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

Abstract

We investigate the group gradings on the algebra of upper triangular matrices over an arbitrary field, viewed as a Lie algebra. These results were obtained a few years early by the same authors. We provide streamlined proofs, and present a complete classification of isomorphism classes of the gradings. We also provide a classification of the practical isomorphism classes of the gradings, which is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is 2, a topic that was neglected in previous works.

Graded polynomial identities for the Lie algebra of upper triangular matrices of order 3

“…Theorem 1 (Theorem 2.8 of [3]). The G-graded identities for an elementary Ggrading ǫ on U T n follow from all f µ where µ runs over the ǫ-bad sequences of length at most n. The classification of isomorphism classes of elementary gradings on U T (−) n was obtained in [10] (see [14] as well). An elementary grading on U T (−) n , as in the associative case, is uniquely defined by a sequence η ∈ G n−1 .…”