Models of the Lie algebra \(F_4\)
The exceptional simple Lie algebra of type \(F_4\) contains seven proper semisimple subalgebras of rank \(4\). Their types are: \(B_4\), \(C_3\oplus A_1\), \(2A_2\), \(A_3\oplus A_1\), \(D_4\), \(C_2\oplus 2A_1\) and \(4A_1\). Given any complex simple Lie algebra \(L\) and a semisimple subalgebra \(S\) of maximal rank, the decomposition of \(L\) as a direct sum of irreducible modules for \(S\) is shown to constitute a group grading of \(L\) over the abelian group \({\mathbb Z}\Phi/{\mathbb Z}\Phi'\) with \(L_0=S\), where \(\Phi\) (respectively \(\Phi'\)) denotes the root system of \(L\) (resp. \(S\)). Therefore, for \(F_4\) there appear seven different such group gradings. This is used to construct models of \(F_4\) based on smaller components. Some of these models are well-known: the \({\mathbb Z}_2\)-gradings associated to the symmetric pairs \((F_4,B_4)\) and \((F_4,C_3\oplus A_1)\), the \({\mathbb Z}_2^2\)-grading related to the subalgebra \(D_4\) and the triality construction of the exceptional simple Lie algebras given by \textit{J. M. Landsberg} and \textit{L. Manivel} [Adv. Math. 171, No. 1, 59--85 (2002; Zbl 1035.17016)] or \textit{C. H. Barton} and \textit{A. Sudbery} [Adv. Math. 180, No. 2, 596--647 (2003; Zbl 1077.17011)], or the \({\mathbb Z}_2^3\)-grading related to the subalgebra of type \(4A_1\) obtained from a refinement of the previous one by the reviewer [Rev. Mat. Iberoam. 23, No. 1, 57--84 (2007; Zbl 1145.17005)]. Precise models based on the remaining three possibilities and very basic linear algebra constructions are given in this paper, as well as many comments on the relationships among the different gradings and on the extension to other exceptional simple Lie algebras.
scientific article; zbMATH DE number 5275767
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