Métodos de clasificación de álgebras con anulador no nulo

Abstract

The classification of algebras is an important and an interesting problem in Modern Algebra. There are algebraic classifications, geometric classifications, degeneration level classifications and some other. In this essay, we focus on the algebraic classification, that is, on the problem of finding all the algebras module isomorphisms of a certain dimension. Specifically, in the classification of algebras with non null annihilator. To this end, we make use of one type of algebra extensions: the so-called annihilator extensions. This concept has been studied in depth in Theory of Lie Algebras, due to its numerous applications, especially outstanding in Physics. Due to this remarkable interest, the study of annihilator extensions of Lie algebras has a long history. However, the use of this notion to classify algebraically different classes of algebras is relatively recent, and that’s the center of our study. As a result of our research, we obtain a procedure to algebraically classify all algebras, of a certain class defined by polynomial identities, of dimension n with a mdimensional annihilator, using the classification of algebras of dimensión n-m. In addition, we apply this procedure in different specific cases, obtaining the classification of the n-dimensional algebras with (n - 2)-dimensional annihilator, the classification of the n-dimensional anticommutative algebras with (n - 3)-dimensional annihilator and the classification of the n-dimensional non-malcev binary-Lie algebras with (n - 4)-dimensional annihilator.