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Gradings on the Albert algebra and on \(\mathfrak{f}_4\). (English) Zbl 1281.17035
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).

MSC:
17C40 Exceptional Jordan structures
17B25 Exceptional (super)algebras
17B70 Graded Lie (super)algebras
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