Completion of restricted Lie algebras (Q1327511)

The authors study a class of Lie algebras which are analogues of pro-\(p\) groups. They denote by \({\mathcal F}_ p\) the class of all finite- dimensional restricted Lie algebras over a finite field of characteristic \(p\) whose \(p\)-map is nilpotent. A restricted ideal \(I\) of a restricted Lie algebra \(L\) is said to be an \({\mathcal F}_ p\)-ideal if \(L/I\) is in \({\mathcal F}_ p\). A pro-\(p\) algebra is a compact Hausdorff topological Lie \(p\)-algebra whose open \({\mathcal F}_ p\)-ideals form a neighborhood base for 0. The authors' primary focus is the structure of topologically finitely generated pro-\(p\) algebras. Their main results call to mind those of \textit{A. Lubotzky} and \textit{A. Mann} [J. Algebra 105, 506-515 (1987; Zbl 0626.20022)] concerning pro-\(p\) groups of finite rank. They also establish a close relationship between the rank of a pro-\(p\) group and the rank of its associated pro-\(p\) algebra.