Módulos irredutíveis de dimensão 3 sobre zero álgebras e bases de Gröbner

Abstract

In this work, we describe the Irreducible Modules of dimension 3 in zero algebras, in the class of commutative and power associative algebras of nilindex four, using the theory of Gröbner bases. The approach consists of exploring the product of the algebra over the module, represented by matrices $3 \times 3$ by fixing a base of the module. The objective is to identify the matrices, excluding those related by conjugation. Even though the classification of irreducible modules of dimension 3 over the zero algebra of dimension two is known, we propose a computational method that uses the Gröbner bases to obtain this classification. During the classification process, we define the affine manifold of nilpotent matrices. However, realizing that all polynomials that arise in the proposed classification are homogeneous, it is more appropriate to work with the projective space instead of the affine space. We present a computational procedure in the SageMath algebraic system to calculate and simplify this process. Although the Gröbner basis obtained for $3 \times 3$ matrices is small, the SageMath program does not have parallel executable support. As a result, the computational capacity of the cluster, made up of 240 cores, was equivalent to that of a common laptop. Therefore, with the serial version, it was not possible to complete the classification.