Revista Matemática Iberoamericana


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Volume 25, Issue 3, 2009, pp. 841–908
DOI: 10.4171/RMI/585

Published online: 2009-12-31

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

Cristina Draper Fontanals[1] and Cándido Martín González[2]

(1) Universidad de Málaga, Spain
(2) Universidad de Málaga, Spain

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).

Keywords: Exceptional Lie algebra, grading, algebraic group, Weyl group

Draper Fontanals Cristina, Martín González Cándido: Gradings on the Albert algebra and on $\mathfrak{f}_4$. Rev. Mat. Iberoamericana 25 (2009), 841-908. doi: 10.4171/RMI/585