Jordan algebras of Gelfand-Kirillov dimension one
Let \(A\) be a finitely generated nonassociative (i.e., not necessarily associative) algebra over a field \(K\) and let \(V\) be a finite dimensional vector subspace of \(A\) generating \(A\). If \(V^n\) denotes the subspace spanned by the products of at most \(n\) elements in \(V\), then the limit superior of the sequence \({\log \dim V^n \over \log n}\) is called the Gelfand-Kirillov dimension of \(A\). The finitely generated associative algebras of Gelfand-Kirillov dimension one have been studied by \textit{L. W. Small}, \textit{J. T. Stafford} and \textit{R. B. Warfield}, jun. [Math. Proc. Camb. Philos. Soc. 97, 407-414 (1985; Zbl 0561.16005)]. The aim of this interesting paper is to get similar results for finitely generated Jordan algebras over fields of characteristic not two. The main results of the paper assert that, as for the associative case, there are no finitely generated Jordan algebras with Gelfand-Kirillov dimension strictly between 1 and 2, that the semiprime Jordan algebras of Gelfand-Kirillov dimension one are finite modules over a finitely generated subalgebra of the associative center and that the McCrimmon radical of any Jordan algebra of Gelfand-Kirillov dimension one is nilpotent. The process of the proof of these results starts with the semiprime case, relating the Gelfand-Kirillov dimension of the Jordan algebra to the one of its (associative) multiplication algebra. The classification of the prime Jordan algebra by \textit{E. I. Zelmanov} [Sib. Math. J. 24, 73-85 (1983); translation from Sib. Mat. Zh. 24, No. 1, 89-104 (1983; Zbl 0534.17009)] is used here. Then graded Jordan algebras are studied in a very technical and detailed way. Finally, the general case is deduced from the graded one, following the methods of Small et al. Since Lie algebras of finite width have finite gradings and are thus related to Jordan systems via the Tits-Kantor-Koecher construction, the authors announce the application of the results in this paper to prounipotent groups and Lie algebras.
scientific article; zbMATH DE number 869811