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arXiv · 1811.05870

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

Mikhail Kochetov·Felipe Yasumura
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Abstract

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.

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Mikhail Kochetov, Felipe Yasumura. 2019-05-14. Group gradings on the Lie and Jordan algebras of block-triangular matrices. https://doi.org/10.1016/j.jalgebra.2019.07.020

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