On invariants of \(F_ 4\) and the center of the Albert algebra
type: Item
instance of: scholarly article
title: On invariants of \(F_ 4\) and the center of the Albert algebra
zbMATH Open document ID: 0851.17033
author: A. V. Il'tyakov
author: Ivan Shestakov
published in: Journal of Algebra
publication date: 1996-11-28T00:00:00Z
review text: Some simple linear algebraic groups appear as groups of automorphisms of certain simple algebras. The classical example is the group \(\text{PSL} (n)\), which is isomorphic to the automorphism group of the associative algebra of \(n\times n\) matrices. The exceptional groups \(G_2\) and \(F_4\) appear also as the automorphism groups of the algebra of octonions (which is not associative, but alternative) and of the Albert algebra (which is a Jordan algebra). Let \(A\) be a central simple nonassociative algebra over a field \(F\) and let \(G\) be its automorphism group. Given a basis \(\{e_1, \dots, e_m\}\) of \(A\) and variables \(X= \{x_{ij}: 1\leq i\leq m\), \(1\leq j\leq n\}\), the subalgebra of \(F(X) \otimes_F A\) generated by the ``generic elements'' \(a_j= \sum^m_{i=1} x_{ij} \otimes e_i\) \((j=1, \dots, n)\) will be denoted by \({\mathcal F}_n (A)\). In case \(F\) is algebraically closed of characteristic zero, \textit{A. V. Iltyakov} [Geom. Dedicata 58, 327-333 (1996)] found the remarkable result that if the minimal number of generators of \(A\) is \(\leq n\) and \(G\) acts diagonally on \(nA= A\times \cdots \times A\), then the field of rational invariants \(F(nA)^G\) coincides with the quotient field \(K_n(A)\) of the centroid of the ``free affine algebra'' \({\mathcal F}_n (A)\). The paper under review is devoted to proving that if \({\mathbf A}\) is the split Albert algebra over a field \(F\) of characteristic \(\neq 2,3\), then \(K_n ({\mathbf A})\) is a purely transcendental extension of \(F\). The critical value is \(n=3\), which is the minimal number of generators of \({\mathbf A}\), and here the authors make use of the properties of the Albert algebra to get an explicit family of algebraically independent elements which generate \(K_3 ({\mathbf A})\) over \(F\). The same kind of result was known for the split Cayley algebra by the work of \textit{S. V. Polikarpov} and \textit{I. P. Shestakov} [Algebra Logic 29, 458-466 (1990); translation from Algeb…
reviewed by: Alberto Elduque
Mathematics Subject Classification ID: 17C30
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 868023
zbMATH Keywords: free affine algebra
zbMATH Keywords: quotient field
zbMATH Keywords: split Albert algebra
zbMATH Keywords: purely transcendental extensions
zbMATH Keywords: rationality
zbMATH Keywords: rational invariants
zbMATH Keywords: algebraic group
MaRDI profile type: Publication
full work available at URL: jabr.1996.0039
OpenAlex ID: W1969551731
DOI: 10.1006/JABR.1996.0039
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description: scientific article; zbMATH DE number 868023