Self-Injective Jacobian Algebras from Postnikov Diagrams

摘要

We study a finite-dimensional algebra Lambda from a Postnikov diagram D in a disk, obtained from the dimer algebra of Baur-King-Marsh by factoring out the ideal generated by the boundary idempotent. Thus, Lambda is isomorphic to the stable endomorphism algebra of a cluster tilting module T is an element of CM(B) introduced by Jensen-King-Su in order to categorify the cluster algebra structure of C[Gr(k)(C-n)]. We show that Lambda is self-injective if and only if D has a certain rotational symmetry. In this case, Lambda is the Jacobian algebra of a self-injective quiver with potential, which implies that its truncated Jacobian algebras in the sense of Herschend-Iyama are 2-representation finite. We study cuts and mutations of such quivers with potential leading to some new 2-representation finite algebras.