A non-computational approach to the gradings on \(\mathfrak f_4\)
A grading by an abelian group \(G\) of a Lie algebra \({\mathcal L}\) is a decomposition \(\Gamma: {\mathcal L}=\oplus_{g\in G}{\mathcal L}_g\) as a direct sum of vector subspaces, such that \([{\mathcal L}_g,{\mathcal L}_h]\subseteq {\mathcal L}_{gh}\) for any \(g,h\in G\). Given two gradings \(\Gamma: {\mathcal L}=\bigoplus_{g\in G}{\mathcal L}_g\) and \(\Gamma': {\mathcal L}=\oplus_{h\in H}{\mathcal L}_h'\), \(\Gamma\) is a refinement of \(\Gamma'\) (or \(\Gamma'\) a coarsening of \(\Gamma\)) if any homogeneous space of \(\Gamma'\) is a sum of homogeneous spaces in \(\Gamma\). \(\Gamma\) is fine if it admits no proper refinement. Any grading is a coarsening of a fine grading. Over algebraically closed fields of characteristic zero, the classification of the fine gradings for the classical Lie algebras was completed by the reviewer [J. Algebra 324, No. 12, 3532--3571 (2010; Zbl 1213.17030)], for \(G_2\) it was obtained independently by \textit{C.~Draper} and \textit{C.~Martín} [Linear Algebra Appl. 418, No. 1, 85--111 (2006; Zbl 1146.17027)] and by \textit{Y.~A.~Bahturin} and \textit{M.~V.~Tvalavadze} [Commun. Algebra 37, No. 3, 885--893 (2009; Zbl 1190.17005)]. For \(F_4\) it was obtained by \textit{C.~Draper} and \textit{C.~Martín} [Rev. Mat. Iberoam. 25, No. 3, 841--908 (2009; Zbl 1281.17035)] in a very technical paper based on many concrete computations. The paper under review gives a different proof of the classification of the fine gradings on \(F_4\). The homogeneous components of any such grading are given by the common eigenspaces for the action of the elements of a maximal quasitorus of the algebraic group \(\text{Aut}({\mathcal L})\). Known results about algebraic groups are used to check that any such quasitorus is either a maximal torus or contains either a nontoral \(2\)-group or a nontoral \(3\)-group. In case it contains a nontoral \(2\)-group, the quasitorus is shown to live in a subgroup of \(\text{Aut}({\mathcal L})\) isomorphic to \(\text{Spin}(9)\). Otherwise, the quasitorus is shown to be isomorphic to \({\mathbb Z}_3^3\). Eventually, this proves that there are exactly four different fine gradings, up to equivalence, by the groups \({\mathbb Z}^4\), \({\mathbb Z}\times{\mathbb Z}_2^3\), \({\mathbb Z}_2^5\) and \({\mathbb Z}_3^3\). A different approach, valid over fields of characteristic not two, and based on computing first the gradings of the Albert algebra (exceptional Jordan algebra) from structural results, has been used by the reviewer and \textit{M. Kochetov} [Rev. Mat. Iberoam. 28, No.~3, 775--815 (2012; Zbl 1303.17019)]. Also, the fine gradings of \(E_6\) have been recently classified by \textit{C.~Draper} and \textit{A.~Viruel} [``Fine gradings on \(\mathfrak{e}_6\)'', \url{arXiv:1207.6690}]. (Also send to Math. Rev.)
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