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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].

##### MSC:
 22E46 Semisimple Lie groups and their representations 17B25 Exceptional (super)algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
##### Keywords:
representation theory; simple Lie algebras; Hasse diagrams