Gradings on the Albert algebra and on \(\mathfrak{f}_4\) (Q2655932)

Summary: We study group gradings on the Albert algebra and on the exceptional simple Lie algebra \(\mathfrak{f}_4\) over algebraically closed fields of characteristic zero. The immediate precedent of this work is the authors' article [Linear Algebra Appl. 418, No. 1, 85--111 (2006; Zbl 1146.17027)] where we described (up to equivalence) all the gradings on the exceptional simple Lie algebra \(\mathfrak{g}_2\). In the cases of the Albert algebra and \(\mathfrak{f}_4\), 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 \(\mathfrak{f}_4\) (also three of them fine).