra structures on the path algebra mathbb{k}Q for an arbitrary quiver Q, extending the classifications known for Hopf quivers, arbitrary quivers in restricted settings, and acyclic quivers with two vertices.

Background

The paper places its study within the broader problem of classifying weak bialgebra structures on path algebras of quivers. Although several partial results are citedincluding classifications for particular quivers and for acyclic quivers with two verticesa complete treatment for arbitrary quivers has not been obtained.

The paper addresses the special case in which the quiver has no arrows, so that its path algebra is the commutative semisimple algebra $\Bbbk^{\oplus n}$. It proves correspondences between bialgebra structures and finite monoids, and between weak bialgebra structures and finite small categories, thereby contributing to but not resolving the general quiver classification problem.