Graduações e identidades com involução em álgebras Matriciais.

Abstract

In this work we solve two problems: the rst one is to prove that for any group grading on a block-triangular matrix algebra, over an arbitrary eld, the Jacobson radical is a graded ideal. As observed by F. Yukihide this yields the classi cation of the group gradings on these algebras and con rms a conjecture made by A. Valenti and M. Zaicev in 2007. The second is, assuming that F is a eld of characteristic zero, to prove that there is a group grading on UTm(F), called the nest, such that every grading that admits graded involutions is one of its coarsening, and this graded involution is equivalent to the re ection or symplectic involution on UTm(F). For this grading, we will exhibit a basis for their graded identities with involution and we will determine the asymptotic growth of their sequence of codimensions. Furthermore, we will study the algebra UT3(F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the nest and a Z2-grading. We determine a basis for the graded (Z2,∗)-identities, in addition we compute the codimension sequence.