An algorithm for the periodicity of deformed preprojective algebras of Dynkin types $\mathbb{E}_6$, $\mathbb{E}_7$ and $\mathbb{E}_8$

TL;DR

This paper develops an algorithmic method to prove that all deformed preprojective algebras of Dynkin types E6, E7, and E8 are periodic, and classifies them up to isomorphism, revealing characteristic-dependent existence.

Contribution

It introduces a numeric algorithm that completes the proof of periodicity for these algebras and classifies them, extending previous theoretical results.

Findings

Deformed preprojective algebras of types E6, E7, E8 are periodic.

Non-trivial deformed preprojective algebras of types E7 and E8 exist only in characteristic 2.

The algorithm classifies such algebras up to isomorphism.

Abstract

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 $\mathbb{E}_7$ and $\mathbb{E}_8$ exist only in characteristic 2. As a consequence, we show that deformed preprojective algebras of Dynkin types $\mathbb{E}_6$, $\mathbb{E}_7$ and $\mathbb{E}_8$ 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 $\varphi_f: P^f(\mathbb{E}_n) \to P(\mathbb{E}_n)$ 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.