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Hasse graphs and parabolic subalgebras of exceptional Lie algebra \(f_4\). (English) Zbl 1102.22012
Slovák, Jan (ed.) et al., The proceedings of the 24th winter school “Geometry and physics”, Srní, Czech Republic, January 17–24, 2004. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 75, 293-308 (2005).
A \([k]\)-graded Lie algebra is understood as a Lie algebra \(\mathfrak g\) equipped with the grading \[ \mathfrak g=\mathfrak g_{-k}\oplus\dots\oplus\mathfrak g_{-1}\oplus \mathfrak g_0\oplus\mathfrak g_1\oplus\dots\oplus\mathfrak g_k \] (i.e., \([\mathfrak g_i,\mathfrak g_j]\subset\mathfrak g_{i+j}\)) together with the requirement that the subalgebra \(\mathfrak g_-=\mathfrak g_{-k}\oplus\dots\oplus\mathfrak g_{-1}\) is generated by \(\mathfrak g_{-1}\). The corresponding graded parabolic Lie subalgebra is \(\mathfrak p=\mathfrak g_0\oplus\dots\oplus\mathfrak g_k\).
In the paper, the graded structure and the Hasse diagrams of graded parabolic subalgebras of the exceptional simple Lie algebra \(\mathfrak f_4\) are studied up to grading 4.
For the entire collection see [Zbl 1074.53001].

22E46 Semisimple Lie groups and their representations
17B25 Exceptional (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)