Group gradings on the Lie and Jordan algebras of block-triangular matrices

JOURNAL OF ALGEBRA

卷:537

Article

Kochetov, Mikhail¹  Yasumura, Felipe Yukihide² 

[1] Mem Univ Newfoundland, Dept Math & Stat, St John, NF A1C 5S7, Canada

[2] Univ Estadual Maringa, Dept Math, Maringa, Parana, Brazil

关键词: Graded algebra;    Block-triangular matrices;    Classification of gradings;   

DOI  :  10.1016/j.jalgebra.2019.07.020

来源: Elsevier

【 摘 要 】

We classify up to isomorphism all gradings by an arbitrary group G on the Lie algebras of zero-trace upper block-triangular matrices over an algebraically closed field of characteristic 0. It turns out that the support of such a grading always generates an abelian subgroup of G. Assuming that G is abelian, our technique also works to obtain the classification of G-gradings on the upper block-triangular matrices as an associative algebra, over any algebraically closed field. These gradings were originally described by A. Valenti and M. Zaicev in 2012 (assuming characteristic 0 and G finite abelian) and classified up to isomorphism by A. Borges et al. in 2018. Finally, still assuming that G is abelian, we classify G-gradings on the upper block-triangular matrices as a Jordan algebra, over an algebraically closed field of characteristic 0. It turns out that, under these assumptions, the Jordan case is equivalent to the Lie case. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free