Graduações e identidades graduadas nas álgebras das matrizes triangulares superiores em blocos.

Abstract

In this work we seek to solve two central problems. The first is the description of the isomorphism classes of upper block triangular matrix algebras graded by a finite abelian group over an algebraically closed field of characteristic zero. Under the same hypothesis, A. Valenti and M. Zaicev proved that any grading on an algebra of upper block-triangular matrices is isomorphic to the tensor product A ⊗ B of an elementary grading A on an upper block-triangular matrix algebra and a division grading B on a matrix algebra. We prove that this result holds without the hypothesis that the group is finite. The second problem is show that the graded identities in A ⊗ B, determine, up to isomorphism, A⊗B. We reduce this question to the case of elementary gradings on algebras of upper block-triangular matrix which was previously studied by O. M. Di Vincenzo and E. Spinelli.