Propriedade de Specht e identidades para álgebras de Jordan.

Abstract

The Jordan algebra of the symmetric matrices of order two over a field has exactly two natural gradings by the group Z2. In this work, presented in five chapters, we exhibit bases for 2-graded polynomial identities for these two grading when the base field is infinite and of characteristic dierent from 2. For a so-called "scalar grading" the result is extended to the case of Jordan algebras of a non-degenerate symmetric bilinear form, denote by B and Bn when its vector spaces have infinite and finite dimensions, respectively. In this case, over a field of characteristic zero, we also show that the ideal of all the 2-graded identities of Bn satisfies the Specht property. Moreover, we study the description all possible G-gradings on Jordan algebra of a bilinear form. Finally, we determine a basis for the identities of the Jordan algebra of a degenerate bilinear form with a n-dimensional vector space of rank n − 1.