Graduações na álgebra de Grassmann pelo grupo cíclico infinito: identidades e PI-equivalência

Abstract

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic zero and Z be the infinite cyclic group. Throughout the development of Kemer’s Theory, the algebra E plays a crucial role. In recent years, the abelian gradings on E and the respective graded identities have been studied in several articles, and it is still a very fertile topic at the research level. Therefore, the focus of our dissertation is to study recent results regarding gradings on E by the group Z. We will study results on the construction of gradings on E and, using methods from elementary number theory, we will describe the Z-graded polynomial identities for the so-called 2-induced Z-gradings on E of full support. As a consequence of this fact, we will show some examples of Z-gradings in E, which are PI-equivalent, but not Z-isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent but not isomorphic as graded algebras. Furthermore, we will introduce the notion of central Z-gradings on E and show that their Z-graded polynomial identities are related to the Z2-graded polynomial identities of E.