Superálgebras minimais e a classificação das variedades minimais de crescimento exponencial

Abstract

Drensky, in the 1980s, conjectured that a variety of algebras V is minimal if and only if Id(V) is a product of verbally prime T-ideals. Giambruno and Zaicev, in 2003, proved that conjecture and presented a classification of minimal varieties having exponential growth. The main goal of this dissertation is to show this classification establishing interesting relations between minimal superalgebras, minimal varieties and verbally prime T-ideals. We will also see relevant results that relate the finite dimensional superalgebras to the minimal ones. It is noteworthy that in order to prove our main result, we work with important concepts of the PI-theory, including the Grassmann envelope and the superenvelope of a superalgebra. We finish the dissertation with a brief discussion concerning the minimal supervarieties.