Identidades Polinomiais ℤ2-Graduadas para as Álgebras M1,1(E) e UT2(F) via representações de grupos

Abstract

In this essay we will briefly study the concept of Algebra. We will introduce a little of Group Representation Theory, looking specifically at Young's Theory, which allows us to present explicitly the decomposition of the group algebra FSn into simple subalgebras, where Sn is the symmetric group of order n!. We will also talk about Polynomial Identities and Graded Polynomial Identities, and some pertinent PI-Theory's results. We will relate Symmetrical Groups Representation Theories with PI-Theory. We will show all the Z2-graded polynomial identities for the algebras M2(F) and M1,1(E), where E is the Grassmann Algebra infinitely generated over a field F of characteristic zero. Finally, we will present all G-gradings possibilities for the algebra UT2(F), of the upper triangular matrices of order two with entries in a field of characteristic zero (we will see that, up to isomorphisms, there are only two possibilities), moreover, we will find all the G-graded polynomial identities for this algebra and we will show a numerical sequence involving the graded cocaracteres.