Jordan algebras and \(F_4\) bundles over the affine plane
type: Item
instance of: scholarly article
title: Jordan algebras and \(F_4\) bundles over the affine plane
zbMATH Open document ID: 0893.17026
author: Maneesh Thakur
author: R. Parimala
author: Venapally Suresh
published in: Journal of Algebra
publication date: 1998-08-27T00:00:00Z
review text: 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.
reviewed by: Alberto Elduque
Mathematics Subject Classification ID: 17C40
Mathematics Subject Classification ID: 14L40
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 1106625
zbMATH Keywords: Jordan algebra
zbMATH Keywords: \(F_4\)-bundle
zbMATH Keywords: affine plane
zbMATH Keywords: anisotropic algebraic group
MaRDI profile type: Publication
full work available at URL: jabr.1997.7161
OpenAlex ID: W2023968833
Wikidata QID: Q105474490
cites work: On reduced exceptional simple Jordan algebras
cites work: Locally polynomial algebras are symmetric algebras
cites work: Q5605675
cites work: Fibres quadratiques et composantes connexes réelles
cites work: Semisimple subalgebras of semisimple Lie algebras
cites work: Some groups of transformations defined by Jordan algebras. II. Groups of type F4.
cites work: Q5622399
cites work: Non-free projective modules over H(X,Y) and stable bundles over \(P_ 2(\)C)
cites work: Q4314102
cites work: Q3998356
cites work: Indecomposable quadratic spaces over the affine plane
cites work: A local global principle for quadratic forms over polynomial rings
cites work: Quadratic and Hermitian forms in additive and Abelian categories
cites work: Principal bundles on affine space and bundles on the projective line
cites work: Principal bundles on the affine line
cites work: Q3267480
cites work: Cayley algebra bundles onA2K
DOI: 10.1006/JABR.1997.7161
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description: scientific article; zbMATH DE number 1106625