On the structure of graded Lie algebras of order 3
Volume 162 / 2020
Colloquium Mathematicum 162 (2020), 245-262
MSC: 17B99, 17B65, 17B05.
DOI: 10.4064/cm7849-7-2019
Published online: 6 May 2020
Abstract
We study the structure of graded Lie algebras of order 3 of arbitrary dimension and over an arbitrary field ${\mathbb K}$. We show that any of such algebras $L$ with a symmetric $G$-support is of the form $L = U + \sum _jI_j$ with $U$ a subspace of $L_1$ and any $I_j$ a well-defined graded ideal of $L$ satisfying $[I_{j}^{\bar 0},I_k] =\{I_{j}^{\bar i},I_{k}^{\bar i}, L^{\bar i}\}= 0$ for $\bar i \in \{ \bar 1, \bar 2\}$ if $j \neq k$. Under certain conditions, it is shown that $L = \bigoplus _{k } I_k ,$ where each $I_k$ is a gr-simple graded ideal of $L$ satisfying $[I_{j},I_k^{\bar 0}] =\{I_{j}^{\bar i},I_{k}^{\bar i}, L^{\bar i}\}= 0$ for $\bar i \in \{ \bar 1, \bar 2\}$ if $j \neq k$.
