Algebraic properties of face algebras

TL;DR

This paper investigates the algebraic and homological properties of face algebras associated with finite quivers, revealing their structure as path algebras of the Kronecker square of the quiver and linking algebraic properties to graph-theoretic features.

Contribution

It establishes an isomorphism between Hayashi's face algebra and the path algebra of the Kronecker square of a quiver, connecting algebraic properties to graph-theoretic characteristics.

Findings

Face algebra is isomorphic to the path algebra of the Kronecker square of the quiver.

Algebraic properties of face algebras are characterized by properties of the underlying quiver.

Homological properties are derived from the graph-theoretic structure of the quiver.

Abstract

Prompted an inquiry of Manin on whether a coacting Hopf-type structure $H$ and an algebra $A$ that is coacted upon share algebraic properties, we study the particular case of $A$ being a path algebra $\Bbbk Q$ of a finite quiver $Q$ and $H$ being Hayashi's face algebra $\mathfrak{H}(Q)$ attached to $Q$. This is motivated by the work of Huang, Wicks, Won, and the second author, where it was established that the weak bialgebra coacting universally on $\Bbbk Q$ (either from the left, right, or both sides compatibly) is $\mathfrak{H}(Q)$. For our study, we define the Kronecker square $\widehat{Q}$ of $Q$, and show that $\mathfrak{H}(Q) \cong \Bbbk \widehat{Q}$ as unital algebras. Then we obtain ring-theoretic and homological properties of $\mathfrak{H}(Q)$ in terms of graph-theoretic properties of $Q$ by way of $\widehat{Q}$.