Gradings on the algebra of triangular matrices as a Lie algebra: revisited
Plamen Koshlukov,
Felipe Yukihide Yasumura
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 investigat… Show more
“…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 .…”
We compute the graded polynomial identities and its graded codimension sequence for the elementary gradings of 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 .…”
We compute the graded polynomial identities and its graded codimension sequence for the elementary gradings of the Lie algebra of upper triangular matrices of order 3.
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