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.

References in corpus (7)

• Kemer's Theorem for Affine PI Algebras Over a Field of Characteristic Zero
• Group gradings on the Jordan algebra of upper triangular matrices
• Graded polynomial identities as identities of universal algebras
• Graded identities of block-triangular matrices
• Graded Isomorphisms on Upper Block Triangular Matrix Algebras
• On the combinatorics of commutators of Lie algebras
• ${\mathbb{Z}}_2$-graded polynomial identities for the Jordan algebra of $2\times 2$ upper triangular matrices