Jordan algebras and \(F_4\) bundles over the affine plane
Let \(k\) be a field of characteristic \(\neq 2,3\) admitting a reduced exceptional Jordan algebra without nilpotent elements \(J\) and let \(G={\operatorname{Aut}} J\). The absence of nilpotent elements is shown to be equivalent to the algebraic group \(G\) being anisotropic. Then an infinite family of mutually non-isomorphic Jordan algebras \(J^{[i]}\) over the affine plane \({\mathbb A}_k^2={\operatorname{Spec}}(k[X,Y])\) is introduced satisfying that they specialize to \(J\) at any point and the associated principal \(G\)-bundles \(P_{J^{[i]}}\) admit no reduction of the structure group to any proper connected reductive subgroup of \(G\). A theorem of the same kind was proved by \textit{M. S. Raghunathan} [Math. Ann. 285, 309-332 (1989; Zbl 0661.14016)] for anistropic algebraic groups with the exceptions of \(G_2\) and \(F_4\). It is also announced in the paper that the case in which \(J\) is a division exceptional Jordan algebra has been dealt with elsewhere, thus settling the \(F_4\) case left in Raghunathan's paper. Key tools in the proof of the result above are the study of extensions of both Cayley and Jordan algebras from the affine to the projective plane and some patching techniques to assemble pieces defined over open subsets.
scientific article; zbMATH DE number 1106625
scientific article