Narrow algebras and groups (Q1296181)

This is a survey on algebras of Gelfand-Kirillov dimension 1 and related pro-\(p\)-groups, also some new results are proved. First, the authors study associative algebras and Lie algebras in characteristic zero. Pro-\(p\)-groups and \(\mathbb N\)-graded Lie algebras cannot be simple. Instead appears the notion of being just-infinite, namely, not having non-trivial normal subgroups (ideals) of infinite index (codimension). Understanding just-infinite filtered or \(\mathbb N\)-graded Lie algebras of Gelfand-Kirillov dimension 1 and pro-\(p\)-groups of finite width is still a formidable challenge. The authors propose a conjecture on a structure of just-infinite filtered Lie algebras \(L=L_1\supset L_2\dots\) over an algebraically closed field of zero characteristic such that \(\dim L_i/L_{i+1}\) are uniformly bounded. As a partial result the authors classify just-infinite \(\mathbb N\)-graded Lie algebras \(L=\bigoplus_{i\geq 1}L_i\) in characteristic zero under assumption \(\dim L_i=d\), \(i\geq n_0\), with ``polynomial multiplication''. Namely, they prove that such algebras are commensurable with the Witt algebra \(W_1\) or with the positive part of some loop algebra. (Two algebras are said commensurable if they contain isomorphic subalgebras of finite codimension). Second, results on pro-\(p\)-groups, finite \(p\)-groups, and graded Lie algebras in characteristic \(p\) are surveyed. Main objects here are groups and algebras of finite coclass.