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

Abstract

We investigate the group gradings on the algebras of upper triangular matrices over an arbitrary field, viewed as Lie algebras. Classification results were obtained in 2017 by the same authors when the base field has characteristic different from 2. In this paper we provide streamlined proofs of these results. Moreover we present a complete classification of isomorphism classes of the group gradings on these algebras over an arbitrary field. Recall that two graded Lie algebras $L_1$ and $L_2$ are practically-isomorphic if there exists an (ungraded) algebra isomorphism $L_1\to L_2$ that induces a graded-algebra isomorphism $L_1/z\left(L_1\right)\to L_2/z\left(L_2\right)$. We provide a classification of the practically-isomorphism classes of the group gradings on the Lie algebra of upper triangular matrices. The latter classification 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.