On invariants of \(F_ 4\) and the center of the Albert algebra
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 Algebra Logika 29, 709-723 (1990; Zbl 0760.17001)] and for the algebra of \(n\times n\) matrices with \(n\leq 4\), obtained by \textit{E. Formanek} [J. Algebra 62, 304-319 (1980; Zbl 0437.16013)]. As consequences of this result of rationality it is proven that over any algebraically closed field \(F\) of characteristic zero, the field of rational invariants \(F(n {\mathbf A})^G\) (for \({\mathbf A}\) the Albert algebra and \(G\) its group of automorphisms) is a rational function field over \(F\) for any \(n\) (including \(n=1,2\)) and also, if \(V\) is the minimal irreducible representation of the simple algebraic group of type \(F_4\), which coincides with the subspace of trace zero elements in the Albert algebra, then \(F(n V)^G\) is a rational function field over \(F\), too.
scientific article; zbMATH DE number 868023