Graduações de grupo na álgebra das matrizes triangulares superiores

Abstract

In this work we study group gradings on the upper triangular matrices algebra UTn(F), which have several applications in the PI-algebra theory. Our main purpose is to exhibit a description of all _nite gradings of UTn(F) by a group G up to isomorphism. To begin with, we restrict to the case where the base _eld F is algebraically closed of characteristic zero and the group G is _nite abelian. Using the method of group representation we present an explicit description of the duality between G-actions and G-gradings on an associative algebra, and such duality plays an important role in the proof of the main result presented for this case. Finally, the proof of the general result, for an arbitrary base _eld F and an arbitrary group G, accomplished by an alternative approach deeply based on semi-simplicity properties of rings is presented.