Group gradings on upper block triangular matrices

Abstract

It was proved by Valenti and Zaicev, in 2011, that, if G is an abelian group and K is an algebraically closed field of characteristic zero, then any G-grading on the algebra of upper block triangular matrices over

Graded identities and isomorphisms on algebras of upper block-triangular matrices

“…Next we prove that the above theorem holds for arbitrary abelian groups. We remark that in [14] it was proved that this result holds in general provided that the base field K is of characteristic zero or that char K is strictly greater than dim U T (d 1 , . .…”

Superinvolutions on block-triangular matrix algebras

“…. , d m ) of upper block-triangular matrices were described in [12] under the hypothesis that the underlying field F is of characteristic zero or that char F > dim U. We present next the results on isomorphisms and anti-automorphisms on G-gradings on such algebras that will be used.…”

“…, q ′ iσi(si) ), where k = ⌊ m 2 ⌋. Note that the entries of γ satisfy the equalities (12). Theorem 4.2 implies that there exists an isomorphism from U to the elementary grading on U T (d 1 , .…”

“…Algebras of upper-triangular matrices and matrix algebras are particular cases of algebras of upper block-triangular matrices. The group gradings and graded involutions for these algebras were studied in [3], [6] and [12] and the group gradings for the Lie and Jordan algebras of block-triangular matrices were classified in [10].…”

Gradings on the algebra of triangular matrices as a Lie algebra: revisited

“…Their related graded PI-properties have been a subject of several recent studies, see [10,16,15,5,13,14,17], to cite only a few examples. The group gradings on the upper block-triangular matrices were computed in [33,6,35]. Here we recall the sophisticated theory developed by A. Kemer in the eighties.…”