An algorithm for the periodicity of deformed preprojective algebras of Dynkin types E6, E7 and E8

• Published in: Applied mathematics and computation Vol. 409
• Main Author: Białkowski, Jerzy
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.11.2021
• Subjects: Deformed preprojective algebra; Periodic algebra; Preprojective algebra; Self-injective algebra; System of equations; Preprojective algebra; Deformed preprojective algebra; Periodic algebra; System of equations; Self-injective algebra
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2021.126289

•We construct a numeric algorithm for completing the proof of a conjecture asserting that all deformed preprojective algebras of generalized Dynkin type are periodic.•We obtain an algorithmic procedure showing that non-trivial deformed preprojective algebras of Dynkin types E7 and E8 exist only in characteristic 2.•We show that deformed preprojective algebras of Dynkin types E6,E7 and E8 are periodic. We construct a numeric algorithm for completing the proof of a conjecture asserting that all deformed preprojective algebras of generalized Dynkin type are periodic. In particular, we obtain an algorithmic procedure showing that non-trivial deformed preprojective algebras of Dynkin types E7 and E8 exist only in characteristic 2. As a consequence, we show that deformed preprojective algebras of Dynkin types E6,E7 and E8 are periodic and we obtain an algorithm for a classification of such algebras, up to algebra isomorphism. We do it by a reduction of the conjecture to a solution of a system of equations associated with the problem of the existence of a suitable algebra isomorphism φf:Pf(En)→P(En) described in Theorem 2.1. One also shows that our algorithmic approach to the conjecture is also applicable to the classification of the mesh algebras of generalized Dynkin type.