Gradings on the Albert algebra and on $\mathfrak{f}_4$

Abstract

We study group gradings on the Albert algebra and on the exceptional simple Lie algebra $\frak{f}_4$ over algebraically closed fields of characteristic zero. The immediate precedent of this work is [Draper, C. and Martin, C.: Gradings on $\frak{g}_2$. Linear Algebra Appl. 418 (2006), no. 1, 85-111] where we described (up to equivalence) all the gradings on the exceptional simple Lie algebra $\frak{g}_2$. In the cases of the Albert algebra and $\frak{f}_4$, we look for the nontoral gradings finding that there are only eight nontoral nonequivalent gradings on the Albert algebra (three of them being fine) and nine on $\frak{f}_4$ (also three of them fine).