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On a cubical generalization of preprojective algebras Minamoto, Hiroyuki


In this abstract $K$ denotes a field of char $K = 0$ and $Q$ denotes a finite acyclic quiver. Recall that the preprojective algebra $\Pi(Q) = K\overline{Q}/(\rho )$ of a quiver $Q$ is the path algebra $K \overline{Q}$ of the double quiver $\overline{Q}$ of $Q$ with the mesh relation $\rho=\sum_{\alpha \in Q_{1}} \alpha \alpha^{*} - \alpha^{*} \alpha$. It is an important mathematical object having rich representation theory and plenty of applications. In this joint work with M. Herschend, we study a cubical generalization $\Lambda= \Lambda(Q) := K \overline{Q}/([a, \rho] \mid a \in \overline{Q}_{1})$ where $[-,+]$ is the commutator. We note that our algebra $\Lambda$ is a special case of algebras $\Lambda_{\lambda, \mu}$ introduced by Etingof-Rains, which is a special case of algebras $\Lambda_{P}$ introduced by Cachazo-Katz-Vafa. However, our algebra $\Lambda$ of very special case has intriguing properties, among other things it provides the universal Auslander-Reiten triangle for $K Q$. We may equip $\Lambda$ with a grading by setting $\deg \alpha = 0, \deg \alpha^{*} : =1$ for $\alpha \in Q_{1}$. We introduce an algebra to be $A = A(Q) := \begin{pmatrix} K Q & \Lambda_{1} \\ 0 & K Q \end{pmatrix}$ where $\Lambda_{1}$ is the degree $1$-part of $\Lambda$. Our results combining with other existing results show that the algebras $A(Q)$ and $\Lambda (Q)$ are one-dimensional higher versions of $K Q$ and $\Pi (Q)$.

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