An algorithm for the periodicity of deformed preprojective algebras of Dynkin types [formula omitted], [formula omitted] and [formula omitted

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 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.