Variedades minimais de crescimento quadrático e a álgebra verbalmente prima M2(E)

Abstract

This work has two independent goals: the first is to classify the minimal varieties of quadratic growth and the second is to get results about the verbally prime Falgebra M2(E), where E is the Grassmann algebra of infinite dimension and F is a field of characteristic zero. For the first objective, it was necessary to present a finite generating set for the T-ideal of one subalgebra of the algebra of upper triangular matrices 3 × 3, denoted by M7, describing the sequence of codimensions cn(M7)}n_1, the cocharacter _n(M7) and the sequence of colengths ln(M7), for all n _ 1. This algebra appeared for the first time in a work of Giambruno and La Mattina, in 2005, where they classified the algebras with linear or onstant growth of codimensions. For M2(E) we initially developed a method to construct central polynomials of a particular degree of this algebra, from the central polynomials of M2(F) with the same degree. This method was based on results involving the explicit decomposition of FSn on its irreducible Sn-modules. Since this construction is algoritmic, we made the implementation of this method using the software GAP.In the sequel, considering the Z2-graduation M = M2(E) =_E 00 E__ _0 EE 0_, we determine all the graded identities of degree _ 5 using the relation between the theory of representations of the group GLn × GLn and the symmetric group Sn. Finally, we describe the complementary T2-space of the T2-ideal Id2(M) in the T2-space C 2(M) of the Z2-graded central polynomials of M.