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Algebraic properties of face algebras

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 $\widehat{Q}$.