Z2-graduações da álgebra de Grassmann: construção, PI-equivalência e isomorfismos

Abstract

The focus of our dissertation is to develope a study on the Z2-gradings of the infinitedimensional Grassmann algebra E. The homogeneous Z2-gradings and their Z2-graded identities are already well known in the literature, see (VINCENZO; SILVA, 2009), (CENTRONE, 2011) e (GONÇALVES, 2018). Nevertheless, the construction of nonhomogeneous Z2-gradings demands the use of the duality between these structures and automorphisms of order ≤ 2 acting on E. Through this, we will study the non-homogeneous Z2-gradings, producing results on their construction. Next, we will investigate under what conditions a non-homogeneous Z2-grading is isomorphic to the canonical Z2-grading of E. Finally, we will provide a Z2-grading on E in which there is no non-zero element of the space L homogeneous, giving a negative answer to the conjecture presented in (GUIMARÃES; KOSHLUKOV, 2023).